EP2164711B1 - Representation system - Google Patents

Description

The invention relates to a representation arrangement for security papers, value documents, electronic display devices or other data carriers for representing one or more predetermined three-dimensional body (s).

Data carriers, such as valuables or identity documents, but also other valuables, such as branded goods, are often provided with security elements for the purpose of security, which permit verification of the authenticity of the data carrier and at the same time serve as protection against unauthorized reproduction. Data carriers in the context of the present invention are in particular banknotes, stocks, bonds, certificates, vouchers, checks, high-quality admission tickets, but also other forgery-prone papers, such as passports and other identity documents, credit cards, health cards and product security elements such as labels, seals, packaging and the like. The term "data carrier" in the following includes all such objects, documents and product protection means.

The security elements may be in the form of, for example, a security thread embedded in a banknote, a tearing thread for product packaging, an applied security strip, a cover sheet for a banknote having a through opening or a self-supporting transfer element, such as a patch or label after its manufacture is applied to a document of value.

Security elements with optically variable elements, which give the viewer a different image impression at different viewing angles, play a special role, since they can not be reproduced even with high-quality color copying machines. The security elements can be equipped with security features in the form of diffractive optical effective micro- or nanostructures, such as with conventional embossed holograms or other hologram-like diffraction structures, as described for example in the publications EP 0 330 733 A1 or EP 0 064 067 A1 are described.

From the publication US 5 712 731 A the use of a moiré magnification arrangement is known as a security feature. The security device described therein has a regular array of substantially identical printed microimages of up to 250 μm in size and a regular two-dimensional array of substantially identical spherical microlenses. The microlens array has substantially the same pitch as the microimage array. When the micro-image array is viewed through the microlens array, one or more enlarged versions of the microimages are created to the viewer in the areas where the two arrays are substantially in register.

The basic mode of operation of such moiré magnification arrangements is described in the article " The moire magnifier ", MC Hutley, R. Hunt, RF Stevens and P. Savander, Pure Appl. Opt. 3 (1994), pp. 133-142 , described. In short, moiré magnification thereafter refers to a phenomenon that occurs when viewing a raster of identical image objects through a lenticular of approximately the same pitch. As with any pair of similar rasters, this results in a moiré pattern, which in this case appears as an enlarged and possibly rotated image of the repeated elements of the image raster.

On this basis, the present invention seeks to avoid the disadvantages of the prior art and in particular to provide a generic Darstellungsanordnung that offers a lot of leeway in the design of the motif images to be considered.

This object is achieved by the representation arrangement with the features of the independent claims. A security paper and a data carrier with such representations are given in the independent claims. Further developments of the invention are the subject of the dependent claims.

• einem Motivbild, das in eine Mehrzahl von Zellen eingeteilt ist, in denen jeweils abgebildete Bereiche des vorgegebenen Körpers angeordnet sind,
• einem Betrachtungsraster aus einer Mehrzahl von Betrachtungselementen zur Darstellung des vorgegebenen Körpers bei der Betrachtung des Motivbilds mithilfe des Betrachtungsrasters,
• wobei das Motivbild mit seiner Einteilung in eine Mehrzahl von Zellen eine Bildfunktion m(x,y) aufweist, die gegeben ist durch $$\mathrm{m}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{f}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{K}}\\ {\mathrm{y}}_{\mathrm{K}}\\ {\mathrm{z}}_{\mathrm{K}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)\end{array}\right)\cdot \mathrm{g}\left(\mathit{x},\mathit{y}\right),$$
  
  mit $$\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{K}}\\ {\mathrm{y}}_{\mathrm{K}}\end{array}\right)=\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\mathit{V}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)\cdot \left(\left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)\right)\mathrm{modW}\right)-{\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)-{\mathrm{w}}_{\mathrm{c}}\left(\mathrm{x},\mathrm{y}\right)\right)$$
  
  $${\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{d}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{d}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right){\mathrm{und\; w}}_{\mathrm{c}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{c}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right),$$
  
  wobei
• die Einheitszelle des Betrachtungsrasters durch Gitterzellenvektoren ${\mathrm{w}}_1=\left(\begin{array}{c}{w}_{11}\\ w_{21}\end{array}\right)$
  
  und ${\mathrm{w}}_2=\left(\begin{array}{c}{w}_{12}\\ w_{22}\end{array}\right)$
  
  beschrieben und in der Matrix W = $\left(\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& w_{22}\end{array}\right)$
  
  zusammengefasst ist, und x_m und y_m die Gitterpunkte des W-Gitters bezeichnen,
• der Vergrößerungsterm V(x,y, x_m,y_m) entweder ein Skalar $\mathrm{V}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)=\left(\frac{{\mathrm{z}}_{\mathrm{K}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)}{\mathrm{e}}-\mathrm{1}\right)$
  
  ist, mit dem effektiven Abstand des Betrachtungsrasters vom Motivbild e, oder eine Matrix V(x,y, x_m,y_m) =(A(x,y, x_m,y_m) - I) ist, wobei die Matrix $\mathrm{A}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{11}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)& {\mathrm{a}}_{\mathrm{12}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)\\ {\mathrm{a}}_{\mathrm{21}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)& {\mathrm{a}}_{\mathrm{22}}\left(\mathrm{z},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)\end{array}\right)$
  
  ein gewünschtes Vergrößerungs- und Bewegungsverhalten des vorgegebenen Körpers beschreibt und I die Einheitsmatrix ist,
• der Vektor (c₁(x,y), c₂(x,y)) mit 0 ≤ c₁(x,y), c₂(x,y) < 1 die relative Position des Zentrums der Betrachtungselemente innerhalb der Zellen des Motivbilds angibt,
• der Vektor (d₁(x,y), d₂(x,y)) mit 0 ≤ d₁(x,y), d₂(x,y) < 1 eine Verschiebung der Zellengrenzen im Motivbild darstellt, und
• g(x,y) eine Maskenfunktion zur Einstellung der Sichtbarkeit des Körpers ist.

According to a first aspect of the invention, a generic representation arrangement includes a raster image arrangement for displaying a given three-dimensional body, which is given by a body function f (x, y, z)

• a motif image which is divided into a plurality of cells, in each of which imaged areas of the predetermined body are arranged,
• a viewing grid of a plurality of viewing elements for displaying the predetermined body when viewing the motif image using the viewing grid,
• wherein the motif image with its division into a plurality of cells has an image function m (x, y), which is given by $$\mathrm{m}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{f}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{K}}\\ {\mathrm{y}}_{\mathrm{K}}\\ {\mathrm{z}}_{\mathrm{K}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)\end{array}\right)\cdot \mathrm{G}\left(\mathit{x},\mathit{y}\right).$$
  
  With $$\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{K}}\\ {\mathrm{y}}_{\mathrm{K}}\end{array}\right)=\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\mathit{V}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)\cdot \left(\left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)\right)\mathrm{ModW}\right)-{\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)-{\mathrm{w}}_{\mathrm{c}}\left(\mathrm{x},\mathrm{y}\right)\right)$$
  
  $${\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{d}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{d}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right){\mathrm{and\; w}}_{\mathrm{c}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{c}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right).$$
  
  in which
• the unit cell of the viewing grid by grid cell vectors ${\mathrm{w}}_1=\left(\begin{array}{c}{w}_{11}\\ w_{21}\end{array}\right)$
  
  and ${\mathrm{w}}_2=\left(\begin{array}{c}{w}_{12}\\ {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{22}\end{array}\right)$
  
  described and in the matrix W = $\left(\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{22}\end{array}\right)$
  
  and x _m and y _m denote the grid points of the W grid,
• the magnification term V (x, y, x _m , y _m ) is either a scalar $\mathrm{V}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)=\left(\frac{{\mathrm{z}}_{\mathrm{K}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)}{\mathrm{e}}-\mathrm{1}\right)$
  
  is, with the effective distance of the viewing grid from the motif image e, or a matrix V (x, y, x _m , y _m ) = (A (x, y, x _m , y _m ) - I), where the matrix $\mathrm{A}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{11}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)& {\mathrm{a}}_{\mathrm{12}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)\\ {\mathrm{a}}_{\mathrm{21}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)& {\mathrm{a}}_{\mathrm{22}}\left(\mathrm{z},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)\end{array}\right)$
  
  Describes a desired magnification and movement behavior of the given body and I is the unit matrix,
• the vector (c ₁ (x, y), c ₂ (x, y)) with 0≤c ₁ (x, y), c ₂ (x, y) <1, the relative position of the center of the viewing elements within the cells of the Motiv image indicates
• the vector (d ₁ (x, y), d ₂ (x, y)) with 0 ≦ d ₁ (x, y), d ₂ (x, y) <1 represents a shift of the cell boundaries in the motif image, and
• g (x, y) is a mask function for adjusting the visibility of the body.

As far as possible, scalars and vectors with lowercase letters, matrices with uppercase letters are used in this description. Arranged on arrow symbols to identify vectors was omitted for the sake of clarity. In addition, it is generally clear to a person skilled in the art whether an occurring variable represents a scalar, a vector or a matrix, or whether several of these possibilities come into consideration. For example, the magnification term V can represent either a scalar or a matrix, so that no unique name with lowercase or uppercase letters is possible. In the context, however, it always becomes clear whether a scalar, a matrix, or both alternatives come into question.

The invention generally relates to the generation of three-dimensional images and to three-dimensional images with varying image contents when the viewing direction is changed. The three-dimensional images are referred to as body in the context of this description. The term "body" refers in particular to point sets, line systems or patches in three-dimensional space, by which three-dimensional "bodies" are described by mathematical means.

For z _k (x, y, x _m , y _m ), ie the z coordinate of a common point of the visual line with the body, more than one value can be considered, from which a value is formed or selected according to rules to be determined becomes. This selection can be made, for example, by specifying an additional characteristic function, as explained below using the example of an opaque body and a transparency step function given in addition to the body function f.

The representation arrangement according to the invention contains a raster image arrangement in which a motif (or the predetermined body) appears to float individually or not necessarily as an array in front of or behind the image plane or penetrates it. The illustrated three-dimensional image moves in tilting the security element, which is formed by the superimposed motif image and the viewing grid, in predetermined by the magnification and movement matrix A directions. The motif image is not produced photographically, not even by an exposure grating, but is mathematically constructed with a modulo algorithm, whereby a variety of different magnification and motion effects can be generated, which are described in more detail below.

In the above-mentioned known moiré magnifier, the image to be displayed consists of individual motifs which are arranged periodically in a grid. The motif image to be viewed through the lenses represents a greatly reduced version of the image to be displayed, with the area associated with each individual motif corresponding at most to approximately one lens cell. Due to the small size of the lens cells, only relatively simple entities can be considered as individual motifs. In contrast, the illustrated three-dimensional image in the "modulo mapping" described here is generally a single image; it does not necessarily have to be composed of a grid of periodically repeated individual motifs. The illustrated three-dimensional image may represent a complex, high-resolution frame.

Subsequently, the name component "Moiré" is used for embodiments in which the moiré effect is involved, in the use of the name component "modulo" a moiré effect is not necessarily involved. The name component "Mapping" indicates any illustrations, while the name component "Magnifier" indicates that not any illustrations but only enlargements are involved.

First, let us briefly consider the modulo operation occurring in the image function m (x, y), from which the modulo magnification arrangement derives its name. For a vector s and an invertible 2x2 matrix W, the expression s mod W as a natural extension of the usual scalar modulo operation represents a reduction of the vector s into the fundamental mesh of the lattice described by the matrix W (the "phase" of the vector s within the grid W).

• Sei $q=\left(\begin{array}{c}{q}_1\\ q_2\end{array}\right)={\mathrm{W}}^{-1}\mathrm{s}$
  
  und q_i = n_i + pi mit ganzzahligem n_i ∈ Z und 0 ≤ p _i < 1 (i=1,2), oder sei mit anderen Worten n_i = floor(q_i) und p_i = q_i mod 1. Dann ist s=Wq = (n₁w₁ + n₂w₂) + (p₁w₁ + p₂w₂), wobei (n₁w₁ + n₂w₂) ein Punkt auf dem Gitter WZ² ist und $$\mathrm{s\; mod\; W}={\mathrm{p}}_{\mathrm{1}}{\mathrm{w}}_{\mathrm{1}}+{\mathrm{p}}_{\mathrm{2}}{\mathrm{w}}_{\mathrm{2}}$$
  
• in der Grundmasche des Gitters liegt und die Phase von s bezüglich des Gitters W angibt.

Formally, the expression s mod W can be defined as follows:

• Be $q=\left(\begin{array}{c}{q}_1\\ q_2\end{array}\right)={\mathrm{W}}^{-1}\mathrm{s}$
  
  and q _i = n _i + pi with integer n _i ∈ Z and 0 ≤ p _i <1 (i = 1,2), or in other words n _i = floor (q _i ) and p _i = q _i mod 1 Then s = Wq = (n ₁ w ₁ + n ₂ w ₂ ) + (p ₁ w ₁ + p ₂ w ₂ ), where (n ₁ w ₁ + n ₂ w ₂ ) is a point on the grating WZ ² is and $$\mathrm{s\; mod\; W}={\mathrm{p}}_{\mathrm{1}}{\mathrm{w}}_{\mathrm{1}}+{\mathrm{p}}_{\mathrm{2}}{\mathrm{w}}_{\mathrm{2}}$$
  
• is in the basic mesh of the grating and indicates the phase of s with respect to the grating W.

In a preferred embodiment of the representation arrangement of the first aspect of the invention, the magnification term is represented by a matrix V (x, y, X _m , y _m ) = (A (x, y, x _m , y _m ) - I) with a ₁₁ (x, y, x _m , y _m ) ₌ z _k (x, y, x _m , y _m ) / e, so that the raster image arrangement represents the given body when viewing the subject image with the eye distance in the x direction. More generally, the magnification term may be represented by a matrix V (x, y, x _m , y _m ) = (A (x, y, x _m , y _m ) -I) with (a ₁₁ cos ² ψ + (a ₁₂ ⁺ a ₂₁ ) cosψ sinψ + a ₂₂ sin ² ψ) = z _k (x, y, x _m , y _m ) / e, so that the raster image arrangement represents the given body when viewing the subject image with eye relief in the direction ψ to the x-axis ,

In an advantageous development of the invention, in addition to the body function f (x, y, z), a transparency step function t (x, y, z) is given, where t (x, y, z) is equal to 1 when the body f (FIG. x, y, z) obscures the background at the location (x, y, z) and otherwise equals 0. For the viewing direction essentially in the direction of the z axis, the smallest value for z _k (x, y, x _m , y _m ) for which t (x, y, z _K ) is not equal to zero is the body front side to look at from the outside.

Alternatively, for z _K (x, y, x _m , y _m ), the largest value for which t (x, y, z _K ) is not equal to zero can also be taken. In this case, a deeply reversed (pseudoscopic) image is created, with the back of the body viewed from the inside.

In all variants, the values z _k (x, y, x _m , y _m ) can assume positive or negative values or also be 0 depending on the position of the body with respect to the plane of the drawing (penetrating behind or in front of the plane of the drawing or the plane of the drawing).

• einem Motivbild, das in eine Mehrzahl von Zellen eingeteilt ist, in denen jeweils abgebildete Bereiche des vorgegebenen Körpers angeordnet sind,
• einem Betrachtungsraster aus einer Mehrzahl von Betrachtungselementen zur Darstellung des vorgegebenen Körpers bei der Betrachtung des Motivbilds mithilfe des Betrachtungsrasters,
• wobei das Motivbild mit seiner Einteilung in eine Mehrzahl von Zellen eine Bildfunktion m(x,y) aufweist, die gegeben ist durch $$\mathrm{m}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{f}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{K}}\\ {\mathrm{y}}_{\mathrm{K}}\end{array}\right)\cdot \mathit{g}\left(\mathit{x},\mathit{y}\right),$$
  
  mit $$\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{K}}\\ {\mathrm{y}}_{\mathrm{K}}\end{array}\right)=\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\mathrm{V}\left(\mathit{x},\mathit{y}\right)\cdot \left(\left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)\right)\mathrm{modW}\right)-{\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)-{\mathrm{w}}_{\mathrm{c}}\left(\mathrm{x},\mathrm{y}\right)\right),$$
  
  $${\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{d}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{d}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$$
  
  und $${\mathrm{w}}_{\mathrm{c}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{c}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right),$$
  
  wobei
• die Einheitszelle des Betrachtungsrasters durch Gitterzellenvektoren ${\mathrm{w}}_1=\left(\begin{array}{c}{w}_{11}\\ w_{21}\end{array}\right)$
  
  und ${\mathrm{w}}_2=\left(\begin{array}{c}{w}_{12}\\ {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{22}\end{array}\right)$
  
  beschrieben und in der Matrix W = $\left(\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{22}\end{array}\right)$
  
  zusammengefasst ist,
• der Vergrößerungsterm V(x,y) entweder ein Skalar $\mathrm{V}\left(\mathrm{x},\mathrm{y}\right)=\left(\frac{\mathrm{z}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}-\mathrm{1}\right)$
  
  ist, mit dem effektiven Abstand des Betrachtungsrasters vom Motivbild e, oder eine Matrix V(x,y) =(A(x,y) - I) ist, wobei die Matrix $\mathrm{A}\left(\mathrm{x},\mathrm{y}\right)=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{11}}\left(\mathrm{x},\mathrm{y}\right)& {\mathrm{a}}_{\mathrm{12}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{a}}_{\mathrm{21}}\left(\mathrm{x},\mathrm{y}\right)& {\mathrm{a}}_{\mathrm{22}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$
  
  ein gewünschtes Vergrößerungs- und Bewegungsverhalten des vorgegebenen Körpers beschreibt, und I die Einheitsmatrix ist,
• der Vektor (c₁(x,y), c₂(x,y)) mit 0 ≤ c₁(x,y), c₂(x,y) < 1 die relative Position des Zentrums der Betrachtungselemente innerhalb der Zellen des Motivbilds angibt,
• der Vektor (d₁(x,y), d₂(x,y)) mit 0 ≤ d₁(x,y), d₂(x,y) < 1 eine Verschiebung der Zellengrenzen im Motivbild darstellt, und
• g(x,y) eine Maskenfunktion zur Einstellung der Sichtbarkeit des Körpers ist.

According to a second aspect of the invention, a generic representation arrangement includes a raster image arrangement for displaying a predetermined three-dimensional body by a height profile with a two-dimensional representation of the body f (x, y) and a height function z (x, y) is given, which contains height / depth information for each point (x, y) of the given body

• a motif image which is divided into a plurality of cells, in each of which imaged areas of the predetermined body are arranged,
• a viewing grid of a plurality of viewing elements for displaying the predetermined body when viewing the motif image using the viewing grid,
• wherein the motif image with its division into a plurality of cells has an image function m (x, y), which is given by $$\mathrm{m}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{f}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{K}}\\ {\mathrm{y}}_{\mathrm{K}}\end{array}\right)\cdot \mathit{G}\left(\mathit{x},\mathit{y}\right).$$
  
  With $$\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{K}}\\ {\mathrm{y}}_{\mathrm{K}}\end{array}\right)=\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\mathrm{V}\left(\mathit{x},\mathit{y}\right)\cdot \left(\left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)\right)\mathrm{ModW}\right)-{\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)-{\mathrm{w}}_{\mathrm{c}}\left(\mathrm{x},\mathrm{y}\right)\right).$$
  
  $${\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{d}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{d}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$$
  
  and $${\mathrm{w}}_{\mathrm{c}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{c}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right).$$
  
  in which
• the unit cell of the viewing grid by grid cell vectors ${\mathrm{w}}_1=\left(\begin{array}{c}{w}_{11}\\ w_{21}\end{array}\right)$
  
  and ${\mathrm{w}}_2=\left(\begin{array}{c}{w}_{12}\\ {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{22}\end{array}\right)$
  
  described and in the matrix W = $\left(\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{22}\end{array}\right)$
  
  is summarized
• the magnification term V (x, y) is either a scalar $\mathrm{V}\left(\mathrm{x},\mathrm{y}\right)=\left(\frac{\mathrm{z}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}-\mathrm{1}\right)$
  
  is, with the effective distance of the viewing grid from the motif image e, or a matrix V (x, y) = (A (x, y) - I), where the matrix $\mathrm{A}\left(\mathrm{x},\mathrm{y}\right)=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{11}}\left(\mathrm{x},\mathrm{y}\right)& {\mathrm{a}}_{\mathrm{12}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{a}}_{\mathrm{21}}\left(\mathrm{x},\mathrm{y}\right)& {\mathrm{a}}_{\mathrm{22}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$
  
  Describes a desired magnification and movement behavior of the given body, and I is the unit matrix,
• the vector (c ₁ (x, y), c ₂ (x, y)) with 0≤c ₁ (x, y), c ₂ (x, y) <1, the relative position of the center of the viewing elements within the cells of the Motiv image indicates
• the vector (d ₁ (x, y), d ₂ (x, y)) with 0 ≦ d ₁ (x, y), d ₂ (x, y) <1 represents a shift of the cell boundaries in the motif image, and
• g (x, y) is a mask function for adjusting the visibility of the body.

This elevation profile model, presented as a second aspect of the invention, uses a two-dimensional drawing f (x, y) of a body to simplify the calculation of the motif image, with an additional z coordinate z (x, y) for each point x, y of the two-dimensional image of the body , y) indicates height / depth information for this point. The two-dimensional drawing f (x, y) is a brightness distribution (grayscale image), a color distribution (color image), a binary distribution (line drawing) or a distribution of other image characteristics such as transparency, reflectivity, density or the like.

gegeben.In an advantageous development of the elevation profile model even two height functions z ₁ (x, y) and z ₂ (x, y) and two angles φ ₁ (x, y) and φ ₂ (x, y) are given and is the magnification term by a matrix V (x, y) = (A (x, y) - I) $$\mathrm{A}\left(\mathrm{x},\mathrm{y}\right)=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{11}}\left(\mathrm{x},\mathrm{y}\right)& {\mathrm{a}}_{\mathrm{12}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{a}}_{\mathrm{21}}\left(\mathrm{x},\mathrm{y}\right)& {\mathrm{a}}_{\mathrm{22}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)=\left(\begin{array}{cc}\frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}& \frac{{\mathrm{z}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\cdot {\cot \mathit{\phi }}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\\ \frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\cdot {\tan \mathit{\phi }}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)& \frac{{\mathrm{z}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\end{array}\right)$$

given.

gegeben ist, so dass beim Drehen der Anordnung bei der Betrachtung die Höhenfunktionen z₁(x,y) und z₂(x,y) des dargestellten Körpers ineinander übergehen.According to a variant it can be provided that two height functions z ₁ (x, y) and z ₂ (x, y) are given and that the magnification term is given by a matrix V (x, y) = (A (x, y) -I ) With $$\mathrm{A}\left(\mathrm{x},\mathrm{y}\right)=\left(\begin{array}{cc}\frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}& \mathrm{0}\\ \mathrm{0}& \frac{{\mathrm{z}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\end{array}\right)$$

is given, so that when turning the arrangement in the consideration, the height functions z ₁ (x, y) and z ₂ (x, y) of the body shown merge into each other.

gegeben. Der dargestellte Körper bewegt sich in dieser Variante bei Betrachtung mit Augenabstand in x-Richtung und Kippen der Anordnung in x-Richtung in Richtung _φ1 zur x-Achse. Beim Kippen in y-Richtung erfolgt keine Bewegung.In another variant, a height function z (x, y) and an angle φ _{1 are} given, and the magnification term is given by a matrix V (x, y) = (A (x, y) -I) $$\mathrm{A}\left(\mathrm{x},\mathrm{y}\right)=\left(\begin{array}{cc}\frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}& \mathrm{0}\\ \frac{{\mathrm{z}}_1\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\cdot {\tan \phi }_1& 1\end{array}\right)$$

given. The illustrated body moves in this variant, when viewed with eye relief in the x-direction and tilting of the array in the x-direction in the direction of _φ 1 to the x-axis. When tilting in the y direction, there is no movement.

mit dem Spalt- bzw. Zylinderachsenabstand d gegeben ist. Die Zylinderlinsenachse liegt dabei in y-Richtung. Alternativ kann das Motivbild auch mit einem Lochblenden- bzw. Linsenarray mit $$\mathrm{W}=\left(\begin{array}{cc}\mathrm{d}& \mathrm{0}\\ \mathrm{d}\cdot \tan \mathit{\beta }& {\mathrm{d}}_{\mathrm{2}}\end{array}\right)$$

mit d₂, β beliebig betrachtet werden.In the latter variant, the viewing grid can also be a split screen, cylindrical lens grid or cylinder cavity mirror grid, the unit cell through $$\mathrm{W}=\left(\begin{array}{cc}\mathrm{<figure-callout\; id="0"\; label="d"\; filenames="imgf0002.png"\; state="\{\{state\}\}">d</figure-callout>}& \mathrm{0}\\ \mathrm{0}& \mathrm{\infty }\end{array}\right)$$

given with the gap or cylinder axis distance d. The cylindrical lens axis lies in the y-direction. Alternatively, the motif image with a pinhole or lens array with $$\mathrm{W}=\left(\begin{array}{cc}\mathrm{d}& \mathrm{0}\\ \mathrm{d}\cdot \tan \mathit{\beta }& {\mathrm{d}}_{\mathrm{2}}\end{array}\right)$$

with d ₂ , β are considered arbitrary.

gegeben, und die passende Matrix A, bei der in Richtung γ keine Vergrößerung oder Verzerrung vorliegt, lautet: $$\mathrm{A}=\left(\begin{array}{cc}\mathrm{cos\gamma }& -\mathrm{sin\gamma }\\ \mathrm{sin\gamma }& \mathrm{cos\gamma }\end{array}\right)\cdot \left(\begin{array}{cc}\frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}& \mathrm{0}\\ \frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\cdot {\tan \mathit{\phi }}_{\mathrm{1}}& \mathrm{1}\end{array}\right)\cdot \left(\begin{array}{cc}\mathrm{cos\gamma }& \mathrm{sin\gamma }\\ -\mathrm{sin\gamma }& \mathrm{cos\gamma }\end{array}\right)\mathrm{.}$$

If in general the cylindrical lens axis lies in any direction γ and if d again denotes the axial spacing of the cylindrical lenses, then the lens grid is through $$\mathrm{W}=\left(\begin{array}{cc}\mathrm{cos\gamma }& -\mathrm{sin\gamma }\\ \mathrm{sin\gamma }& \mathrm{cos\gamma }\end{array}\right)\cdot \left(\begin{array}{cc}\mathrm{<figure-callout\; id="0"\; label="d"\; filenames="imgf0002.png"\; state="\{\{state\}\}">d</figure-callout>}& \mathrm{0}\\ \mathrm{0}& \mathrm{\infty }\end{array}\right)$$

given, and the appropriate matrix A, in which no magnification or distortion in the direction of γ is: $$\mathrm{A}=\left(\begin{array}{cc}\mathrm{cos\gamma }& -\mathrm{sin\gamma }\\ \mathrm{sin\gamma }& \mathrm{cos\gamma }\end{array}\right)\cdot \left(\begin{array}{cc}\frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}& \mathrm{0}\\ \frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\cdot {\tan \mathit{\phi }}_{\mathrm{1}}& \mathrm{1}\end{array}\right)\cdot \left(\begin{array}{cc}\mathrm{cos\gamma }& \mathrm{sin\gamma }\\ -\mathrm{sin\gamma }& \mathrm{cos\gamma }\end{array}\right)\mathrm{,}$$

wobei d₂ , β beliebig sein können.The pattern thus created for the print or embossed image to be created behind a lenticular grid W can be viewed not only with the slit-diaphragm or cylindrical lens array with axis in the direction γ, but also with a pinhole or lens array $$\mathrm{W}=\left(\begin{array}{cc}\mathrm{cos\gamma }& -\mathrm{sin\gamma }\\ \mathrm{sin\gamma }& \mathrm{cos\gamma }\end{array}\right)\cdot \left(\begin{array}{cc}\mathrm{d}& \mathrm{0}\\ \mathrm{d}\cdot \tan \mathit{\beta }& {\mathrm{d}}_{\mathrm{2}}\end{array}\right).$$

where d ₂ , β can be arbitrary.

gegeben, so dass sich der dargestellte Körper bei Betrachtung mit Augenabstand in x-Richtung und Kippen der Anordnung in x-Richtung senkrecht zur x-Achse bewegt. Bei Betrachtung mit Augenabstand in y-Richtung und Kippen der Anordnung in y-Richtung bewegt sich der Körper in Richtung φ₂ zur x-Achse.Another variant describes an orthoparallactic 3D effect. In this variant, two height functions z ₁ (x, y) and z ₂ (x, y) and an angle φ _{2 are} given and the magnification term is given by a matrix V (x, y) = (A (x, y) -I ) With $$\mathrm{A}\left(\mathrm{x},\mathrm{y}\right)=\left(\begin{array}{cc}\mathrm{0}& \frac{{\mathrm{z}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\cdot {\cot \mathit{\phi }}_{\mathrm{2}}\\ \frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}& \frac{{\mathrm{z}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\end{array}\right).\mathrm{A}\left(\mathrm{x},\mathrm{y}\right)=\left(\begin{array}{cc}\mathrm{0}& \frac{{\mathrm{z}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\\ \frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}& \mathrm{0}\end{array}\right){\mathrm{if\; \phi }}_{\mathrm{2}}=\mathrm{0}$$

given, so that the body shown when viewed with eye relief in the x-direction and tilting of the array in the x-direction perpendicular to the x-axis moves. When viewing with eye relief in the y direction and tilting the arrangement in the y-direction, the body moves in the direction of φ ₂ to the x-axis.

• ein Motivbild, das in eine Mehrzahl von Zellen eingeteilt ist, in denen jeweils abgebildete Bereiche des vorgegebenen Körpers angeordnet sind, und
• ein Betrachtungsraster aus einer Mehrzahl von Betrachtungselementen zur Darstellung des vorgegebenen Körpers bei der Betrachtung des Motivbilds mithilfe des Betrachtungsrasters,
• wobei das Motivbild mit seiner Einteilung in eine Mehrzahl von Zellen eine Bildfunktion m(x,y) aufweist, die gegeben ist durch $$\mathrm{m}\left(\mathrm{x},\mathrm{y}\right)={\mathrm{f}}_{\mathrm{j}}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{K}}\\ {\mathrm{y}}_{\mathrm{K}}\end{array}\right)\cdot \mathit{g}\left(\mathit{x},\mathit{y}\right),\mathrm{mit}$$
  
  $$\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{K}}\\ {\mathrm{y}}_{\mathrm{K}}\end{array}\right)=\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathit{V}}_{\mathrm{j}}\cdot \left(\left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)\right)\mathrm{modW}\right)-{\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)-{\mathrm{w}}_{\mathrm{c}}\left(\mathrm{x},\mathrm{y}\right)\right),$$
  
  $${\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{d}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{d}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$$
  
  und $${\mathrm{w}}_{\mathrm{c}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{c}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right),$$
  
  wobei für j der kleinste oder der größte Index zu nehmen ist, für den ${\mathrm{t}}_{\mathrm{j}}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{K}}\\ {\mathrm{y}}_{\mathrm{K}}\end{array}\right)$
  
  ungleich Null ist, und wobei
• die Einheitszelle des Betrachtungsrasters durch Gitterzellenvektoren ${\mathrm{w}}_1=\left(\begin{array}{c}{w}_{11}\\ w_{21}\end{array}\right)$
  
  und ${\mathrm{w}}_2=\left(\begin{array}{c}{w}_{12}\\ {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{22}\end{array}\right)$
  
  beschrieben und in der Matrix $\mathrm{W}=$
  
  $\left(\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{22}\end{array}\right)$
  
  zusammengefasst ist,
• der Vergrößerungsterm V_j entweder ein Skalar ${\mathrm{V}}_{\mathrm{j}}=\left(\frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}-\mathrm{1}\right)$
  
  ist, mit dem effektiven Abstand des Betrachtungsrasters vom Motivbild e, oder eine Matrix V_j =(A_j - I) ist, wobei die Matrix ${\mathrm{A}}_{\mathrm{j}}=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{j}\mathrm{11}}& {\mathrm{a}}_{\mathrm{j}\mathrm{12}}\\ {\mathrm{a}}_{\mathrm{j}\mathrm{21}}& {\mathrm{a}}_{\mathrm{<figure-callout\; id="22"\; label="j"\; filenames="imgf0001.png"\; state="\{\{state\}\}">j</figure-callout>}\mathrm{22}}\end{array}\right)$
  
  ein gewünschtes Vergrößerungs- und Bewegungsverhalten des vorgegebenen Körpers beschreibt, und I die Einheitsmatrix ist,
• der Vektor (c₁(x,y), c₂(x,y)) mit 0 ≤ c₁(x,y), c₂(x,y) < 1 die relative Position des Zentrums der Betrachtungselemente innerhalb der Zellen des Motivbilds angibt,
• der Vektor (d₁(x,y), d₂(x,y)) mit 0 ≤ d₁(x,y), d₂(x,y) < 1 eine Verschiebung der Zellengrenzen im Motivbild darstellt, und
• g(x,y) eine Maskenfunktion zur Einstellung der Sichtbarkeit des Körpers ist.

According to a third aspect of the invention, a generic representation arrangement comprises a raster image arrangement for displaying a predetermined three-dimensional body, which is characterized by n sections fi (x, y) and n transparency step functions tj (x, y) with j = 1,. when viewed with eye relief in the x direction, the sections each lie at a depth z _j , z _j > z _ji . Depending on the position of the body with respect to the plane of the drawing (penetrating behind or in front of the plane of the drawing or the drawing plane), z _j can be positive or negative or even 0. f _j (x, y) is the image function of the j-th intersection, and the transparency step function t _j (x, y) equals 1 if the intersection j obscures objects behind it (x, y) and is otherwise equal to 0. The representation arrangement contains

• a motif image, which is divided into a plurality of cells, in each of which imaged areas of the predetermined body are arranged, and
• a viewing grid of a plurality of viewing elements for displaying the predetermined body when viewing the motif image using the viewing grid,
• wherein the motif image with its division into a plurality of cells has an image function m (x, y), which is given by $$\mathrm{m}\left(\mathrm{x},\mathrm{y}\right)={\mathrm{f}}_{\mathrm{j}}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{K}}\\ {\mathrm{y}}_{\mathrm{K}}\end{array}\right)\cdot \mathit{G}\left(\mathit{x},\mathit{y}\right).\mathrm{With}$$
  
  $$\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{K}}\\ {\mathrm{y}}_{\mathrm{K}}\end{array}\right)=\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathit{V}}_{\mathrm{j}}\cdot \left(\left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)\right)\mathrm{ModW}\right)-{\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)-{\mathrm{w}}_{\mathrm{c}}\left(\mathrm{x},\mathrm{y}\right)\right).$$
  
  $${\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{d}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{d}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$$
  
  and $${\mathrm{w}}_{\mathrm{c}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{c}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right).$$
  
  where j is the smallest or the largest index for which ${\mathrm{t}}_{\mathrm{j}}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{K}}\\ {\mathrm{y}}_{\mathrm{K}}\end{array}\right)$
  
  is not zero, and where
• the unit cell of the viewing grid by grid cell vectors ${\mathrm{w}}_1=\left(\begin{array}{c}{w}_{11}\\ w_{21}\end{array}\right)$
  
  and ${\mathrm{w}}_2=\left(\begin{array}{c}{w}_{12}\\ {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{22}\end{array}\right)$
  
  described and in the matrix $\mathrm{W}=$
  
  $\left(\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{22}\end{array}\right)$
  
  is summarized
• the magnification term V _j either a scalar ${\mathrm{V}}_{\mathrm{j}}=\left(\frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}-\mathrm{1}\right)$
  
  is, with the effective distance of the viewing grid from the motif image e, or a matrix V _j = (A _j - I), where the matrix ${\mathrm{A}}_{\mathrm{j}}=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{j}\mathrm{11}}& {\mathrm{a}}_{\mathrm{j}\mathrm{12}}\\ {\mathrm{a}}_{\mathrm{j}\mathrm{21}}& {\mathrm{a}}_{\mathrm{<figure-callout\; id="22"\; label="j"\; filenames="imgf0001.png"\; state="\{\{state\}\}">j</figure-callout>}\mathrm{22}}\end{array}\right)$
  
  Describes a desired magnification and movement behavior of the given body, and I is the unit matrix,
• the vector (c ₁ (x, y), c ₂ (x, y)) with 0≤c ₁ (x, y), c ₂ (x, y) <1, the relative position of the center of the viewing elements within the cells of the Motiv image indicates
• the vector (d ₁ (x, y), d ₂ (x, y)) with 0 ≦ d ₁ (x, y), d ₂ (x, y) <1 represents a shift of the cell boundaries in the motif image, and
• g (x, y) is a mask function for adjusting the visibility of the body.

ungleich Null ist, so erhält man ein Bild, das die Körpervorderseite von außen zeigt. Wird dagegen der größte Index genommen, für den ${\mathrm{t}}_{\mathrm{j}}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{K}}\\ {\mathrm{y}}_{\mathrm{K}}\end{array}\right)$

ungleich Null ist, so erhält man ein tiefenumgekehrtes (pseudoskopisches) Bild, das die Körperrückseite von innen zeigt.If the index j is selected, the smallest index is taken for which ${\mathrm{t}}_{\mathrm{j}}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{K}}\\ {\mathrm{y}}_{\mathrm{K}}\end{array}\right)$

is not equal to zero, you get an image that shows the front of the body from the outside. In contrast, the largest index is taken for the ${\mathrm{t}}_{\mathrm{j}}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{K}}\\ {\mathrm{y}}_{\mathrm{K}}\end{array}\right)$

is not equal to zero, we obtain a deeply reversed (pseudoscopic) image showing the back of the body from the inside.

When cutting level model of the third invention aspect of the three-dimensional body to simplify the calculation of the motif image by n sections fj (x, y) and n transparency step functions is t _j (x, y) with j = 1, ... n, predetermined, when viewed with eye relief in the x-direction each lie at a depth z _j , z _j > z _j-1 . _fj (x, y) is the image function of the jth section which can indicate a brightness distribution (grayscale image), a color distribution (color image), a binary distribution (line drawing) or other image properties such as transparency, reflectivity, density or the like , The transparency step function t _j (x, y) is equal to 1 if the cut j at the location (x, y) obscures objects behind it and is otherwise equal to 0.

gegeben, so dass sich beim Drehen der Anordnung der Tiefeneindruck des dargestellten Körpers um den Änderungsfaktor k ändert.In an advantageous embodiment of the sectional plane model, a change factor k is given other than 0 and the magnification term is given by a matrix V _j = (Aj - I) $${\mathrm{A}}_{\mathrm{j}}=\left(\begin{array}{cc}\frac{{\mathrm{<figure-callout\; id="0"\; label="z\; j\; e"\; filenames="imgf0002.png"\; state="\{\{state\}\}">z\; </figure-callout>}}_{\mathrm{j}}}{\mathrm{e}}& \mathrm{0}\\ \mathrm{0}& \mathrm{k}\cdot \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\end{array}\right)$$

given, so that the rotation of the arrangement of the depth impression of the body shown by the change factor k changes.

gegeben, so dass sich der dargestellte Körper bei Betrachtung mit Augenabstand in x-Richtung und Kippen der Anordnung in x-Richtung in Richtung φ₁ zur x-Achse bewegt und bei Betrachtung mit Augenabstand in y-Richtung und Kippen der Anordnung in y-Richtung in Richtung φ₂ zur x-Achse bewegt und um den Änderungsfaktor k in der Tiefendimension gedehnt ist.In an advantageous variant, a change factor k other than 0 and two angles φ ₁ and φ _{2 are} given, and the magnification term is given by a matrix V _j = (A _{j -I} ) $${\mathrm{A}}_{\mathrm{j}}=\left(\begin{array}{cc}\frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}& \mathrm{k}\cdot \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\cdot {\cot \mathit{\phi }}_{\mathrm{2}}\\ \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\cdot {\tan \mathit{\phi }}_{\mathrm{1}}& \mathrm{k}\cdot \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\end{array}\right)$$

given, so that the body shown when viewed with eye relief in the x-direction and tilting the array in the x-direction in the direction of φ ₁ to the x-axis moves and when viewed with eye relief in the y-direction and tilting of the array in the y-direction moved in the direction of φ ₂ to the x-axis and is stretched by the change factor k in the depth dimension.

gegeben, so dass sich der dargestellte Körper bei Betrachtung mit Augenabstand in x-Richtung und Kippen der Anordnung in x-Richtung in Richtung φ₁ zur x-Achse bewegt und beim Kippen in y-Richtung keine Bewegung erfolgt.According to a further advantageous variant, an angle φ _{1 is} predetermined and the magnification term is given by a matrix V _j = (A _{j -I} ) $${\mathrm{A}}_{\mathrm{j}}=\left(\begin{array}{cc}\frac{\mathrm{zj}}{\mathrm{e}}& \mathrm{0}\\ \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\cdot {\tan \mathit{\phi }}_{\mathrm{1}}& \mathrm{1}\end{array}\right)$$

given, so that the body shown moves when viewed with eye relief in the x direction and tilting of the array in the x direction in the direction of φ ₁ to the x-axis and no tilting occurs in the y-direction.

gegeben. Wie bereits oben in Zusammenhang mit dem zweiten Erfindungsaspekt beschrieben, kann auch hier das Motivbild mit einem Lochblenden- bzw. Linsenarray mit $\mathrm{W}=\left(\begin{array}{cc}\mathrm{d}& \mathrm{0}\\ \mathrm{d}\cdot \tan \mathit{\beta }& {\mathrm{d}}_{\mathrm{2}}\end{array}\right)$

mit d₂, β beliebig, betrachtet werden, oder mit einem Zylinderlinsenraster, bei dem die Zylinderlinsen-Achsen in beliebiger Richtung γ liegen. Die durch Drehung um einen Winkel γ erhaltene Gestalt von W und A wurde oben bereits explizit angegeben.In the latter variant, the viewing grid can also be a split screen or cylinder lens grid with the gap or cylinder axis spacing be d. If the cylindrical lens axis lies in the y direction, the unit cell of the viewing grid is through $$\mathrm{W}=\left(\begin{array}{cc}\mathrm{<figure-callout\; id="0"\; label="d"\; filenames="imgf0002.png"\; state="\{\{state\}\}">d</figure-callout>}& \mathrm{0}\\ \mathrm{0}& \mathrm{\infty }\end{array}\right)$$

given. As already described above in connection with the second aspect of the invention, the motif image with a pinhole or lens array can also be used here $\mathrm{W}=\left(\begin{array}{cc}\mathrm{d}& \mathrm{0}\\ \mathrm{d}\cdot \tan \mathit{\beta }& {\mathrm{d}}_{\mathrm{2}}\end{array}\right)$

with d ₂ , β are arbitrarily considered, or with a cylindrical lens grid, in which the cylindrical lens axes lie in any direction γ. The shape of W and A obtained by rotation through an angle γ has already been explicitly stated above.

wenn φ = 0 gegeben, so dass sich der dargestellte Körper beim waagrechten Kippen senkrecht zur Kipprichtung und beim senkrechten Kippen in Richtung φ zur x-Achse bewegt.According to a further advantageous variant, a change factor k is not equal to 0 and an angle φ is given and the magnification term is given by a matrix V _j = (Aj - I) $${\mathrm{A}}_{\mathrm{j}}=\left(\begin{array}{cc}\mathrm{0}& \mathrm{k}\cdot \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\cdot \cot \mathit{\phi }\\ \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}& \mathrm{k}\cdot \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\end{array}\right).{\mathrm{A}}_{\mathrm{j}}=\left(\begin{array}{cc}\mathrm{0}& \mathrm{k}\cdot \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\\ \frac{{\mathrm{<figure-callout\; id="0"\; label="z\; j\; e"\; filenames="imgf0002.png"\; state="\{\{state\}\}">z\; </figure-callout>}}_{\mathrm{j}}}{\mathrm{e}}& \mathrm{0}\end{array}\right)$$

if φ = 0 given, so that the body shown moves horizontally tilting perpendicular to the tilting direction and the vertical tilting in the direction φ to the x-axis.

gegeben, so dass sich der dargestellte Körper unabhängig von der Kipprichtung stets in Richtung φ₁ zur x-Achse bewegt.In a further variant, a change factor k other than 0 and an angle φ _{1 are} given and the magnification term is given by a matrix V _j = (A _{j -I} ) $${\mathrm{A}}_{\mathrm{j}}=\left(\begin{array}{cc}\frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}& \mathrm{k}\cdot \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\cdot {\cot \mathit{\phi }}_{\mathrm{1}}\\ \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\cdot {\tan \mathit{\phi }}_{\mathrm{1}}& \mathrm{k}\cdot \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\end{array}\right)$$

given, so that the body shown always moves independently of the tilting direction in the direction of φ ₁ to the x-axis.

In all of the aspects of the invention, the viewing elements of the viewing grid are preferably arranged periodically or locally periodically, with the local period parameters preferably changing only slowly in relation to the periodicity length in the latter case. The periodicity length or the local periodicity length is preferably between 3 μm and 50 μm, preferably between 5 μm and 30 μm, particularly preferably between about 10 μm and about 20 μm. An abrupt change in the periodicity length is also possible if it was previously kept constant or nearly constant over a length which is large in comparison to the periodicity length, for example for more than 20, 50 or 100 periodicity lengths.

The viewing elements can be formed in all aspects of the invention by non-cylindrical microlenses, in particular by microlenses with a circular or polygonal limited base surface, or by elongated cylindrical lenses whose extension in the longitudinal direction more than 250 microns, preferably more than 300 microns, more preferably more than 500 μm and in particular more than 1 mm. In further preferred variants of the invention, the viewing elements are pinhole apertures, slotted apertures, apertured apertured or slit apertures, aspheric lenses, Fresnel lenses, GRIN (Gradient Refraction Index) lenses, zone plates, holographic lenses, concave mirrors, Fresnel mirrors, zone mirrors, or other focusing or focusing elements also formed with a masking effect.

größer ist als die Einheitszelle des Betrachtungsrasters W. Der Träger einer Funktion bezeichnet dabei in üblicher Weise die abgeschlossene Hülle des Bereichs, in dem die Funktion nicht Null ist. Auch für das Schnittebenen-Modell sind die Träger der Schnittbilder $${\mathit{f}}_{\mathit{j}}\left(\left(\mathrm{A}-\mathrm{I}\right)\cdot \left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)\right)$$

vorzugsweise größer als die Einheitszelle des Betrachtungsrasters W.In preferred embodiments of the height profile model it is provided that the carrier of the image function $$\mathit{f}\left(\left(\mathrm{A}-\mathrm{I}\right)\cdot \left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)\right)$$

is greater than the unit cell of the viewing grid W. The carrier of a function referred to in the usual way, the closed envelope of the area in which the function is not zero. Also for the cutting plane model are the carriers of the sectional images $${\mathit{f}}_{\mathit{j}}\left(\left(\mathrm{A}-\mathrm{I}\right)\cdot \left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)\right)$$

preferably larger than the unit cell of the viewing grid W.

The illustrated three-dimensional image has, in advantageous embodiments, no periodicity, ie is a representation of a single 3D motif.

In an advantageous variant of the invention, the viewing grid and the motif image of the presentation arrangement are firmly connected to one another and thus form a security element with a viewing grid and motif image arranged at a distance one above the other. The motif image and the viewing grid are advantageously arranged on opposite surfaces of an optical spacer layer. The security element may in particular be a security thread, a tear thread, a security tape, a security strip, a patch or a label for application to a security paper, value document or the like. The total thickness of the security element is preferably below 50 μm, preferably below 30 μm and particularly preferably below 20 μm.

According to another, likewise advantageous variant of the invention, the viewing grid and the motif image of the presentation arrangement are at different Positioning a disk arranged that the viewing grid and the motif image for self-authentication are superimposed and form a security element in the superimposed state. The viewing grid and the motif image are superimposed in particular by bending, folding, bending or folding the data carrier.

According to a further, likewise advantageous variant of the invention, the motif image is displayed by an electronic display device and the viewing grid for viewing the displayed motif image is firmly connected to the electronic display device. Instead of being firmly connected to the electronic display device, the viewing grid can also be a separate viewing grid, which can be brought onto or in front of the electronic display device for viewing the displayed motif image.

In the context of this description, the security element can thus be formed both as a permanent security element by a viewing grid and motif image fixedly connected to one another, as well as by a spatially separated viewing grid and an associated motif image, wherein the two elements form a temporarily present security element when superimposed. Statements in the description about the movement behavior or the visual impression of the security element relate both to firmly connected permanent security elements and to superimposed temporary security elements.

In all variants of the invention, the cell boundaries in the motif image may advantageously be spatially independent, so that the vector (d ₁ (x, y), d ₂ (x, y)) occurring in the image function m (x, y) is constant. Alternatively, the cell boundaries in the motif image may also be spatially dependent. Especially For example, the motif image may have two or more subregions with different, each constant cell grid.

A location-dependent vector (d ₁ (x, y), d ₂ (x, y)) can also be used to define the outline of the cells in the motif image. For example, instead of parallelogram-shaped cells, it is also possible to use cells with a different uniform shape which match one another in such a way that the area of the motif image is filled up completely (tiling of the surface of the motif image). By choosing the location-dependent vector (d ₁ (x, y), d ₂ (x, y)), the cell shape can be set as desired. As a result, the designer has particular influence on which viewing angles subject jumps occur.

The motif image can also be subdivided into different regions, in which the cells each have identical shape, while the cell shapes differ in the different regions. This causes parts of the motif, which are assigned to different areas, to jump at different tilt angles when tilting the security element. If the areas with different cells are large enough that they are visible to the naked eye, additional visible information can be accommodated in the security element in this way. On the other hand, if the areas are microscopic, ie can only be seen with magnifying aids, additional hidden information can be accommodated in the security element in this way, which can serve as a higher-level security feature.

Furthermore, a location-dependent vector (d ₁ (x, y), d ₂ (x, y)) can also be used to generate cells, all of which are mutually different in shape differ. As a result, it is possible to generate a completely individual security feature that can be tested, for example, by means of a microscope.

The mask function g occurring in the image function m (x, y) of all variants of the invention is advantageously identical in many cases. 1. In other, likewise advantageous configurations, the mask function g is zero in subareas, in particular in edge regions of the cells of the motif image, and then limits the solid angle range under which the three-dimensional image can be seen. In addition to an angle constraint, the mask function may also describe an image field constraint in which the three-dimensional image is not visible, as explained in greater detail below.

In advantageous embodiments of all variants of the invention, it is further provided that the relative position of the center of the viewing elements within the cells of the motif image is location-independent, ie the vector (c ₁ (x, y), c ₂ (x, y)) is constant. In other embodiments, however, it may also be appropriate to make the relative position of the center of the viewing elements within the cells of the motif image location-dependent, as explained in more detail below.

According to a development of the invention, the motif image for enhancing the three-dimensional visual impression is filled with Fresnel structures, Blazegittern or other optically active structures.

In the aspects of the invention described so far, the raster image arrangement of the representation arrangement always represents a single three-dimensional image. In further aspects, the invention also encompasses configurations in which a plurality of three-dimensional images are displayed simultaneously or alternately.

• einem Motivbild, das in eine Mehrzahl von Zellen eingeteilt ist, in denen jeweils abgebildete Bereiche der vorgegebenen Körper angeordnet sind,
• einem Betrachtungsraster aus einer Mehrzahl von Betrachtungselementen zur Darstellung der vorgegebenen Körper bei der Betrachtung des Motivbilds mithilfe des Betrachtungsrasters,
• wobei das Motivbild mit seiner Einteilung in eine Mehrzahl von Zellen eine Bildfunktion m(x,y) aufweist, die gegeben ist durch m(x, y) = F(h ₁,h ₂ ,...h _N), mit den beschreibenden Funktionen $${\mathrm{h}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y}\right)={\mathrm{f}}_{\mathrm{i}}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{iK}}\\ {\mathrm{y}}_{\mathrm{iK}}\\ {\mathrm{z}}_{\mathrm{iK}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)\end{array}\right)\cdot {\mathit{g}}_{\mathit{i}}\left(\mathit{x},\mathit{y}\right),$$
  
  mit $$\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{iK}}\\ {\mathrm{y}}_{\mathrm{iK}}\end{array}\right)=\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathit{V}}_{\mathit{i}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)\cdot \left(\left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)\right)\mathrm{modW}\right)-{\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)-{\mathrm{w}}_{\mathrm{ci}}\left(\mathrm{x},\mathrm{y}\right)\right)$$
  
  $${\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{d}}_{\mathrm{i}\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{d}}_{\mathrm{i}\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$$
  
  und $${\mathrm{w}}_{\mathrm{ci}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{i}\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{c}}_{\mathrm{i}\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right),$$
  
• wobei F(h ₁,h ₂ ,...h _N) eine Masterfunktion ist, die eine Verknüpfung der N beschreibenden Funktionen h_i(x,y) angibt, und wobei
• die Einheitszelle des Betrachtungsrasters durch Gitterzellenvektoren ${\mathrm{w}}_1=\left(\begin{array}{c}{w}_{11}\\ w_{21}\end{array}\right)$
  
  und ${\mathrm{w}}_2=\left(\begin{array}{c}{w}_{12}\\ {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{22}\end{array}\right)$
  
  beschrieben und in der Matrix $\mathrm{W}=$
  
  $\left(\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{22}\end{array}\right)$
  
  zusammengefasst ist, und x_m und y_m die Gitterpunkte des W-Gitters bezeichnen,
• die Vergrößerungsterme V_i(x,y,x_m,y_m) entweder Skalare ${\mathrm{V}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)=\left(\frac{{\mathrm{z}}_{\mathrm{iK}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)}{\mathrm{e}}-\mathrm{1}\right)$
  
  sind, mit dem effektiven Abstand des Betrachtungsrasters vom Motivbild e, oder Matrizen V_i(x,y,x_m,y_m) =(A_i(x,y,x_m,y_m) - I) sind, wobei die Matrizen ${\mathrm{A}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{i}\mathrm{11}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)& {\mathrm{a}}_{\mathrm{i}\mathrm{12}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)\\ {\mathrm{a}}_{\mathrm{i}\mathrm{21}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)& {\mathrm{a}}_{\mathrm{i}\mathrm{22}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)\end{array}\right)$
  
  jeweils ein gewünschtes Vergrößerungs- und Bewegungsverhalten des vorgegebenen Körpers f_i beschreiben und I die Einheitsmatrix ist,
• die Vektoren (c_i1(x,y), c_i2(x,y)) mit 0 ≤ c_i1 (x,y),c_i2(x,y) < 1 für den Körper f_i jeweils die relative Position des Zentrums der Betrachtungselemente innerhalb der Zellen i des Motivbilds angeben,
• die Vektoren (d_i1(x,y), d_i2(x,y)) mit 0 ≤ d_i1(x,y),d_i2(x,y) < 1 jeweils eine Verschiebung der Zellengrenzen im Motivbild darstellen, und
• g_i(x,y) Maskenfunktionen zur Einstellung der Sichtbarkeit des Körpers f_i sind.

A representation according to a fourth aspect of the invention corresponding to the general perspective of the first aspect of the invention includes a raster image arrangement for displaying a plurality of predetermined three-dimensional bodies, represented by body functions f _i (x, y, z), i = 1,2,.. N≥1 are given, with

• a motif image, which is divided into a plurality of cells, in each of which imaged areas of the predetermined body are arranged,
• a viewing grid of a plurality of viewing elements for displaying the predetermined bodies when viewing the motif image using the viewing grid,
• wherein the motif image with its division into a plurality of cells has an image function m (x, y), which is given by m (x, y) = F ( h ₁ , h ₂ , ... h _{N )} , with the descriptive features $${\mathrm{H}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y}\right)={\mathrm{f}}_{\mathrm{i}}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{iK}}\\ {\mathrm{y}}_{\mathrm{iK}}\\ {\mathrm{z}}_{\mathrm{iK}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)\end{array}\right)\cdot {\mathit{G}}_{\mathit{i}}\left(\mathit{x},\mathit{y}\right).$$
  
  With $$\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{iK}}\\ {\mathrm{y}}_{\mathrm{iK}}\end{array}\right)=\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathit{V}}_{\mathit{i}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)\cdot \left(\left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)\right)\mathrm{ModW}\right)-{\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)-{\mathrm{w}}_{\mathrm{ci}}\left(\mathrm{x},\mathrm{y}\right)\right)$$
  
  $${\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{d}}_{\mathrm{i}\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{d}}_{\mathrm{i}\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$$
  
  and $${\mathrm{w}}_{\mathrm{ci}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{i}\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{c}}_{\mathrm{i}\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right).$$
  
• where F (h ₁ , h ₂ , ... h _N ) is a master function indicating a combination of the N descriptive functions h _i (x, y), and where
• the unit cell of the viewing grid by grid cell vectors ${\mathrm{w}}_1=\left(\begin{array}{c}{w}_{11}\\ w_{21}\end{array}\right)$
  
  and ${\mathrm{w}}_2=\left(\begin{array}{c}{w}_{12}\\ {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{22}\end{array}\right)$
  
  described and in the matrix $\mathrm{W}=$
  
  $\left(\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{22}\end{array}\right)$
  
  and x _m and y _m denote the grid points of the W grid,
• the magnification terms V _i (x, y, x _m , y _m ) are either scalars ${\mathrm{V}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)=\left(\frac{{\mathrm{z}}_{\mathrm{iK}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)}{\mathrm{e}}-\mathrm{1}\right)$
  
  are, with the effective distance of the viewing grid from the motif image e, or matrices V _i (x, y, x _m , y _m ) = (A _i (x, y, x _m , y _m ) - I), where the matrices ${\mathrm{A}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{i}\mathrm{11}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)& {\mathrm{a}}_{\mathrm{i}\mathrm{12}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)\\ {\mathrm{a}}_{\mathrm{i}\mathrm{21}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)& {\mathrm{a}}_{\mathrm{i}\mathrm{22}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)\end{array}\right)$
  
  each describe a desired magnification and movement behavior of the given body f _i and I is the unit matrix,
• the vectors (c _i1 (x, y), c _i2 (x, y)) with 0 ≤ c _i1 (x, y), c _i2 (x, y) <1 for the body f _i respectively the relative position of the center indicate the viewing elements within the cells i of the motif image,
• the vectors (d _i1 (x, y), d _i2 (x, y)) with 0 ≦ d _i1 (x, y), d _i2 (x, y) <1 represent respectively a shift of the cell boundaries in the motif image, and
• g _i (x, y) are mask functions for adjusting the visibility of the body f _i .

For z _ik (x, y, x _m , y _m ), ie the z-coordinate of a common point of the visual line with the body f _i , more than one value can be considered, from which a value is formed or selected according to rules to be determined , In an opaque body, for example, in addition to the body function f _i (x, y, z), a transparency step function (characteristic function) t _i (x, y, z) may be predetermined, where t _i (x, y, z) is equal to 1 is when the body f _i (x, y, z) at the location (x, y, z) obscures the background and otherwise equals 0. For viewing direction substantially in the direction of the z-axis, the smallest value for z _ik (x, y, x _m , y _m ) for which t _i (x, y, z _ik ) is not equal to 0 is, if one has want to look at the body front.

The values z _ik (x, y, x _m , y _m ) may take positive or negative values, or be 0, depending on the position of the body with respect to the plane of the drawing (penetrating behind or in front of the plane of the drawing or the plane of the drawing).

In an advantageous development of the invention, in addition to the body functions f _i (x, y, z), transparency step functions t _i (x, y, z) are given, where t _i (x, y, z) is equal to 1 when the Body f _i (x, y, z) at the location (x, y, z) obscures the background and otherwise equals 0. For viewing direction substantially in the direction of the z-axis, the smallest value for z _ik (x, y, x _m , y _m ) for which t _i (x, y, z _k ) is not equal to zero must be taken to be z Front body of the body to look at f _i from the outside. Alternatively, for z _ik (x, y, x _m , y _m ), the largest value may be taken for which t _i (x, y, z _k ) is non-zero in order to view the body back of body f _i from the inside ,

• einem Motivbild, das in eine Mehrzahl von Zellen eingeteilt ist, in denen jeweils abgebildete Bereiche der vorgegebenen Körper angeordnet sind,
• einem Betrachtungsraster aus einer Mehrzahl von Betrachtungselementen zur Darstellung der vorgegebenen Körper bei der Betrachtung des Motivbilds mithilfe des Betrachtungsrasters,
• wobei das Motivbild mit seiner Einteilung in eine Mehrzahl von Zellen eine Bildfunktion m(x,y) aufweist, die gegeben ist durch $$m\left(x,y\right)=F\left(h_1,h_2,{\dots h}_N\right),$$
  
  mit den beschreibenden Funktionen $${\mathrm{h}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y}\right)={\mathrm{f}}_{\mathrm{i}}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{iK}}\\ {\mathrm{y}}_{\mathrm{iK}}\end{array}\right)\cdot {\mathit{g}}_{\mathrm{i}}\left(\mathit{x},\mathit{y}\right),$$
  
  mit $$\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{iK}}\\ {\mathrm{y}}_{\mathrm{iK}}\end{array}\right)=\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathit{V}}_{\mathit{i}}\left(\mathrm{x},\mathrm{y}\right)\cdot \left(\left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)\right)\mathrm{modW}\right)-{\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)-{\mathrm{w}}_{\mathrm{ci}}\left(\mathrm{x},\mathrm{y}\right)\right)$$
  
  $${\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{d}}_{\mathrm{i}\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{d}}_{\mathrm{i}\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$$
  
  und $${\mathrm{w}}_{\mathrm{ci}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{i}\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{c}}_{\mathrm{i}\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right),$$
  
• wobei F(h ₁,h ₂ ,...h_N ) eine Masterfunktion ist, die eine Verknüpfung der N beschreibenden Funktionen h_i(x,y) angibt, und wobei
• die Einheitszelle des Betrachtungsrasters durch Gitterzellenvektoren ${\mathrm{w}}_1=\left(\begin{array}{c}{w}_{11}\\ w_{21}\end{array}\right)$
  
  und ${\mathrm{w}}_2=\left(\begin{array}{c}{w}_{12}\\ {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{22}\end{array}\right)$
  
  beschrieben und in der Matrix $\mathrm{W}=$
  
  $\left(\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{22}\end{array}\right)$
  
  zusammengefasst ist,
• die Vergrößerungsterme V_i(x,y) entweder Skalare ${\mathrm{V}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y}\right)=\left(\frac{{\mathrm{z}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}-\mathrm{1}\right)$
  
  sind, mit dem effektiven Abstand des Betrachtungsrasters vom Motivbild e, oder Matrizen V_i(x,y) =(A_i(x,y) - I) sind, wobei die Matrizen ${\mathrm{A}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y}\right)=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{i}\mathrm{11}}\left(\mathrm{x},\mathrm{y}\right)& {\mathrm{a}}_{\mathrm{i}\mathrm{12}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{a}}_{\mathrm{i}\mathrm{21}}\left(\mathrm{x},\mathrm{y}\right)& {\mathrm{a}}_{\mathrm{i}\mathrm{22}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$
  
  jeweils ein gewünschtes vergrößerungs- und Bewegungsverhalten des vorgegebenen Körpers f_i beschreiben und I die Einheitsmatrix ist,
• die Vektoren (c_i1(x,y), c_i2(x,y)) mit 0 ≤ c_i1(x,y), c_i2(x,y) < 1 für den Körper f_i jeweils die relative Position des Zentrums der Betrachtungselemente innerhalb der Zellen i des Motivbilds angeben,
• die Vektoren (d_i1(x,y), d_i2(x,y)) mit 0 ≤ d_i1(x,y), d_i2(x,y) < 1 jeweils eine Verschiebung der Zellengrenzen im Motivbild darstellen, und
• g_i(x,y) Maskenfunktionen zur Einstellung der Sichtbarkeit des Körpers f_i sind.

A depiction arrangement according to a fifth aspect of the invention corresponding to the height profile model of the second aspect of the invention contains a raster image arrangement for depicting a plurality of predefined ones Three-dimensional body, which are given by height profiles with two-dimensional representations of the body f _i (x, y), i = 1,2, ... N, with N≥1 and by height functions z _i (x, y), each for each point (x, y) of the given body f _{i contains} a height / depth information, with

• a motif image, which is divided into a plurality of cells, in each of which imaged areas of the predetermined body are arranged,
• a viewing grid of a plurality of viewing elements for displaying the predetermined bodies when viewing the motif image using the viewing grid,
• wherein the motif image with its division into a plurality of cells has an image function m (x, y), which is given by $$m\left(x,y\right)=F\left(H_1,H_2,{...H}_N\right).$$
  
  with the descriptive functions $${\mathrm{H}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y}\right)={\mathrm{f}}_{\mathrm{i}}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{iK}}\\ {\mathrm{y}}_{\mathrm{iK}}\end{array}\right)\cdot {\mathit{G}}_{\mathrm{i}}\left(\mathit{x},\mathit{y}\right).$$
  
  With $$\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{iK}}\\ {\mathrm{y}}_{\mathrm{iK}}\end{array}\right)=\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathit{V}}_{\mathit{i}}\left(\mathrm{x},\mathrm{y}\right)\cdot \left(\left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)\right)\mathrm{ModW}\right)-{\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)-{\mathrm{w}}_{\mathrm{ci}}\left(\mathrm{x},\mathrm{y}\right)\right)$$
  
  $${\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{d}}_{\mathrm{i}\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{d}}_{\mathrm{i}\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$$
  
  and $${\mathrm{w}}_{\mathrm{ci}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{i}\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{c}}_{\mathrm{i}\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right).$$
  
• where F (h ₁ , h ₂ , ... h _N ) is a master function indicating a combination of the N descriptive functions h _i (x, y), and where
• the unit cell of the viewing grid by grid cell vectors ${\mathrm{w}}_1=\left(\begin{array}{c}{w}_{11}\\ w_{21}\end{array}\right)$
  
  and ${\mathrm{w}}_2=\left(\begin{array}{c}{w}_{12}\\ {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{22}\end{array}\right)$
  
  described and in the matrix $\mathrm{W}=$
  
  $\left(\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{22}\end{array}\right)$
  
  is summarized
• the magnification terms V _i (x, y) are either scalars ${\mathrm{V}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y}\right)=\left(\frac{{\mathrm{z}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}-\mathrm{1}\right)$
  
  are, with the effective distance of the viewing grid from the motif image e, or matrices V _i (x, y) = (A _i (x, y) - I), where the matrices ${\mathrm{A}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y}\right)=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{i}\mathrm{11}}\left(\mathrm{x},\mathrm{y}\right)& {\mathrm{a}}_{\mathrm{i}\mathrm{12}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{a}}_{\mathrm{i}\mathrm{21}}\left(\mathrm{x},\mathrm{y}\right)& {\mathrm{a}}_{\mathrm{i}\mathrm{22}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$
  
  each describe a desired enlargement and movement behavior of the given body f _i and I is the unit matrix,
• the vectors (c _i1 (x, y), c _i2 (x, y)) with 0 ≤ c _i1 (x, y), c _i2 (x, y) <1 for the body f _i respectively the relative position of the center indicate the viewing elements within the cells i of the motif image,
• the vectors (d _i1 (x, y), d _i2 (x, y)) with 0 ≦ d _i1 (x, y), d _i2 (x, y) <1 represent respectively a shift of the cell boundaries in the motif image, and
• g _i (x, y) are mask functions for adjusting the visibility of the body f _i .

• einem Motivbild, das in eine Mehrzahl von Zellen eingeteilt ist, in denen jeweils abgebildete Bereiche der vorgegebenen Körper angeordnet sind,
• einem Betrachtungsraster aus einer Mehrzahl von Betrachtungselementen zur Darstellung der vorgegebenen Körper bei der Betrachtung des Motivbilds mithilfe des Betrachtungsrasters,
• wobei das Motivbild mit seiner Einteilung in eine Mehrzahl von Zellen eine Bildfunktion m(x,y) aufweist, die gegeben ist durch $$m\left(x,y\right)=F\left(h_{11},{\mathrm{<figure-callout\; id="12"\; label="h"\; filenames="imgf0001.png"\; state="\{\{state\}\}">h</figure-callout>}}_{12},\dots ,h_{1n_1},h_{21},{\mathrm{<figure-callout\; id="22"\; label="h"\; filenames="imgf0001.png"\; state="\{\{state\}\}">h</figure-callout>}}_{22},\dots ,h_{2n_2},\dots ,h_{N1},h_{N2},\dots ,h_{{\mathit{Nn}}_N}\right),$$
  
  mit den beschreibenden Funktionen $${\mathrm{h}}_{\mathrm{ij}}={\mathrm{f}}_{\mathrm{ij}}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{iK}}\\ {\mathrm{y}}_{\mathrm{iK}}\end{array}\right)\cdot {\mathit{g}}_{\mathrm{ij}}\left(\mathit{x},\mathit{y}\right),$$
  
  mit $$\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{iK}}\\ {\mathrm{y}}_{\mathrm{iK}}\end{array}\right)=\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathrm{V}}_{\mathrm{ij}}\cdot \left(\left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)\right)\mathrm{modW}\right)-{\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)-{\mathrm{w}}_{\mathrm{ci}}\left(\mathrm{x},\mathrm{y}\right)\right)$$
  
  ${\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{d}}_{\mathrm{i}\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{d}}_{\mathrm{i}\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$
  
  und ${\mathrm{w}}_{\mathrm{ci}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{i}\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{c}}_{\mathrm{i}\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right),$
  
  wobei für ij jeweils das Indexpaar zu nehmen ist, für das ${\mathrm{t}}_{\mathrm{ij}}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{iK}}\\ {\mathrm{y}}_{\mathrm{iK}}\end{array}\right)$
  
  ungleich Null und Z_ij minimal bzw. maximal ist, und
• wobei F(h ₁₁,h ₁₂,...,h _{1n 1} ,h ₂₁,h ₂₂,...,h _{2n 2} ,...,h _N1,h _N2,...h_NnN ) eine Masterfunktion ist, die eine Verknüpfung der beschreibenden Funktionen h_ij(x,y) angibt, und wobei
• die Einheitszelle des Betrachtungsrasters durch Gitterzellenvektoren ${\mathrm{w}}_1=\left(\begin{array}{c}{w}_{11}\\ w_{21}\end{array}\right)$
  
  und ${\mathrm{w}}_2=\left(\begin{array}{c}{w}_{12}\\ {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{22}\end{array}\right)$
  
  beschrieben und in der Matrix $\mathrm{W}=$
  
  $\left(\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{22}\end{array}\right)$
  
  zusammengefasst ist,
• die Vergrößerungsterme V_ij entweder Skalare ${\mathrm{V}}_{\mathrm{ij}}=\left(\frac{{\mathrm{z}}_{\mathrm{ij}}}{\mathrm{e}}-\mathrm{1}\right)$
  
  -1 sind, mit dem effektiven Abstand des Betrachtungsrasters vom Motivbild e, oder Matrizen V_ij =(A_ij - I) sind, wobei die Matrizen ${\mathrm{A}}_{\mathrm{ij}}=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{ij}\mathrm{11}}& {\mathrm{a}}_{\mathrm{ij}\mathrm{12}}\\ {\mathrm{a}}_{\mathrm{ij}\mathrm{21}}& {\mathrm{a}}_{\mathrm{<figure-callout\; id="22"\; label="ij"\; filenames="imgf0001.png"\; state="\{\{state\}\}">ij</figure-callout>}\mathrm{22}}\end{array}\right)$
  
  jeweils ein gewünschtes Vergrößerungs- und Bewegungsverhalten des vorgegebenen Körpers f_i beschreiben und I die Einheitsmatrix ist,
• die Vektoren (c_i1(x,y), c_i2(x,y)) mit 0 ≤c_i1(x,y),c_i2 (x,y) < 1 für den Körper f_i jeweils die relative Position des Zentrums der Betrachtungselemente innerhalb der Zellen i des Motivbilds angeben,
• die Vektoren (d_i1(x,y), d_i2(x,y)) mit 0 ≤ d_i1(x,y), d_i2(x,y) < 1 jeweils eine Verschiebung der Zellengrenzen im Motivbild darstellen, und
• g_ij(x,y) Maskenfunktionen zur Einstellung der Sichtbarkeit des Körpers f_i sind.

A display arrangement according to a sixth aspect of the invention corresponding to the sectional plane model of the third aspect of the invention includes a raster image arrangement for displaying a plurality (N≥1) of predetermined ones three-dimensional body, each by n _i sections f _ij (x, y) and n _i transparency step functions t _ij (x, y) with i = 1,2, ... N and j = 1,2, ... n _i , where the intersections of the body i are each at a depth z _ij when viewed at eye relief in the x-direction, and f _ij (x, y) is the image function of the j-th intersection of the i-th body and the transparency step function t _ij (x, y) is equal to 1, if the section j of the body i at the point (x, y) obscures objects behind it and otherwise equals 0, with

• a motif image, which is divided into a plurality of cells, in each of which imaged areas of the predetermined body are arranged,
• a viewing grid of a plurality of viewing elements for displaying the predetermined bodies when viewing the motif image using the viewing grid,
• wherein the motif image with its division into a plurality of cells has an image function m (x, y), which is given by $$m\left(x,y\right)=F\left(H_{11}.{\mathrm{<figure-callout\; id="12"\; label="H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">H</figure-callout>}}_{12}.....H_{1n_1}.H_{21}.{\mathrm{<figure-callout\; id="22"\; label="H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">H</figure-callout>}}_{22}.....H_{2n_2}.....H_{N1}.H_{N2}.....H_{{\mathit{nn}}_N}\right).$$
  
  with the descriptive functions $${\mathrm{H}}_{\mathrm{ij}}={\mathrm{f}}_{\mathrm{ij}}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{iK}}\\ {\mathrm{y}}_{\mathrm{iK}}\end{array}\right)\cdot {\mathit{G}}_{\mathrm{ij}}\left(\mathit{x},\mathit{y}\right).$$
  
  With $$\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{iK}}\\ {\mathrm{y}}_{\mathrm{iK}}\end{array}\right)=\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathrm{V}}_{\mathrm{ij}}\cdot \left(\left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)\right)\mathrm{ModW}\right)-{\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)-{\mathrm{w}}_{\mathrm{ci}}\left(\mathrm{x},\mathrm{y}\right)\right)$$
  
  ${\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{d}}_{\mathrm{i}\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{d}}_{\mathrm{i}\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$
  
  and ${\mathrm{w}}_{\mathrm{ci}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{i}\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{c}}_{\mathrm{i}\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right).$
  
  where for ij in each case the index pair is to be taken, for the ${\mathrm{t}}_{\mathrm{ij}}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{iK}}\\ {\mathrm{y}}_{\mathrm{iK}}\end{array}\right)$
  
  equal to zero and Z _ij is minimum or maximum, and
• where F (h _11, h _12,. .., h _{1 n 1} , H _21, h _22. , , , h _{2 n 2} , . , , , h _{N 1} , h _{N 2} , ... h _{Nn N} ) is a master function indicating a concatenation of the descriptive functions h _ij (x, y), and wherein
• the unit cell of the viewing grid by grid cell vectors ${\mathrm{w}}_1=\left(\begin{array}{c}{w}_{11}\\ w_{21}\end{array}\right)$
  
  and ${\mathrm{w}}_2=\left(\begin{array}{c}{w}_{12}\\ {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{22}\end{array}\right)$
  
  described and in the matrix $\mathrm{W}=$
  
  $\left(\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{22}\end{array}\right)$
  
  is summarized
• the magnification terms V _{ij are} either scalars ${\mathrm{V}}_{\mathrm{ij}}=\left(\frac{{\mathrm{z}}_{\mathrm{ij}}}{\mathrm{e}}-\mathrm{1}\right)$
  
  Are -1, with the effective distance of the viewing grid from the motif image e, or matrices V _ij = (A _ij - I), where the matrices ${\mathrm{A}}_{\mathrm{ij}}=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{ij}\mathrm{11}}& {\mathrm{a}}_{\mathrm{ij}\mathrm{12}}\\ {\mathrm{a}}_{\mathrm{ij}\mathrm{21}}& {\mathrm{a}}_{\mathrm{<figure-callout\; id="22"\; label="ij"\; filenames="imgf0001.png"\; state="\{\{state\}\}">ij</figure-callout>}\mathrm{22}}\end{array}\right)$
  
  each describe a desired magnification and movement behavior of the given body f _i and I is the unit matrix,
• the vectors (c _i1 (x, y), c _i2 (x, y)) with 0 ≤c _i1 (x, y), c _i2 (x, y) <1 for the body f _i, respectively, the relative position of the center indicate the viewing elements within the cells i of the motif image,
• the vectors (d _i1 (x, y), d _i2 (x, y)) with 0 ≦ d _i1 (x, y), d _i2 (x, y) <1 represent respectively a shift of the cell boundaries in the motif image, and
• g _ij (x, y) are mask functions for adjusting the visibility of the body f _i.

All versions f made during the first three aspects of the invention for single body also apply to the plurality of bodies f _i of the general raster image arrays of the fourth to sixth aspect of the invention. In particular, at least one (or even all) of the descriptive functions of the fourth, fifth or sixth aspect of the invention may be designed as indicated above for the image function m (x, y) of the first, second or third aspect of the invention.

Advantageously, the raster image arrangement represents a swap image, a motion image or a morph image. The mask functions g _i or g _ij can in particular define a strip-like or checkerboard-like change of the visibility of the body f _i . An image sequence can advantageously take place when tilting along a predetermined direction; in this case expediently strip-like mask functions g _i and g _ij used, so the mask features that are, for each i, only in a traveling within the unit cell strips equal to zero. In the general case, however, it is also possible to select mask functions which allow a sequence of images to take place by means of curved, meandering or spiral tilting movements.

While in alternating images (tilt images) or other motion pictures ideally only one three-dimensional image is simultaneously visible, the invention also includes designs in which for the viewer two or more three-dimensional images (body) f _i are simultaneously visible. The master function F advantageously represents the sum function, the maximum function, an OR operation, an XOR operation or another logical operation.

The motif image is present in particular in an embossed or printed layer. According to an advantageous development of the invention, the security element has in all aspects an opaque cover layer for covering the raster image arrangement by area. Thus, no modulo magnification effect occurs within the covered area, so that the optically variable effect can be combined with conventional information or with other effects. This cover layer is advantageously in the form of patterns, characters or codes before and / or has recesses in the form of patterns, characters or codes.

If the motif image and the viewing grid are arranged on opposite surfaces of an optical spacer layer, the spacer layer may comprise, for example, a plastic film and / or a lacquer layer.

The permanent security element itself in all aspects of the invention preferably represents a security thread, a tear-open thread, a security strip, a security strip, a patch or a label for application to a security paper, value document or the like. In an advantageous embodiment, the security element can form a transparent or recessed area Span the disk. Different appearances can be realized on different sides of the data carrier. Also two-sided designs come into question, in which both sides of a motif image viewing grid are arranged.

The raster image arrangements according to the invention can be combined with other security features, for example with diffractive structures, with hologram structures in all variants, metallized or non-metallized, with sub-wavelength structures, metallized or non-metallized, with subwavelength gratings, with layer systems which show a color change on tilting, semitransparent or opaque , with diffractive optical elements, with refractive optical elements, such as prismatic beam formers, with special hole shapes, with safety features with specifically set electrical conductivity, with incorporated materials with magnetic coding, with substances with phosphorescent, fluorescent or luminescent effect, with safety features based on liquid crystals , with matt structures, with micromirrors, with elements with louvre effect or with sawtooth structures. Further security features with which the raster image arrangements according to the invention can be combined are disclosed in the document WO 2005/052650 A2 stated on pages 71 to 73; These are included in the present description.

In all aspects of the invention, the image contents of individual cells of the motif image can be interchanged with one another after the determination of the image function m (x, y).

The invention also includes methods of making the display assemblies of the first to sixth aspects of the invention wherein a motif image is calculated from one or more predetermined three-dimensional bodies. The procedure and the required mathematical relationships for the general perspective, the height profile model and the sectional plane model have already been indicated above and are also explained in more detail by the following exemplary embodiments.

The size of the motif picture elements and the viewing elements is within the scope of the invention typically about 5 to 50 microns, so that the influence of the modulo magnification arrangement on the thickness of the security elements can be kept low. The production of such small lens arrays and such small images is for example in the document DE 10 2005 028162 A1 described, the disclosure of which is included in the present application in this respect.

A typical procedure is the following: For the production of microstructures (microlenses, micromirrors, microimage elements), techniques of semiconductor structuring can be used, for example photolithography or electron beam lithography. A particularly suitable method is to expose the structures in photoresist by means of a focused laser beam. Subsequently, the structures, which may have binary or more complex three-dimensional cross-sectional profiles, are exposed with a developer. As an alternative method laser ablation can be used.

The original obtained in one of these ways can be further processed into a stamping tool, with the help of the structures, for example, by embossing in UV varnish, thermoplastic embossing or by the in the document WO 2008/00350 A1 microtip technique described can be duplicated. The latter technique is a micro-gravure technique that combines the benefits of printing and embossing technologies. Details of this micro-gravure printing process and the associated advantages of the document WO 2008/00350 A1 are removed, the disclosure content of which is included in the present application in this respect.

For the final product, there are a number of different variants in question: metal-coated embossed structures, coloring by metallic nanostructures, embossing in colored UV lacquer, micro gravure printing according to the publication WO 2008/00350 A1 , the coloring of the embossed structures and subsequent doctoring of the embossed film, or else in the German patent application 10 2007 062 089.8 described method for selectively transferring an imprint material on elevations or depressions of an embossed structure. Alternatively, the subject image may be written directly into a photosensitive layer with a focused laser beam.

The microlens array can also be fabricated by laser ablation or grayscale lithography. Alternatively, a binary exposure can take place, wherein the lens shape is formed only later by melting of photoresist ("thermal reflow"). From the original, as in the case of the microstructure array, an embossing tool can be produced with the aid of which mass production can take place, for example by embossing in UV lacquer or thermoplastic embossing.

If the modulo-magnifier principle or modulo-mapping principle is used for decorative articles (eg greeting cards, pictures as wall decorations, curtains, table supports, key chains, etc.) or the decoration of products, the size of the images and lenses to be introduced is about 50 to 1000 microns. In this case, the motif images to be introduced can be printed in color using conventional printing methods, such as offset printing, gravure printing, letterpress printing, screen printing, or digital printing methods, such as inkjet printing or laser printing.

The modulo-magnifier principle or modulo-mapping principle according to the invention can also be used for three-dimensional computer and television images which are generally shown on an electronic display device. The size of the images to be introduced and the size of the lenses in the lens array to be mounted in front of the screen in this case is about 50 to 500 microns. The screen resolution should be at least an order of magnitude better, so that high-resolution screens are required for this application.

Finally, the invention also includes a security paper for the production of security or value documents, such as banknotes, checks, identity cards, documents or the like, with a representation arrangement of the type described above. The invention further includes a data carrier, in particular a branded article, a value document, a decorative article, such as a package, postcards or the like with a representation arrangement of the type described above. The viewing grid and / or the motif image of the presentation arrangement can be arranged over the entire surface, on partial surfaces or in a window region of the data carrier.

The invention also relates to an electronic display device having an electronic display device, in particular a computer or television screen, a control device and a display device of the type described above. The control device is designed and configured to display the motif image of the display device on the electronic display device. The viewing grid for viewing the displayed motif image can be connected to the electronic display device or can be a separate viewing grid, which can be brought onto or in front of the electronic display device for viewing the displayed motif image.

All variants described can be carried out with two-dimensional lens grids in grating arrangements of any lower or higher symmetry or in cylindrical lens arrangements. All arrangements can also be calculated for curved surfaces, as basically in the document WO 2007/076952 A2 described, the disclosure of which is included in the present application in this respect.

Further embodiments and advantages of the invention are explained below with reference to the figures. For better clarity, a scale and proportioned representation is omitted in the figures.

Fig.1

eine schematische Darstellung einer Banknote mit einem eingebetteten Sicherheitsfaden und einem aufgeklebten Transferelement,

Fig. 2

schematisch den Schichtaufbau eines erfindungsgemäßen Sicherheitselements im Querschnitt,

Fig. 3

schematisch eine Seitenansicht eines darzustellenden Körpers im Raum, der perspektivisch in einer Motivbildebene dargestellt werden soll, und

Fig. 4

für das Höhenprofil-Modell in (a) eine zweidimensionale Darstellung f(x,y) eines darzustellenden Würfels in Zentralprojektion, in (b) die zugehörige Höhen/Tiefeninformation z(x,y) in Graucodierung und in (c) die mithilfe dieser Vorgaben berechnete Bildfunktion m(x,y).

Show it:

Fig.1

a schematic representation of a banknote with an embedded security thread and a glued transfer element,

Fig. 2

schematically the layer structure of a security element according to the invention in cross-section,

Fig. 3

schematically a side view of a body to be displayed in space, which is to be shown in perspective in a scene image plane, and

Fig. 4

for the height profile model in (a) a two-dimensional representation f (x, y) of a cube to be displayed in central projection, in (b) the associated height / depth information z (x, y) in Gray coding and in (c) the image function m (x, y) calculated using these constraints.

The invention will now be explained using the example of security elements for banknotes. Fig. 1 shows a schematic representation of a banknote 10, which is provided with two security elements 12 and 16 according to embodiments of the invention. The first security element represents a security thread 12 that emerges in certain window areas 14 on the surface of the banknote 10, while it is embedded in the intervening areas inside the banknote 10. The second security element is formed by a glued transfer element 16 of any shape. The security element 16 can also be designed in the form of a cover film, which is arranged over a window area or a through opening of the banknote. The security element may be designed for viewing in supervision, review or viewing both in supervision and in review.

Both the security thread 12 and the transfer element 16 may include a modulo magnification arrangement according to an embodiment of the invention. The mode of operation and the production method according to the invention for such arrangements will be described in more detail below with reference to the transfer element 16.

Fig. 2 schematically shows the layer structure of the transfer element 16 in cross section, wherein only the parts of the layer structure required for the explanation of the principle of operation are shown. The transfer element 16 includes a carrier 20 in the form of a transparent plastic film, in the embodiment of an approximately 20 micron thick polyethylene terephthalate (PET) film.

The upper side of the carrier film 20 is provided with a grid-like arrangement of microlenses 22 which form on the surface of the carrier film a two-dimensional Bravais grid with a preselected symmetry. The Bravais grating, for example, have a hexagonal lattice symmetry. However, other, in particular lower symmetries and thus more general forms, such as the symmetry of a parallelogram grating, are also possible.

The spacing of adjacent microlenses 22 is preferably chosen as small as possible in order to ensure the highest possible area coverage and thus a high-contrast representation. The spherically or aspherically configured microlenses 22 preferably have a diameter between 5 μm and 50 μm and in particular a diameter between only 10 μm and 35 μm and are therefore not visible to the naked eye. It is understood that in other designs, larger or smaller dimensions come into question. For example, for modulo magnification arrangements, the microlenses may have a diameter between 50 μm and 5 mm for decoration purposes, while dimensions below 5 μm may also be used in modulo magnification arrangements which are intended to be decipherable only with a magnifying glass or a microscope.

On the underside of the carrier film 20, a motif layer 26 is arranged, which contains a divided into a plurality of cells 24 motif image with motif picture elements 28.

The optical thickness of the carrier film 20 and the focal length of the microlenses 22 are coordinated so that the motif layer 26 is located approximately at a distance of the lens focal length. The carrier film 20 thus forms a optical spacer layer, which ensures a desired, constant distance of the microlenses 22 and the motif layer 26 with the motif image.

To explain the operation of the modulo magnification arrangements according to the invention Fig. 3 very schematically a side view of a body 30 in space, the perspective in the scene image plane 32, which is also referred to below as drawing plane to be displayed.

The body 30 is generally described by a body function f (x, y, z) and a transparency step function t (x, y, z), where the z-axis is perpendicular to the plane of the drawing spanned by the x and y axes 32 stands. The body function f (x, y, z) indicates a characteristic property of the body at the position (x, y, z), for example a brightness distribution, a color distribution, a binary distribution or other body properties, such as transparency, reflectivity, density or the like , In general, therefore, it can represent not only a scalar function but also a vector-valued function of the location coordinates x, y and z. The transparency step function t (x, y, z) is equal to 1 if the body conceals the background at the location (x, y, z) and is otherwise, ie in particular if the body is at the location (x, y, z). z) is transparent or absent, equal to 0.

It is understood that the three-dimensional image to be displayed may comprise not only a single object but also a plurality of three-dimensional objects which need not necessarily be related. The term "body" used in this description is used in the sense of any three-dimensional structure and includes structures having one or more separate three-dimensional objects.

The arrangement of the microlenses in the lens plane 34 is described by a two-dimensional Bravais grating whose unit cell is indicated by vectors w ₁ and w ₂ (with the components w ₁₁ , w ₂₁ , and w ₁₂ , w ₂₂ , respectively). In compact notation, the unit cell may be indicated in matrix form by a lenticular array W: $$W=\left(w_1,w_2\right)=\left(\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& {\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{22}\end{array}\right)$$

The lenticular matrix W is often referred to simply as a lens matrix or lenticular array hereinafter. Instead of the term lens plane, the term pupil plane is also used below. The positions x _m , y _m in the pupil plane designated below as pupil positions represent the grid points of the W grid in the lens plane 34.

In the lens plane 34, instead of lenses 22, it is also possible, for example, to use pinholes on the principle of the pinhole camera.

Also all other types of lenses and imaging systems, such as aspheric lenses, cylindrical lenses, slit diaphragms, apertured apertured or slit diaphragms, Fresnel lenses, GRIN (Gradient Refraction Index) lenses, zoned diffraction lenses, holographic lenses, concave mirrors, Fresnel mirrors, zone mirrors and other elements with focussing or also fading effect, can be used as viewing elements in the viewing grid.

Basically, in addition to elements with focussing effect elements with ausblendender effect (hole or slit, even mirror surfaces behind hole or slit) are used as viewing elements in the viewing grid.

When using a concave mirror array and in other inventively used specular viewing grids, the observer looks through the partially transparent in this case motif image on the underlying mirror array and sees the individual small mirror as light or dark points, from which builds the image to be displayed. The motif image is generally so finely structured that it can only be seen as a veil. The formulas described for the relationships between the image to be displayed and the motif image apply, even if this is not mentioned in detail, not only for lenticular, but also for mirror grid. It is understood that in the inventive use of concave mirrors in place of the lens focal length, the mirror focal length occurs.

In the inventive application of a mirror array instead of a lens array is in Fig. 2 to think of the viewing direction from below, and in Fig. 3 In the mirror array arrangement, the levels 32 and 34 are interchanged. The description of the invention is based on lens grids, which are representative of all other viewing grids used in the invention.

With respect to again Fig. 3 e denotes the lens focal length (in general, the effective distance e takes into account the lens data and the refractive index of the medium between the lens grid and the motif grid). A point (x _k, y _k, z _k) of the body 30 located in the room is in the plane 32 to the pupil position (x _m, y _m, 0) shown in perspective.

Of the body to be sampled value f (x _k, y _k, z _k (x, y, x _m, y _m)) is applied to the location (x, y, e) in the plane 32 registered, where (x _k, y _k , z _k (x, y, x _m , y _m )) the common point of the body 30 with the characteristic function t (x, y, z) and view line [(xm, y _m , 0), (x, y, e)] with the smallest z value.

In this case, a possible sign of z is taken into account, so that not the point with the smallest z-value, but the point with the most negative z-value is selected.

wobei für z_k(x,y,x_m,y_m) der kleinste Wert zu nehmen ist, für den t(x,y,z_k) ungleich 0 ist.If one first considers only one body standing in space without any movement effects when the magnification arrangement is tilted, the motif image in the motif plane 32, which generates a representation of the desired body when viewed through the lenticular grid W arranged in the lens plane 34, is represented by an image function m (x , y), which according to the invention is given by: $$\mathrm{f}\left(\begin{array}{l}\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\left(\frac{{\mathrm{z}}_{\mathrm{K}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)}{\mathrm{e}}-\mathrm{1}\right)\cdot \left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)\mathrm{ModW}\right)-\mathrm{w}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{1}}\\ {\mathrm{c}}_{\mathrm{2}}\end{array}\right)\right)\\ {\mathrm{z}}_{\mathrm{K}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)\end{array}\right)=\mathrm{f}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{K}}\\ {\mathrm{y}}_{\mathrm{K}}\\ {\mathrm{z}}_{\mathrm{K}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)\end{array}\right)\mathrm{,}$$

where z _k (x, y, x _m , y _m ) is the smallest value for which t (x, y, z _k ) is not equal to 0.

The vector (c ₁ , c ₂ ), which in the general case can be location-dependent, that is by (c ₁ (x, y), c ₂ (x, y)) with 0 ≤ c ₁ (x, y), c ₂ (x, y) <1, indicates the relative position of the center of the viewing elements within the cells of the motif image.

The computation of z _k (x, y, x _m , y _m ) is generally very expensive, since in the lenticular image 10 000 to 1 000 000 and more positions (x _m , y _m ) are taken into account. Further below, therefore, some methods are shown in which z _K becomes independent of (x _m , y _m ) (height profile model) or even independent of (x, y, x _m , y _m ) becomes (sectional plane model).

First, however, a generalization to the above formula is presented, in which not only standing in space body are shown, but in which the body appearing in the lenticular device changes in depth when changing the viewing direction. For this purpose, instead of the scalar magnification v = z (x, y, x _m , y _m ) / e, an enlargement and motion matrix A (x, y, x _m , y _m ) is used in which the term v = z (x , y, x _m , y _m ) / e is included.

For the image function m (x, y) then results $$\mathrm{f}\left(\begin{array}{l}\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\left(\mathrm{A}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)-\mathrm{I}\right)\cdot \left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)\mathrm{ModW}\right)-\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{1}}\\ {\mathrm{c}}_{\mathrm{2}}\end{array}\right)\right)\\ {\mathrm{z}}_{\mathrm{K}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)\end{array}\right)=\mathrm{f}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{K}}\\ {\mathrm{y}}_{\mathrm{K}}\\ {\mathrm{z}}_{\mathrm{K}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)\end{array}\right)$$

stellt die Rasterbildanordnung den vorgegebenen Körper bei Betrachtung des Motivbilds mit Augenabstand in x-Richtung dar. Soll die Rasterbildanordnung den vorgegebenen Körper bei Betrachtung des Motivbilds mit Augenabstand in Richtung ψ zur x-Achse darstellen, so werden die Koeffizienten von A so gewählt, dass $$\left({\mathrm{a}}_{\mathrm{11}}{\cos }^{\mathrm{2}}\mathrm{\psi }+\left({\mathrm{a}}_{\mathrm{12}}+{\mathrm{a}}_{\mathrm{21}}\right)\mathrm{cos\psi \; sin\psi }+{\mathrm{a}}_{\mathrm{22}}{\sin }^{\mathrm{2}}\mathrm{\psi }\right)={\mathrm{z}}_{\mathrm{K}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)/\mathrm{e}$$

erfüllt ist.With $${\mathrm{a}}_{\mathrm{11}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)={\mathrm{z}}_{\mathrm{K}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)/\mathrm{e}$$

If the raster image arrangement is to represent the given body when viewing the motif image with the eye distance in the direction ψ to the x-axis, then the coefficients of A are chosen such that $$\left({\mathrm{a}}_{\mathrm{11}}{\cos }^{\mathrm{2}}\mathrm{\psi }+\left({\mathrm{a}}_{\mathrm{12}}+{\mathrm{a}}_{\mathrm{21}}\right)\mathrm{cos\psi \; sin\psi }+{\mathrm{a}}_{\mathrm{22}}{\sin }^{\mathrm{2}}\mathrm{\psi }\right)={\mathrm{z}}_{\mathrm{K}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)/\mathrm{e}$$

is satisfied.

Height profile model

In order to simplify the calculation of the motif image, the height profile is based on a two-dimensional drawing f (x, y) of a body, with an additional z-coordinate z (x, y) given for each point x, y of the two-dimensional image of the body How far is this point in the real body away from the drawing plane 32? z (x, y) can assume both positive and negative values.

For illustration shows Fig. 4 (a) a two-dimensional representation 40 of a cube in central projection, wherein at each pixel (x, y) a gray value f (x, y) is given. Of course, instead of a central projection, it is also possible to use a parallel projection or another projection method that is particularly easy to produce. The two-dimensional representation f (x, y) can also be a fantasy image; what is important is that each pixel has a height in addition to the gray (or more generally color, transparency, reflectivity, density, etc.) information - / depth information z (x, y) is assigned. Such height representation 42 is in Fig. 4 (b) shown schematically in gray coding, with the front lying pixels of the cube white, further behind pixels gray or black are shown.

In the case of a pure magnification, the information of f (x, y) and z (x, y) results for the image function $$\mathrm{m}\left(\mathrm{x},\mathrm{y}\right)=\mathit{f}\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\left(\frac{\mathrm{z}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}-\mathrm{1}\right)\cdot \left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)\mathrm{mod\; W}\right)-\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{1}}\\ {\mathrm{c}}_{\mathrm{2}}\end{array}\right)\right)\right)\mathrm{,}$$

die Darstellung eines hinter der Zeichenebene dreidimensional erscheinenden Würfels erzeugt. Fig. 4 (c) shows the image function m (x, y) of the motif image 44 calculated in this way, and the matching scaling when viewed with a lenticular grid $$\mathrm{W}=\left(\begin{array}{cc}\mathrm{2}\mathrm{<figure-callout\; id="0"\; label="mm"\; filenames="imgf0002.png"\; state="\{\{state\}\}">mm</figure-callout>}& \mathrm{0}\\ \mathrm{0}& \mathrm{2}\mathrm{mm}\end{array}\right)$$

the representation of a behind the plane of three-dimensional appearing cube generated.

wobei die Vergrößerungs- und Bewegungsmatrix A(x,y) im allgemeinen Fall durch $$\mathrm{A}\left(\mathrm{x},\mathrm{y}\right)=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{11}}\left(\mathrm{x},\mathrm{y}\right)& {\mathrm{a}}_{\mathrm{12}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{a}}_{\mathrm{21}}\left(\mathrm{x},\mathrm{y}\right)& {\mathrm{a}}_{\mathrm{22}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)=\left(\begin{array}{cc}\frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}& \frac{{\mathrm{z}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\cdot {\cot \mathit{\phi }}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\\ \frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\cdot {\tan \mathit{\phi }}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)& \frac{{\mathrm{z}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\end{array}\right)$$

gegeben ist. Zur Illustration seien einige Spezialfälle behandelt:If not only bodies in space are to be displayed, but the bodies appearing in the lenticular apparatus change in depth when the viewing direction changes, an enlargement and movement matrix A (instead of the magnification v = z (x, y) / e) x, y): $$\mathrm{m}\left(\mathrm{x},\mathrm{y}\right)=\mathit{f}\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\left(\mathrm{A}\left(\mathrm{x},\mathrm{y}\right)-\mathrm{I}\right)\cdot \left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)\mathrm{mod\; W}\right)-\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{1}}\\ {\mathrm{c}}_{\mathrm{2}}\end{array}\right)\right)\right).$$

wherein the magnification and motion matrix A (x, y) in the general case by $$\mathrm{A}\left(\mathrm{x},\mathrm{y}\right)=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{11}}\left(\mathrm{x},\mathrm{y}\right)& {\mathrm{a}}_{\mathrm{12}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{a}}_{\mathrm{21}}\left(\mathrm{x},\mathrm{y}\right)& {\mathrm{a}}_{\mathrm{22}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)=\left(\begin{array}{cc}\frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}& \frac{{\mathrm{z}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\cdot {\cot \mathit{\phi }}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\\ \frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\cdot {\tan \mathit{\phi }}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)& \frac{{\mathrm{z}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\end{array}\right)$$

given is. For illustration, consider some special cases:

Example 1:

erhält. Beim Drehen der Anordnung bei der Betrachtung gehen die Höhenfunktionen z₁(x,y) und z₂(x,y) des dargestellten Körpers ineinander über.Two height functions z ₁ (x, y) and z ₂ (x, y) are given, so that the magnification and motion matrix A (x, y) is the shape $$\mathrm{A}\left(\mathrm{x},\mathrm{y}\right)=\left(\begin{array}{cc}\frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}& \mathrm{0}\\ \mathrm{0}& \frac{{\mathrm{z}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\end{array}\right)$$

receives. When rotating the arrangement as viewed, the height functions z ₁ (x, y) and z ₂ (x, y) of the displayed body merge into one another.

Example 2:

erhält. Beim Drehen der Anordnung bei der Betrachtung gehen die Höhenfunktionen des dargestellten Körpers ineinander über. Die beiden Winkel φ₁ und φ₂ haben folgende Bedeutung:Two height functions z ₁ (x, y) and z ₂ (x, y) and two angles φ ₁ and φ ₂ are specified so that the magnification and motion matrix A (x, y) is the shape $$\mathrm{A}\left(\mathrm{x},\mathrm{y}\right)=\left(\begin{array}{cc}\frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}& \frac{{\mathrm{z}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\cdot {\cot \mathit{\phi }}_{\mathrm{2}}\\ \frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\cdot {\tan \mathit{\phi }}_{\mathrm{1}}& \frac{{\mathrm{z}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\end{array}\right)$$

receives. When turning the arrangement in the viewing, the height functions of the body shown merge into each other. The two angles φ ₁ and φ ₂ have the following meaning:

In normal viewing (eye distance direction in x-direction) one sees the body in the height relief z ₁ (x, y) and when tilting the arrangement in the x-direction the body moves in the direction φ ₁ to the x-axis.

When rotated by 90 ° viewing (eye distance direction in y-direction) you can see the body in height relief z ₂ (x, y) and when tilting the arrangement in the y-direction, the body moves in the direction of φ ₂ to the x-axis.

Example 3:

erhält. Bei normaler Betrachtung (Augenabstandsrichtung in x-Richtung) und Kippen der Anordnung in x-Richtung bewegt sich der Körper in Richtung φ₁ zur x-Achse. Beim Kippen in y-Richtung erfolgt keine Bewegung.An altitude function z (x, y) and an angle φ _{1 are} specified so that the magnification and motion matrix A (x, y) is the shape $$\mathrm{A}\left(\mathrm{x},\mathrm{y}\right)=\left(\begin{array}{cc}\frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}& \mathrm{0}\\ \frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\cdot {\tan \mathit{\phi }}_{\mathrm{1}}& \mathrm{1}\end{array}\right)$$

receives. During normal viewing (eye distance direction in the x-direction) and tilting of the arrangement in the x-direction, the body moves in the direction φ ₁ to the x-axis. When tilting in the y direction, there is no movement.

mit dem Spalt- bzw. Zylinderachsenabstand d gegeben ist, oder mit einem Lochblenden- bzw. Linsenarray mit $\mathrm{W}=\left(\begin{array}{cc}\mathrm{d}& \mathrm{0}\\ \mathrm{d}\cdot \tan \mathit{\beta }& {\mathrm{d}}_{\mathrm{2}}\end{array}\right)$

mit d₂, β beliebig.In this embodiment, the consideration is also possible with a suitable cylindrical lens grid, for example with a split screen or cylindrical lens grid, the unit cell through $$\mathrm{W}=\left(\begin{array}{cc}\mathrm{<figure-callout\; id="0"\; label="d"\; filenames="imgf0002.png"\; state="\{\{state\}\}">d</figure-callout>}& \mathrm{0}\\ \mathrm{0}& \mathrm{\infty }\end{array}\right)$$

given with the gap or cylinder axis distance d, or with a pinhole or lens array with $\mathrm{W}=\left(\begin{array}{cc}\mathrm{d}& \mathrm{0}\\ \mathrm{d}\cdot \tan \mathit{\beta }& {\mathrm{d}}_{\mathrm{2}}\end{array}\right)$

with d ₂ , β arbitrary.

ist die passende Matrix A, bei der in Richtung γ keine Vergrößerung oder Verzerrung vorliegt: $$\mathrm{W}=\left(\begin{array}{cc}\mathrm{cos\; \gamma }& -\mathrm{sin\; \gamma }\\ \mathrm{sin\; \gamma }& \mathrm{cos\; \gamma }\end{array}\right)\cdot \left(\begin{array}{cc}\frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}& \mathrm{0}\\ \frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\cdot {\tan \mathit{\phi }}_{\mathrm{1}}& \mathrm{1}\end{array}\right)\cdot \left(\begin{array}{cc}\mathrm{cos\; \gamma }& \mathrm{sin\; \gamma }\\ -\mathrm{sin\; \gamma }& \mathrm{cos\; \gamma }\end{array}\right)\mathrm{.}$$

In a cylindrical lens axis in any direction γ and with axis distance d, so a lenticular grid $$\mathrm{W}=\left(\begin{array}{cc}\mathrm{cos\; \gamma }& -\mathrm{sin\; \gamma }\\ \mathrm{sin\; \gamma }& \mathrm{cos\; \gamma }\end{array}\right)\cdot \left(\begin{array}{cc}\mathrm{<figure-callout\; id="0"\; label="d"\; filenames="imgf0002.png"\; state="\{\{state\}\}">d</figure-callout>}& \mathrm{0}\\ \mathrm{0}& \mathrm{\infty }\end{array}\right)$$

is the appropriate matrix A with no magnification or distortion in the direction of γ: $$\mathrm{W}=\left(\begin{array}{cc}\mathrm{cos\; \gamma }& -\mathrm{sin\; \gamma }\\ \mathrm{sin\; \gamma }& \mathrm{cos\; \gamma }\end{array}\right)\cdot \left(\begin{array}{cc}\frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}& \mathrm{0}\\ \frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\cdot {\tan \mathit{\phi }}_{\mathrm{1}}& \mathrm{1}\end{array}\right)\cdot \left(\begin{array}{cc}\mathrm{cos\; \gamma }& \mathrm{sin\; \gamma }\\ -\mathrm{sin\; \gamma }& \mathrm{cos\; \gamma }\end{array}\right)\mathrm{,}$$

wobei d₂, β beliebig sein können.The pattern thus created for the print or embossed image to be created behind a lenticular grid W can be viewed not only with the slit-diaphragm or cylindrical lens array with axis in the direction γ, but also with a pinhole or lens array $\mathrm{W}=\left(\begin{array}{cc}\mathrm{cos\gamma }& -\mathrm{sin\gamma }\\ \mathrm{sin\gamma }& \mathrm{cos\gamma }\end{array}\right)\cdot \left(\begin{array}{cc}\mathrm{d}& \mathrm{0}\\ \mathrm{d}\cdot \tan \mathit{\beta }& {\mathrm{d}}_{\mathrm{2}}\end{array}\right).$

where d ₂ , β can be arbitrary.

Example 4:

wenn φ₂ = 0 erhält. Beim Drehen der Anordnung bei der Betrachtung gehen die Höhenfunktionen des dargestellten Körpers ineinander über.Two height functions z ₁ (x, y) and z ₂ (x, y) and an angle φ _{2 are} specified so that the magnification and motion matrix A (x, y) is the shape $$\mathrm{A}\left(\mathrm{x},\mathrm{y}\right)=\left(\begin{array}{cc}0& \frac{{\mathrm{z}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\cdot {\cot \mathit{\phi }}_{\mathrm{2}}\\ \frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}& \frac{{\mathrm{z}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\end{array}\right).A\left(\mathrm{x},\mathrm{y}\right)=\left(\begin{array}{cc}0& \frac{{\mathrm{z}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\\ \frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}& 0\end{array}\right)$$

if φ ₂ = 0 is obtained. When turning the arrangement in the viewing, the height functions of the body shown merge into each other.

Furthermore, the arrangement has an orthoparallactic 3D effect, wherein the body is in normal viewing (eye distance direction in the x direction) and moved when tilting the arrangement in the x-direction perpendicular to the x-axis.

When rotated by 90 ° viewing (eye distance direction in the y direction) and tilting the arrangement in the y direction, the body moves in the direction of φ ₂ to the x-axis.

A three-dimensional effect comes about here in normal observation (eye distance direction in x-direction) only by movement.

Average level model

In the cut-plane model, the three-dimensional body is given by n cuts _fj (x, y) and n transparency step functions tj (x, y) with j = 1, ... n, for example, to simplify the calculation of the motif image when viewed with eye relief in the x-direction each lie at a depth z _j , z _j > z _j-1 . The A _j matrix must then be chosen such that the upper left coefficient is equal to z _j / e.

In this case, f _j (x, y) is the image function of the jth section indicating a brightness distribution (grayscale image), a color distribution (color image), a binary distribution (line drawing) or other image properties such as transparency, reflectivity, density or the like can. The transparency step function t _j (x, y) is equal to 1 if the intersection j conceals objects behind it at the position (x, y) and is otherwise equal to 0.

wobei j der kleinste Index ist, für den $$t_j\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\left({\mathrm{A}}_{\mathrm{j}}-\mathrm{I}\right)\cdot \left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)\mathrm{mod\; W}\right)-\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{1}}\\ {\mathrm{c}}_{\mathrm{2}}\end{array}\right)\right)\right)$$

ungleich Null ist.For the image function m (x, y) then results $$f_j\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\left({\mathrm{A}}_{\mathrm{j}}-\mathrm{I}\right)\cdot \left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)\mathrm{mod\; W}\right)-\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{1}}\\ {\mathrm{c}}_{\mathrm{2}}\end{array}\right)\right)\right).$$

where j is the smallest index for which $$t_j\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\left({\mathrm{A}}_{\mathrm{j}}-\mathrm{I}\right)\cdot \left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)\mathrm{mod\; W}\right)-\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{1}}\\ {\mathrm{c}}_{\mathrm{2}}\end{array}\right)\right)\right)$$

is not equal to zero.

A woodcut or copper engraving 3D image is obtained, for example, if the sections f _j , t _j are described by several function values in the following way:

fj = black and white (or grayscale value) on the contour line, or black and white (or grayscale) values in different areas of the section, adjacent to the edge, and $${\mathrm{t}}_{\mathrm{j}}=\{\begin{array}{cc}\mathrm{1}& \mathrm{opacity}\ddot{\mathrm{a}}\mathrm{t}\left(\mathrm{opacity}\right)\mathrm{within\; the\; sectional\; figure\; of\; the\; K}\ddot{\mathrm{O}}\mathrm{<figure-callout\; id="0"\; label="rpers"\; filenames="imgf0002.png"\; state="\{\{state\}\}">rpers</figure-callout>}\\ \mathrm{0}& \mathrm{opacity}\ddot{\mathrm{a}}\mathrm{t}\left(\mathrm{opacity}\right)\mathrm{outside\; the\; cut\; figure\; of\; the\; K}\ddot{\mathrm{O}}\mathrm{rpers}\end{array}$$

To illustrate the cutting plane model, here are some special cases:

Example 5:

In the simplest case, the magnification and motion matrix is given by $${\mathrm{A}}_{\mathrm{j}}=\left(\begin{array}{cc}\frac{{\mathrm{<figure-callout\; id="0"\; label="z\; j\; e"\; filenames="imgf0002.png"\; state="\{\{state\}\}">z\; </figure-callout>}}_{\mathrm{j}}}{\mathrm{e}}& \mathrm{0}\\ \mathrm{0}& \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\end{array}\right)=\frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\cdot \mathrm{I}={\mathrm{v}}_{\mathrm{j}}\cdot \mathrm{I}\mathrm{,}$$

In all viewing directions, all eye distance directions, and when rotating the assembly, the depth remains unchanged.

Example 6:

erhält. Beim Drehen der Anordnung ändert sich der Tiefeneindruck des dargestellten Körpers um den Änderungsfaktor k.A change factor k other than 0 is given so that the magnification and motion matrix A _{j is} the shape $${\mathrm{A}}_{\mathrm{j}}=\left(\begin{array}{cc}\frac{{\mathrm{<figure-callout\; id="0"\; label="z\; j\; e"\; filenames="imgf0002.png"\; state="\{\{state\}\}">z\; </figure-callout>}}_{\mathrm{j}}}{\mathrm{e}}& \mathrm{0}\\ \mathrm{0}& \mathrm{k}\cdot \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\end{array}\right)$$

receives. When turning the arrangement, the depth impression of the illustrated body changes by the change factor k.

Example 7:

erhält. Bei normaler Betrachtung (Augenabstandsrichtung in x-Richtung) und Kippen der Anordnung in x-Richtung bewegt sich der Körper in Richtung φ₁ zur x-Achse, bei um 90° gedrehter Betrachtung (Augenabstandsrichtung in y-Richtung) und Kippen der Anordnung in y-Richtung bewegt sich der Körper in Richtung φ₂ zur x-Achse und ist um den Faktor k in der Tiefendimension gedehnt.There will be a change factor k equal to 0 and two angles φ ₁ and φ ₂ given, so that the magnification and movement matrix A _j, the shape $${\mathrm{A}}_{\mathrm{j}}=\left(\begin{array}{cc}\frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}& \mathrm{k}\cdot \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\cdot {\cot \mathit{\phi }}_{\mathrm{2}}\\ \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\cdot \tan {\mathit{\phi }}_{\mathrm{1}}& \mathrm{k}\cdot \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\end{array}\right)$$

receives. When viewed normally (eye distance direction in the x direction) and tilting the arrangement in the x direction, the body moves in the direction of φ ₁ to the x-axis, by 90 ° rotated viewing (eye distance direction in the y direction) and tilting the arrangement in y Direction moves the body in the direction φ ₂ to the x-axis and is stretched by the factor k in the depth dimension.

Example 8:

erhält. Bei normaler Betrachtung (Augenabstandsrichtung in x-Richtung) und Kippen der Anordnung in x-Richtung bewegt sich der Körper in Richtung φ₁ zur x-Achse. Beim Kippen in y-Richtung erfolgt keine Bewegung.An angle φ _{1 is} specified so that the magnification and motion matrix A _{j is} the shape $${\mathrm{A}}_{\mathrm{j}}=\left(\begin{array}{cc}\frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}& \mathrm{0}\\ \frac{z_j}{e}\cdot {\tan \phi }_1& 1\end{array}\right)$$

receives. During normal viewing (eye distance direction in the x-direction) and tilting of the arrangement in the x-direction, the body moves in the direction φ ₁ to the x-axis. When tilting in the y direction, there is no movement.

mit dem Spalt- bzw. Zylinderachsenabstand d gegeben ist.In this embodiment, the consideration is also possible with a suitable cylindrical lens grid, for example with a split screen or cylindrical lens grid, the unit cell through $$\mathrm{W}=\left(\begin{array}{cc}\mathrm{<figure-callout\; id="0"\; label="d"\; filenames="imgf0002.png"\; state="\{\{state\}\}">d</figure-callout>}& \mathrm{0}\\ \mathrm{0}& \mathrm{\infty }\end{array}\right)$$

given with the gap or cylinder axis distance d.

Example 9:

wenn φ = 0 erhält. Beim waagrechten Kippen kippt der dargestellte Körper senkrecht zur Kipprichtung, beim senkrechten Kippen kippt der Körper in Richtung φ zur x-Achse.It will be a change factor k equal to 0 and φ a predetermined angle so that the magnification and movement matrix A _j, the shape $${\mathrm{A}}_{\mathrm{j}}=\left(\begin{array}{cc}\mathrm{0}& \mathrm{k}\cdot \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\cdot \cot \mathit{\phi }\\ \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}& \mathrm{k}\cdot \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\end{array}\right).{\mathrm{A}}_{\mathrm{j}}=\left(\begin{array}{cc}\mathrm{0}& \mathrm{k}\cdot \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\\ \frac{{\mathrm{<figure-callout\; id="0"\; label="z\; j\; e"\; filenames="imgf0002.png"\; state="\{\{state\}\}">z\; </figure-callout>}}_{\mathrm{j}}}{\mathrm{e}}& \mathrm{0}\end{array}\right)$$

if φ = 0. When horizontally tilting the body shown tilts perpendicular to the tilting direction, the vertical tilting tilts the body in the direction of φ to the x-axis.

Example 10:

erhält. Der dargestellte Körper bewegt sich unabhängig von der Kipprichtung stets in Richtung φ₁ zur x-Achse.It will be a change factor k equal to 0 and an angle φ ₁ defined so that the magnification and movement matrix A _j, the shape $${\mathrm{A}}_{\mathrm{j}}=\left(\begin{array}{cc}\frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}& \mathrm{k}\cdot \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\cdot \cot {\mathit{\phi }}_{\mathrm{1}}\\ \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\cdot {\tan \mathit{\phi }}_{\mathrm{1}}& \mathrm{k}\cdot \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\end{array}\right)$$

receives. The body shown always moves independently of the tilting direction in the direction of φ ₁ to the x-axis.

Common configurations

Hereinafter, further embodiments of the invention are shown, which are each explained using the example of the height profile model, in which the body to be displayed according to the above explanation by a two-dimensional drawing f (x, y) and a height indication z (x, y) is shown. However, it is understood that the embodiments described below also in the context of the general perspective and the sectional plane model The two-dimensional function f (x, y) can then be used correspondingly by the three-dimensional functions f (x, y, z) and t (x, y, z) or the sectional images f _j (x, y) and t _j (x, y) are replaced.

mit $$\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{k}}\\ {\mathrm{y}}_{\mathrm{k}}\end{array}\right)=\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\mathit{V}\left(\mathrm{x},\mathrm{y}\right)\cdot \left(\left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)\right)\mathrm{modW}\right)-{\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)-{\mathrm{w}}_{\mathrm{c}}\left(\mathrm{x},\mathrm{y}\right)\right),$$

$${\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{d}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{d}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$$

und $${\mathrm{w}}_{\mathrm{c}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{c}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)\mathrm{.}$$

For the height profile model, the image function m (x, y) is generally given by $$\mathrm{m}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{f}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{k}}\\ {\mathrm{y}}_{\mathrm{k}}\end{array}\right)\cdot \mathrm{G}\left(\mathrm{x},\mathrm{y}\right).$$

With $$\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{k}}\\ {\mathrm{y}}_{\mathrm{k}}\end{array}\right)=\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\mathit{V}\left(\mathrm{x},\mathrm{y}\right)\cdot \left(\left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)\right)\mathrm{ModW}\right)-{\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)-{\mathrm{w}}_{\mathrm{c}}\left(\mathrm{x},\mathrm{y}\right)\right).$$

$${\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{d}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{d}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$$

and $${\mathrm{w}}_{\mathrm{c}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{c}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)\mathrm{,}$$

das gewünschte Vergrößerungs- und Bewegungsverhalten des vorgegebenen Körpers beschreibt, und I die Einheitsmatrix ist. Im Spezialfall einer reinen Vergrößerung ohne Bewegungseffekt ist der Vergrößerungsterm ein Skalar $$\mathrm{V}\left(\mathrm{x},\mathrm{y}\right)=\left(\frac{\mathrm{z}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}-\mathrm{1}\right)\mathrm{.}$$

The magnification term V (x, y) is generally a matrix V (x, y) = (A (x, y) -I), where the matrix $\mathrm{A}\left(\mathrm{x},\mathrm{y}\right)=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{11}}\left(\mathrm{x},\mathrm{y}\right)& {\mathrm{a}}_{\mathrm{12}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{a}}_{\mathrm{21}}\left(\mathrm{x},\mathrm{y}\right)& {\mathrm{a}}_{\mathrm{22}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$

describes the desired magnification and movement behavior of the given body, and I is the unit matrix. In the special case of pure magnification without motion effect, the magnification term is a scalar $$\mathrm{V}\left(\mathrm{x},\mathrm{y}\right)=\left(\frac{\mathrm{z}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}-\mathrm{1}\right)\mathrm{,}$$

The vector (c ₁ (x, y), c ₂ (x, y)) with 0≤c ₁ (x, y), c ₂ (x, y) <1 gives the relative position of the center of the viewing elements within the cells of the motif image. The vector (d ₁ (x, y), d ₂ (x, y)) with 0≤d ₁ (x, y), d ₂ (x, y) <1 represents a shift of the cell boundaries in the motif image, and g (x, y) is a mask function for adjusting the visibility of the body.

Example 11:

For some applications, an angle constraint may be desirable when viewing the motif images, i. The illustrated three-dimensional image should not be visible from all directions or even be recognized only in a small solid angle range.

Such an angle restriction may be particularly advantageous in combination with the alternate frames described below since switching from one subject to another is generally not perceived by both eyes simultaneously. This can lead to an unwanted double image being seen as a superimposition of adjacent image motifs during the switchover. However, if the frames are bordered by an edge of appropriate width, such visually undesirable overlay can be suppressed.

Furthermore, it has been shown that the image quality can slacken significantly under oblique view of the lens array under certain circumstances: While a sharp image can be seen when viewed vertically from the arrangement, the image is blurred in this case with increasing tilt angle and blurred. For this reason, an angle restriction may also be advantageous in the representation of individual images if, in particular, it fades out the surface areas between the lenses, which are only probed through the lenses at relatively high tilt angles. As a result, the three-dimensional image for the viewer disappears when tilted, before it can be perceived blurry.

mit 0 <= k_ij < 1. Dadurch wird von der Gitterzelle (w₁₁, w₂₁), (w₁₂, w₂₂) nur ein Ausschnitt genutzt, und zwar der Bereich k₁₁ · (w₁₁, w₂₁) bis k₁₂ · (w₁₁, w₂₁) in Richtung des ersten Gittervektors und der Bereich k₂₁ · (w₁₂, w₂₂) bis k₂₂ · (w₁₂, w₂₂) in Richtung des zweiten Gittervektors. Als Summe der beiden Randbereiche beträgt die Breite der ausgeblendeten Streifen (k₁₁+(1-k₁₂)) · (w₁₁, w₂₁) bzw. (k₂₁ + (1-k₂₂)) · (w₁₂, w₂₂).Such an angle restriction can be achieved by a mask function g ≠ 1 in the general formula for the motif image m (x, y). A simple example of such a mask function is $$\mathit{G}\left(\begin{array}{c}x\\ y\end{array}\right)=[\begin{array}{ll}\mathrm{1}& \mathit{f}\ddot{\mathit{u}}\mathit{r}\left(\mathrm{x},\mathrm{y}\right)\mathrm{ModW}={\mathrm{t}}_{\mathrm{1}}\left({\mathrm{w}}_{\mathrm{11}},{\mathrm{w}}_{\mathrm{21}}\right)+{\mathrm{t}}_{\mathrm{2}}\left({\mathrm{w}}_{\mathrm{12}},{\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{\mathrm{22}}\right){\mathit{With}\mathrm{k}}_{\mathrm{11}}\le {\mathrm{t}}_{\mathrm{1}}\le {\mathrm{<figure-callout\; id="12"\; label="k"\; filenames="imgf0001.png"\; state="\{\{state\}\}">k</figure-callout>}}_{\mathrm{12}}{\mathit{and}\mathrm{k}}_{\mathrm{21}}\le {\mathrm{t}}_{\mathrm{2}}\le {\mathrm{<figure-callout\; id="22"\; label="k"\; filenames="imgf0001.png"\; state="\{\{state\}\}">k</figure-callout>}}_{\mathrm{22}}\\ \mathrm{0}& \mathit{otherwise}\end{array}$$

with 0 <= k _ij <1. As a result of the grid cell (w ₁₁ , w ₂₁ ), (w ₁₂ , w ₂₂ ) only a section used, namely the range k ₁₁ · (w ₁₁ , w ₂₁ ) to k ₁₂ · (w ₁₁ , w ₂₁ ) in the direction of the first grating vector and the range k ₂₁ · (w ₁₂ , w ₂₂ ) to k ₂₂ · (w ₁₂ , w ₂₂ ) in the direction of the second grating vector. The sum of the two edge regions is the width of the hidden bands (k ₁₁ + (1-k ₁₂ )) · (w ₁₁ , w ₂₁ ) and (k ₂₁ + (1-k ₂₂ )) · (w ₁₂ , w ₂₂ ).

It is understood that the function g (x, y) can generally arbitrarily specify the distribution of occupied and free areas within a cell.

In addition to an angle constraint, mask functions can also define areas in which the three-dimensional image is not visible as a field constraint. The areas where g = 0 may in this case extend over a plurality of cells. For example, the embodiments with adjacent images mentioned below can be described by such macroscopic mask functions. In general, a masking function for image field limitation is given by $$G\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)=[\begin{array}{ll}\mathrm{1}& \mathrm{in\; areas\; where\; the}\mathrm{3}\mathrm{D}-\mathrm{Image\; should\; be\; visible}\\ \mathrm{0}& \mathrm{in\; areas\; where\; the}\mathrm{3}\mathrm{D}-\mathrm{Image\; should\; not\; be\; visible}\end{array}$$

When using a mask function g ≠ 1, for the case of location-independent cell boundaries in the motif image, one obtains m (x, y) from the formula for the image function: $$\mathrm{m}\left(\mathrm{x},\mathrm{y}\right)=\mathit{f}\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\left(\mathrm{A}-\mathrm{I}\right)\cdot \left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)\mathrm{ModW}\right)-\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{1}}\\ {\mathrm{c}}_{\mathrm{2}}\end{array}\right)\right)\right)\cdot \mathit{G}\left(\mathit{x},\mathit{y}\right)\mathrm{,}$$

Example 12:

mit 0 ≤d₁(x,y),d₂(x,y) < 1 dar.In the examples so far described, the vector (d ₁ (x, y), d ₂ (x, y)) was identically zero, the cell boundaries were uniformly distributed over the entire area. In some embodiments, however, it may also be advantageous to shift the grid of the cells in the motif plane in a location-dependent manner in order to achieve special optical effects when changing the viewing direction. With g≡1 the image function m (x, y) is then in the form $$\mathit{f}\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\left(\mathrm{A}-\mathrm{I}\right)\cdot \left(\left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\mathrm{W}\left(\begin{array}{c}{\mathrm{d}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{d}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)\mathrm{ModW}\right)-\mathrm{W}\left(\begin{array}{c}{\mathrm{d}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{d}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)-\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{1}}\\ {\mathrm{c}}_{\mathrm{2}}\end{array}\right)\right)\right)\right)$$

with 0≤d ₁ (x, y), d ₂ (x, y) <1.

Example 13:

mit 0 ≤ c₁ (x, y), c₂ (x, y) < 1 dar. Selbstverständlich können auch hier der Vektor (d₁(x,y), d₂(x,y)) ungleich Null und die Bewegungsmatrix A(x,y) ortsabhängig sein, so dass sich für g ≡ 1 allgemein ergibt: $$\mathit{f}\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\left(\mathrm{A}\left(\mathrm{x},\mathrm{y}\right)-\mathrm{I}\right)\cdot \left(\left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\mathrm{W}\left(\begin{array}{c}{\mathrm{d}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{d}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)\right)\mathrm{modW}\right)-\mathrm{W}\left(\begin{array}{c}{\mathrm{d}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{d}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)-\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{c}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)\right)\right)$$

mit 0 ≤ c₁(x,y),c₂(x,y); d₁(x,y),d₂(x,y) < 1.The vector (c ₁ (x, y), c ₂ (x, y)) may also be a function of the location. With g ≡ 1, the image function m (x, y) then appears in the form $$\mathit{f}\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\left(\mathrm{A}-\mathrm{I}\right)\cdot \left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)\mathrm{ModW}\right)-\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{c}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)\right)\right)$$

with 0 ≤ c ₁ (x, y), c ₂ (x, y) <1. Of course, here too, the vector (d ₁ (x, y), d ₂ (x, y)) may be nonzero and the motion matrix A (x, y) be location-dependent, so that for g ≡ 1 it generally follows: $$\mathit{f}\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\left(\mathrm{A}\left(\mathrm{x},\mathrm{y}\right)-\mathrm{I}\right)\cdot \left(\left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\mathrm{W}\left(\begin{array}{c}{\mathrm{d}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{d}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)\right)\mathrm{ModW}\right)-\mathrm{W}\left(\begin{array}{c}{\mathrm{d}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{d}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)-\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{c}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)\right)\right)$$

with 0≤c ₁ (x, y), c ₂ (x, y); d ₁ (x, y), d ₂ (x, y) <1.

As explained above, the vector (c ₁ (x, y), c ₂ (x, y)) describes the position of the cells in the scene image plane relative to the lens array W, whereby the raster of the lens centers can be considered as the reference point set. If the vector (c ₁ (x, y), c ₂ (x, y)) is a function of the location, this means that changes in (c ₁ (x, y), c ₂ (x, y)) occur in a change in relative positioning between the cells in the scene image plane and the lenses, resulting in variations in the periodicity of the motif picture elements.

For example, a location dependence of the vector (c ₁ (x, y), c ₂ (x, y)) can advantageously be used if a film web is used which carries a lens embossing on the front side with a homogeneous homogeneous pattern W. If a modulo magnification arrangement with location-independent (c ₁ (x, y), c ₂ (x, y)) is impressed on the rear side, it is left to chance, under which viewing angles one recognizes which features, if there is no exact registration between Front and back side embossing is possible. If, however, one varies (c ₁ (x, y), c ₂ (x, y)) transversely to the direction of film travel, then a strip-shaped region is found in the direction of travel of the film, which fulfills the required positioning between front and back side embossing.

In addition, (c ₁ (x, y), c ₂ (x, y)) can also be varied, for example, in the running direction of the film in order to find sections in each strip in the longitudinal direction of the film which have the correct registration. This makes it possible to prevent the appearance of metallized hologram strips or security threads from banknote to banknote.

Example 14:

In a further exemplary embodiment, the three-dimensional image should not only be visible when viewed through a normal hole / lenticular grid, but also when viewed through a slit grid or cylindrical lens grid, wherein a three-dimensional image can be given in particular a non-periodically repeating individual image.

This case can also be described by the general formula for m (x, y), wherein, if the motif image to be applied is not transformed in the gap / cylinder direction with respect to the image to be displayed, a special matrix A is needed, which can be determined as follows:

If the cylinder axis direction lies in the y direction and the cylinder axis distance d, then the slit or cylindrical lens grid is described by: $$\mathrm{W}=\left(\begin{array}{cc}\mathrm{<figure-callout\; id="0"\; label="d"\; filenames="imgf0002.png"\; state="\{\{state\}\}">d</figure-callout>}& \mathrm{0}\\ \mathrm{0}& \mathrm{\infty }\end{array}\right)\mathrm{,}$$

The matching matrix A, with no magnification or distortion in the y-direction, is then: $$\mathrm{A}=\left(\begin{array}{cc}{a}_{11}& 0\\ a_{21}& 1\end{array}\right)=\left(\begin{array}{cc}{v}_1\cdot {\cos \phi }_1& 0\\ v_1\cdot {\sin \phi }_1& 1\end{array}\right)=\left(\begin{array}{cc}\frac{z_1}{e}& 0\\ \frac{z_1}{e}\cdot {\tan \phi }_1& 1\end{array}\right)$$

In this case, the matrix (A-1) in the relationship (A-1) W acts only on the first row of W, so that W can represent an infinitely long cylinder.

wobei auch möglich ist, dass der Träger von $\mathit{f}\left(\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{11}}-1& \mathrm{0}\\ {\mathrm{a}}_{\mathrm{21}}& \mathrm{0}\end{array}\right)\cdot \left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)\right)$

nicht in eine Zelle W passt, und dabei so groß ist, dass das anzulegende Muster in den Zellen keine kompletten zusammenhängenden Bilder zeigt. Das so erzeugte Muster lässt sich nicht nur mit dem Spaltblenden- bzw. Zylinderlinsen-Array $\mathrm{W}=\left(\begin{array}{cc}\mathrm{d}& \mathrm{0}\\ \mathrm{0}& \mathrm{\infty }\end{array}\right)$

betrachten, sondern auch mit einem Lochblenden- bzw.The motif image to be created with the cylinder axis in y-direction then results in: $$\mathit{f}\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{11}}-\mathrm{1}& \mathrm{0}\\ {\mathrm{a}}_{\mathrm{21}}& \mathrm{0}\end{array}\right)\cdot \left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)\mathrm{ModW}\right)-\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{1}}\\ {\mathrm{c}}_{\mathrm{2}}\end{array}\right)\right)\right)=\mathit{f}\left(\left(\begin{array}{c}\mathrm{x}+\left({\mathrm{a}}_{\mathrm{11}}-\mathrm{1}\right)\cdot \left(\left(\mathrm{x\; mod\; d}\right)-\mathrm{d}\cdot {\mathrm{c}}_{\mathrm{1}}\right)\\ \mathrm{y}+{\mathrm{a}}_{\mathrm{21}}\cdot \left(\left(\mathrm{x\; mod\; d}\right)-\mathrm{d}\cdot {\mathrm{c}}_{\mathrm{1}}\right)\hfill \end{array}\right)\right)$$

it being also possible for the wearer of $\mathit{f}\left(\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{11}}-1& \mathrm{0}\\ {\mathrm{a}}_{\mathrm{21}}& \mathrm{0}\end{array}\right)\cdot \left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)\right)$

does not fit into a cell W, and is so large that the pattern to be applied in the cells does not show complete coherent images. The pattern produced in this way can not be used only with the slit or cylindrical lens array $\mathrm{W}=\left(\begin{array}{cc}\mathrm{<figure-callout\; id="0"\; label="d"\; filenames="imgf0002.png"\; state="\{\{state\}\}">d</figure-callout>}& \mathrm{0}\\ \mathrm{0}& \mathrm{\infty }\end{array}\right)$

but also with a pinhole or

wobei d₂ und β beliebig sind.Lens array with $\mathrm{W}=\left(\begin{array}{cc}\mathrm{d}& \mathrm{0}\\ \mathrm{d}\cdot \tan \mathit{\beta }& {\mathrm{d}}_{\mathrm{2}}\end{array}\right).$

where d ₂ and β are arbitrary.

Common arrangements for the representation of several bodies

In the previous embodiments, the modulo magnification arrangement usually represents a single three-dimensional image (body) when viewed. However, the invention also encompasses configurations in which several three-dimensional images are displayed simultaneously or alternately.

In the simultaneous representation, the three-dimensional images can in particular have different movement behavior when tilting the arrangement. In three-dimensional images shown in alternation these can merge into one another in particular when tilting the arrangement. The different images can be independent of each other or content related to each other and represent, for example, a movement.

Here, too, the principle is explained using the example of the height profile model, it being understood again that the embodiments described with appropriate adaptation or replacement of the functions f _i (x, y) also in the context of the general perspective with body functions f _i (x, y, z) and transparency step functions t _i (x, y, z) or in the context of the sectional plane model with sectional images f _ij (x, y) and transparency step functions t _ij (x, y) can be used.

mit den beschreibenden Funktionen $${\mathrm{h}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y}\right)={\mathrm{f}}_{\mathrm{i}}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{iK}}\\ {\mathrm{y}}_{\mathrm{iK}}\end{array}\right)\cdot {\mathit{g}}_{\mathrm{i}}\left(\mathit{x},\mathit{y}\right),$$

mit $$\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{iK}}\\ {\mathrm{y}}_{\mathrm{iK}}\end{array}\right)=\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathit{V}}_{\mathit{i}}\left(\mathrm{x},\mathrm{y}\right)\cdot \left(\left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)\right)\mathrm{modW}\right)-{\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)-{\mathrm{w}}_{\mathrm{ci}}\left(\mathrm{x},\mathrm{y}\right)\right),$$

$${\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{d}}_{\mathrm{i}\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{d}}_{\mathrm{i}\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$$

und $${\mathrm{w}}_{\mathrm{ci}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{i}\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{c}}_{\mathrm{i}\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)\mathrm{.}$$

A multiplicity N≥1 of given three-dimensional bodies is to be represented, which are given by height profiles with two-dimensional representations of the bodies f _i (x, y), i = 1,2, ... N and by height functions z _i (x, y) are each containing a height / depth information for each point (x, y) of the predetermined body f _i . For the height profile model, the image function m (x, y) is then generally given by $$m\left(x,y\right)=F\left(H_1.H_2....H_N\right).$$

with the descriptive functions $${\mathrm{H}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y}\right)={\mathrm{f}}_{\mathrm{i}}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{iK}}\\ {\mathrm{y}}_{\mathrm{iK}}\end{array}\right)\cdot {\mathit{G}}_{\mathrm{i}}\left(\mathit{x},\mathit{y}\right).$$

With $$\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{iK}}\\ {\mathrm{y}}_{\mathrm{iK}}\end{array}\right)=\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathit{V}}_{\mathit{i}}\left(\mathrm{x},\mathrm{y}\right)\cdot \left(\left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)\right)\mathrm{ModW}\right)-{\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)-{\mathrm{w}}_{\mathrm{ci}}\left(\mathrm{x},\mathrm{y}\right)\right).$$

$${\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{d}}_{\mathrm{i}\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{d}}_{\mathrm{i}\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$$

and $${\mathrm{w}}_{\mathrm{ci}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{i}\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{c}}_{\mathrm{i}\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)\mathrm{,}$$

mit dem effektiven Abstand des Betrachtungsrasters vom Motivbild e, oder Matrizen $${\mathrm{V}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y}\right)=\left({\mathrm{A}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y}\right)-\mathrm{I}\right),$$

wobei die Matrizen ${\mathrm{A}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y}\right)=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{i}\mathrm{11}}\left(\mathrm{x},\mathrm{y}\right)& {\mathrm{a}}_{\mathrm{i}\mathrm{12}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{a}}_{\mathrm{i}\mathrm{21}}\left(\mathrm{x},\mathrm{y}\right)& {\mathrm{a}}_{\mathrm{i}\mathrm{22}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$

jeweils das gewünschtes Vergrößerungs- und Bewegungsverhalten des vorgegebenen Körpers f_i beschreiben und I die Einheitsmatrix ist. Die Vektoren (c_i1(x,y), c_i2(x,y)) mit 0 ≤ c _i1(x,y),c_i2(x,y) < 1 geben für den Körper f_i jeweils die relative Position des Zentrums der Betrachtungselemente innerhalb der Zellen i des Motivbilds an. Die Vektoren (d_i1(x,y), d_i2(x,y)) mit 0 ≤ d_i1 (x,y),d_i2 (x,y) < 1 stellen jeweils eine Verschiebung der Zellengrenzen im Motivbild dar, und g_i(x,y) sind Maskenfunktionen zur Einstellung der Sichtbarkeit des Körpers f_i.In this case, F ( h ₁ , h ₂ ,... H _N ) is a master function which specifies a combination of the N descriptive functions h _i (x, y). The magnification terms V _i (x, y) are either scalars $${\mathrm{V}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y}\right)=\left(\frac{{\mathrm{z}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}-\mathrm{1}\right).$$

with the effective distance of the viewing grid from the motif image e, or matrices $${\mathrm{V}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y}\right)=\left({\mathrm{A}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y}\right)-\mathrm{I}\right).$$

where the matrices ${\mathrm{A}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y}\right)=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{i}\mathrm{11}}\left(\mathrm{x},\mathrm{y}\right)& {\mathrm{a}}_{\mathrm{i}\mathrm{12}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{a}}_{\mathrm{i}\mathrm{21}}\left(\mathrm{x},\mathrm{y}\right)& {\mathrm{a}}_{\mathrm{i}\mathrm{22}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$

each describe the desired magnification and movement behavior of the given body f _i and I is the unit matrix. The vectors (c _i1 (x, y), c _i2 (x, y)) with 0 ≦ c _{i 1} (x, y), c _i2 (x, y) <1 give the relative position to the body f _i, respectively of the center of the viewing elements within the cells i of the motif image. The vectors (d _i1 (x, y), d _i2 (x, y)) with _{0≤d i1} (x, y), d _i2 (x, y) <1 represent respectively a shift of the cell boundaries in the motif image, and g _i (x, y) are mask functions for adjusting the visibility of the body f _i .

Example 14:

A simple example of multi-dimensional image (body) designs is a simple tilt image in which two three-dimensional bodies f ₁ (x, y) and f ₂ (x, y) alternate as the security element is tilted in a similar manner. Under what angles the change between the two bodies takes place is determined by the mask functions g ₁ and g. ₂ To prevent - even when viewing with only one eye - both images can be seen simultaneously, the carriers of the functions g ₁ and g _{2 are} chosen disjointly.

wobei für ein schachbrettartiges Wechseln der Sichtbarkeit der beiden Bilder $${\mathit{g}}_1\left(\begin{array}{c}x\\ y\end{array}\right)=[\begin{array}{ll}\mathrm{1}& \mathit{f}\ddot{\mathit{u}}\mathit{r}\left(\mathrm{x},\mathrm{y}\right)\mathrm{modW}={\mathrm{t}}_{\mathrm{1}}\left({\mathrm{w}}_{\mathrm{11}},{\mathrm{w}}_{\mathrm{21}}\right)+{\mathrm{t}}_{\mathrm{2}}\left({\mathrm{w}}_{\mathrm{12}},{\mathrm{w}}_{\mathrm{22}}\right)\mathit{mit}0\le {\mathrm{t}}_{\mathrm{1}},{\mathrm{t}}_2\le 0.5\mathit{oder}0.5\le {\mathrm{t}}_1,{\mathrm{t}}_2\le 1\\ \mathrm{0}& \mathit{sonst}\end{array}$$

$$g_2\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)=[\begin{array}{l}0\mathit{f}\ddot{\mathit{u}}\mathit{r}\left(\mathrm{x},\mathrm{y}\right)\mathrm{modW}={\mathrm{t}}_1\left({\mathrm{w}}_{11},{\mathrm{w}}_{21}\right)+{\mathrm{t}}_2\left({\mathrm{w}}_{12},{\mathrm{w}}_{22}\right)\mathit{mit}0\le {\mathrm{t}}_1,{\mathrm{t}}_2<0.5\mathit{oder}0.5\le {\mathrm{t}}_1,{\mathrm{t}}_2<1\\ 1\mathit{sonst}\end{array}$$

$$g_2\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)=1-g_1\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)$$

gewählt wird. In diesem Beispiel wurden die Grenzen zwischen den Bildbereichen im Motivbild bei 0,5 gewählt, so dass die zu den beiden Bildern f ₁ und f ₂ gehörigen Flächenabschnitte gleich groß sind. Natürlich können die Grenzen im allgemeinen Fall beliebig gewählt werden. Die Lage der Grenzen bestimmt die Raumwinkelbereiche, von denen aus die beiden dreidimensionalen Bilder zu sehen sind.Master function F is the sum function. This results in the image function of the motif image m (x, y): $$\left({\mathit{f}}_1\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\left(\mathrm{A}-\mathrm{I}\right)\cdot \left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)\mathrm{ModW}\right)-\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{1}}\\ {\mathrm{c}}_{\mathrm{2}}\end{array}\right)\right)\right)\right)\cdot \left(G_1\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)\right)\right)+\left({\mathit{f}}_2\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\left(\mathrm{A}-\mathrm{I}\right)\cdot \left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)\mathrm{ModW}\right)-\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{1}}\\ {\mathrm{c}}_{\mathrm{2}}\end{array}\right)\right)\right)\right)\cdot \left(G_2\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)\right)\right)$$

whereby for a checkerboard-like changing the visibility of the two pictures $${\mathit{G}}_1\left(\begin{array}{c}x\\ y\end{array}\right)=[\begin{array}{ll}\mathrm{1}& \mathit{f}\ddot{\mathit{u}}\mathit{r}\left(\mathrm{x},\mathrm{y}\right)\mathrm{ModW}={\mathrm{t}}_{\mathrm{1}}\left({\mathrm{w}}_{\mathrm{11}},{\mathrm{w}}_{\mathrm{21}}\right)+{\mathrm{t}}_{\mathrm{2}}\left({\mathrm{w}}_{\mathrm{12}},{\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{\mathrm{22}}\right)\mathit{With}0\le {\mathrm{t}}_{\mathrm{1}}.{\mathrm{t}}_2\le 0.5\mathit{or}0.5\le {\mathrm{t}}_1.{\mathrm{t}}_2\le 1\\ \mathrm{0}& \mathit{otherwise}\end{array}$$

$$G_2\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)=[\begin{array}{l}0\mathit{f}\ddot{\mathit{u}}\mathit{r}\left(\mathrm{x},\mathrm{y}\right)\mathrm{ModW}={\mathrm{t}}_1\left({\mathrm{w}}_{11},{\mathrm{w}}_{21}\right)+{\mathrm{t}}_2\left({\mathrm{w}}_{12},{\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{22}\right)\mathit{With}0\le {\mathrm{t}}_1.{\mathrm{t}}_2<0.5\mathit{or}0.5\le {\mathrm{t}}_1.{\mathrm{t}}_2<1\\ 1\mathit{otherwise}\end{array}$$

$$G_2\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)=1-G_1\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)$$

is selected. In this example, the boundaries between the image areas in the motif image were chosen to be 0.5, so that the area sections belonging to the two images f ₁ and f _{2 are the} same size. Of course, the limits can be chosen arbitrarily in the general case. The location of the borders determines the solid angle ranges from which the two three-dimensional images can be seen.

$$g_2\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)=[\begin{array}{l}0\mathit{f}\ddot{\mathit{u}}\mathit{r}\left(\mathrm{x},\mathrm{y}\right)\mathrm{modW}={\mathrm{t}}_1\left({\mathrm{w}}_{11},{\mathrm{w}}_{21}\right)+{\mathrm{t}}_2\left({\mathrm{w}}_{12},{\mathrm{w}}_{22}\right)\mathit{mit}0\le {\mathrm{t}}_1<0.5{\mathit{und\; t}}_2\mathit{beliebig}\\ 1\mathit{sonst}\end{array}$$

Instead of a checkered pattern, the displayed images can also alternate in strips, for example by using the following mask functions: $$G_1\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)=[\begin{array}{l}1\mathit{f}\ddot{\mathit{u}}\mathit{r}\left(\mathrm{x},\mathrm{y}\right)\mathrm{ModW}={\mathrm{t}}_1\left({\mathrm{w}}_{11},{\mathrm{w}}_{21}\right)+{\mathrm{t}}_2\left({\mathrm{w}}_{12},{\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{22}\right)\mathit{With}0\le {\mathrm{t}}_1<0.5{\mathit{and}\mathrm{t}}_2\mathit{any}\\ 0\mathit{otherwise}\end{array}$$

$$G_2\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)=[\begin{array}{l}0\mathit{f}\ddot{\mathit{u}}\mathit{r}\left(\mathrm{x},\mathrm{y}\right)\mathrm{ModW}={\mathrm{t}}_1\left({\mathrm{w}}_{11},{\mathrm{w}}_{21}\right)+{\mathrm{t}}_2\left({\mathrm{w}}_{12},{\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{22}\right)\mathit{With}0\le {\mathrm{t}}_1<0.5{\mathit{and\; t}}_2\mathit{any}\\ 1\mathit{otherwise}\end{array}$$

In this case, a change of image information occurs when the security element is tilted along the direction indicated by the vector (w ₁₁ , w ₂₁ ), whereas tilting along the second vector (w ₁₂ , w ₂₂ ) results in no image change. Again, the border was chosen at 0.5, ie the area of the motif image was divided into strips of equal width, which alternately contain the information of the two three-dimensional images.

If the borders of the stripes lie exactly under the lens centers or the lens boundaries, then the solid angle ranges under which the two images can be seen are distributed equally: starting with a vertical view, one of the two three-dimensional images is first seen from the right half of the hemisphere , from the left half of the Hemisphere first, the other three-dimensional image. In general, the boundary between the stripes can of course be arbitrarily set.

Example 15:

In the case of modulo-morphing or modulo-cinema, the various three-dimensional images are directly related, wherein in the case of modulo morphing, a starting image is transformed into a final image over a defined number of intermediate stages, and preferably simple sequences of motion are displayed in the modulo-cinema become.

${\mathit{f}}_{\mathrm{2}}\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)\cdots \cdots {\mathit{f}}_{\mathit{n}}\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)$

und z₁(x,y) ... z_n(x,y) gegeben, die beim Kippen entlang der durch den Vektor (w₁₁, w₂₁) vorgegebenen Richtung nacheinander erscheinen sollen. Um dies zu erreichen, wird mithilfe der Maskenfunktionen g_i eine Einteilung in Streifen gleicher Breite vorgenommen. Wird auch hier w_di = 0 für i=1...n gewählt, und als Masterfunktion F die Summenfunktion verwendet, so ergibt sich für die Bildfunktion des Motivbilds $$\mathrm{m}\left(\mathrm{x},\mathrm{y}\right)={\displaystyle \sum _{i=1}^n}\left(\left(f_i\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\left({\mathrm{A}}_{\mathrm{i}}-\mathrm{I}\right)\cdot \left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)\mathrm{modW}-\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_1\\ {\mathrm{c}}_2\end{array}\right)\right)\right)\right)\cdot g_i\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)\right)$$

$$g_i\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)=[\begin{array}{l}1\mathit{f}\ddot{\mathit{u}}\mathit{r}\left(\mathrm{x},\mathrm{y}\right)\mathrm{modW}={\mathrm{t}}_1\left({\mathrm{w}}_{11},{\mathrm{w}}_{21}\right)+{\mathrm{t}}_2\left({\mathrm{w}}_{12},{\mathrm{w}}_{22}\right)\mathit{mit}\frac{\mathrm{i}-1}{\mathrm{n}}\le {\mathrm{t}}_1<\frac{\mathrm{i}}{\mathrm{n}}{\mathit{und}\mathrm{t}}_2\mathit{beliebig}\\ 0\mathit{sonst}\end{array}$$

The three-dimensional images are in the elevation profile model through images ${\mathit{f}}_{\mathrm{1}}\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right).$

${\mathit{f}}_{\mathrm{2}}\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)\cdots \cdots {\mathit{f}}_{\mathit{n}}\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)$

given and z is ₁ (x, y) ... z _n (x, y), the predetermined during tipping along the by the vector (w _11, w ₂₁₎ direction are to appear in succession. In order to achieve this, the mask functions g _{i are used} to divide into strips of equal width. Is also w _d i = 0 is selected for i = 1 ... n, and used as a master function F the sum function is obtained for the image function of the motif image $$\mathrm{m}\left(\mathrm{x},\mathrm{y}\right)={\displaystyle \underset{i=1}{\overset{n}{\Sigma }}}\left(\left(f_i\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\left({\mathrm{A}}_{\mathrm{i}}-\mathrm{I}\right)\cdot \left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)\mathrm{ModW}-\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_1\\ {\mathrm{c}}_2\end{array}\right)\right)\right)\right)\cdot G_i\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)\right)$$

$$G_i\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)=[\begin{array}{l}1\mathit{f}\ddot{\mathit{u}}\mathit{r}\left(\mathrm{x},\mathrm{y}\right)\mathrm{ModW}={\mathrm{t}}_1\left({\mathrm{w}}_{11},{\mathrm{w}}_{21}\right)+{\mathrm{t}}_2\left({\mathrm{w}}_{12},{\mathrm{<figure-callout\; id="22"\; label="w"\; filenames="imgf0001.png"\; state="\{\{state\}\}">w</figure-callout>}}_{22}\right)\mathit{With}\frac{\mathrm{i}-1}{\mathrm{n}}\le {\mathrm{t}}_1<\frac{\mathrm{i}}{\mathrm{n}}{\mathit{and}\mathrm{t}}_2\mathit{any}\\ 0\mathit{otherwise}\end{array}$$

In general, here too, instead of the regular distribution expressed in the formula, the stripe width can be selected irregularly.

Although it is expedient to retrieve the image sequence by tilting along one direction (linear tilting movement), this is not absolutely necessary. Instead, the morphing or movement effects can also be played, for example, by meander-shaped or spiral-shaped tilting movements.

Example 16:

In the examples 14 and 15 was basically the goal, from a certain viewing direction only ever recognize a single three-dimensional image, but not two or more simultaneously. However, the simultaneous visibility of multiple images within the scope of the invention is also possible and can lead to attractive visual effects. The different three-dimensional images f _i can be treated completely independently of each other. This applies both to the respective image contents, as well as to the apparent position of the depicted objects and their movement in space.

While the image content can be rendered using drawings, the location and motion of the displayed objects in the dimensions of the space are described using the motion matrices A _i . Also, the relative phase of the individual displayed images can be set individually, as expressed by the coefficients c _ij in the general formula for m (x, y). The relative phase controls in which viewing directions the motifs can be recognized. Is the simplicity each selected half for the mask functions g _i, the unit function, the cell boundaries in the subject image are shifted not location-dependent, and is selected as the master function F is the sum function, the result for a number of superimposed three-dimensional images f _i: $$\mathrm{m}\left(\mathrm{x},\mathrm{y}\right)={\displaystyle \underset{i}{\Sigma }}\left(f_i\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\left({\mathrm{A}}_{\mathrm{i}}-\mathrm{I}\right)\cdot \left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)\mathrm{ModW}-\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{i}1}\\ {\mathrm{c}}_{\mathrm{i}2}\end{array}\right)\right)\right)\right)\mathrm{,}$$

When superimposing several images, the use of the sum function as a master function corresponds to an addition of the gray, color, transparency or density values depending on the character of the image function f, the resulting image values typically being set to the maximum value when the maximum value range is exceeded.

However, it may also be more convenient to choose functions other than the sum function for the master function F, such as an OR, an exclusive-OR (XOR), or the maximum function. Other possibilities are to select the signal with the lowest value of function or, as above, to form the sum of all function values meeting at a certain point. If there is a maximum upper limit, for example the maximum exposure intensity of a laser exposure, then one can cut off the sum at this maximum value.

By appropriate visibility links, blending, and overlaying multiple images, e.g. also "3D X-ray images" are shown, wherein an "outer skin" and an "inner skeleton" mixed and superimposed

Example 17:

All embodiments discussed in the context of this description can also be arranged side by side or in one another, for example as exchangeable images or as superimposed images. The boundaries between The image parts do not have to run in a straight line, but can be designed as desired. In particular, the boundaries may be chosen to represent the outlines of symbols or lettering, patterns, shapes of any kind, plants, animals or humans.

The juxtaposed or nested image parts are considered in preferred embodiments with a uniform lens array. In addition, the magnification and motion matrix A of the different image parts may differ, for example, to allow special motion effects of the individual magnified motifs. It may be advantageous to control the phase relationship between the image parts so that the enlarged motifs appear at a defined distance from each other.

Further developments for all embodiments

Using the above-described formulas for the motif image m (x, y), the microstructure plane can be calculated to render a three-dimensional object when viewed using a lenticular grid. This is basically based on the fact that the magnification factor is location-dependent, so the motif fragments in the different cells can have different sizes.

This three-dimensional appearance can be enhanced by filling surfaces of different inclinations with blazed gratings whose parameters differ from each other. A blaze grating is defined by specifying the parameters azimuth angle Φ, period d and inclination α .

This can be clearly explained by means of so-called Fresnel structures: for the visual appearance of a three-dimensional structure is the Reflection of the incident light at the surface of the structure crucial. Since the volume of the body is not critical to this effect, it can be eliminated using a simple algorithm. Round surfaces can be approximated by a large number of small flat surfaces.

When eliminating the volume, care must be taken to ensure that the depth of the structures is within an area that is accessible by the intended manufacturing process and lies within the focus range of the lenses. Moreover, it may be advantageous if the period d of the saw teeth is sufficiently large in order to largely avoid the formation of colored diffraction effects.

This development of the invention is therefore based on combining two methods for generating three-dimensional structures with one another: location-dependent magnification factor and filling with Fresnel structures, blazed gratings or other optically active structures, such as sub-wavelength structures.

When calculating a point in the microstructure plane, not only the value of the height profile at this point (which is included in the magnification at this point) is considered, but also optical properties at this point. In contrast to the cases discussed so far, in which also binary patterns in the microstructure level were sufficient, a three-dimensional structuring of the microstructure level is necessary for realizing this development of the invention.

Example: three-sided pyramid

Due to the location-dependent enlargement, different sized fragments of the three-sided pyramid are accommodated in the cells of the microstructure plane. Each of the three sides is assigned a blaze grating, which differ in their azimuth angle. In the case of a straight equilateral pyramid, the azimuth angles are 0 °, 120 ° and 240 °. All surface areas representing side 1 of the pyramid are equipped with the blaze grating with azimuth 0 ° - regardless of their size defined by the location-dependent A matrix. Correspondingly, pages 2 and 3 of the pyramid are used: they are filled with blaze gratings with azimuth angle 120 ° (page 2) or 240 ° (page 3). By vapor deposition of the resulting three-dimensional microstructure plane with metal (e.g., 50 nm aluminum), the reflectivity of the surface is increased and the 3D effect further enhanced.

Another possibility is the use of light-absorbing structures. Instead of blaze grids, it is also possible to use structures that not only reflect light, but also absorb it to a greater extent. This is usually the case when the aspect ratio depth / width (period or quasi-period) is relatively high, for example 1/1 or 2/1 or higher. The period or quasi-period can range from sub-wavelength structures to microstructures - this also depends on the size of the cells. How dark a surface should appear can be regulated, for example, via the surface density of the structures or the aspect ratio. Surfaces of different inclinations can be assigned structures with different absorption properties.

eindeutig festgelegt. Dabei ist $${\mathrm{pr}}_{\mathrm{XY}}:{\mathrm{R}}^{\mathrm{3}}\to {\mathrm{R}}^{\mathrm{2}},{\mathrm{pr}}_{\mathrm{XY}}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)=\left(\mathrm{x},\mathrm{y}\right)$$

die Projektion auf XY-Ebene, $$<\mathrm{a},\mathrm{b}>$$

stellt das Skalarprodukt dar, wobei <(x, y, z), ez >, das Skalarprodukt von (x, y, z) mit ez = (0, 0,1) die z-Komponente ergibt, und die Mengenschreibweise $$\langle A,x\rangle =\left\{\langle a,x\rangle |a\in A\right\}$$

zur Abkürzung eingeführt wurde. Weiter wird die charakteristische Funktion verwendet, die für eine Menge A gegeben ist durch $${\chi }_A\left(x\right)=\{\begin{array}{cc}1& \mathit{falls\; x}\in A\\ 0& \mathit{sonst}\end{array}$$

und das Lochraster bzw. Linsenraster W = {w ₁, w ₂, w ₃, ...} ist durch eine beliebige diskrete Teilmenge von R³ gegeben.Finally, a generalization of the modulo magnification arrangement is mentioned, in which the lens elements (or generally the viewing elements) need not be arranged in the form of a regular grid, but may be distributed arbitrarily in space at different distances. The motif image designed for viewing with such a general viewing element arrangement can then no longer be described in the modulo notation, but is the following relationship $$\mathrm{m}\left(\mathrm{x},\mathrm{y}\right)={\displaystyle \underset{w\in W}{\Sigma }}{\chi }_{M\left(w\right)}\left(x,y\right)\cdot \left(f_2\circ p_w^{-1}\right)\left(x.y.\min <p_w\left(f_1^{-1}\left(1\right)\right).\cap {\mathit{pr}}_{\mathit{XY}}^{-1}\left(x,y\right){.e}_z>\right)$$

clearly defined. It is $${\mathrm{pr}}_{\mathrm{XY}}:{\mathrm{R}}^{\mathrm{3}}\to {\mathrm{R}}^{\mathrm{2}}.{\mathrm{pr}}_{\mathrm{XY}}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)=\left(\mathrm{x},\mathrm{y}\right)$$

the projection on the XY plane, $$<\mathrm{a}.\mathrm{b}>$$

represents the scalar product, where <(x, y, z), ez>, the scalar product of (x, y, z) with ez = (0, 0,1) gives the z-component, and the set notation $$<A.x>=\left\{<a.x>|a\in A\right\}$$

was introduced for brevity. Further, the characteristic function given for a set A is used $${\chi }_A\left(x\right)=\{\begin{array}{cc}1& \mathit{if\; x}\in A\\ 0& \mathit{otherwise}\end{array}$$

and the lenticular grid W = { w ₁ , w ₂ , w ₃ , ...} is given by any discrete subset of R ³ .

$${\mathrm{p}}_{\mathrm{wm}}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)=\left(\left({\mathrm{z}}_{\mathrm{m}}\mathrm{x}-{\mathrm{x}}_{\mathrm{m}}\mathrm{z}\right)/\left({\mathrm{z}}_{\mathrm{m}}-\mathrm{z}\right),\left({\mathrm{z}}_{\mathrm{m}}\mathrm{y}-{\mathrm{y}}_{\mathrm{m}}\mathrm{z}\right)/\left({\mathrm{z}}_{\mathrm{m}}-\mathrm{z}\right),\left({\mathrm{z}}_{\mathrm{m}}\mathrm{z}\right)/\left({\mathrm{z}}_{\mathrm{m}}-\mathrm{z}\right)\right)$$

The perspective image to the grid point w _m = (x _m , y _m , z _m ) is given by $${\mathrm{p}}_{\mathrm{wm}}/{\mathrm{R}}^{\mathrm{3}}\to {\mathrm{R}}^{\mathrm{3}}.$$

$${\mathrm{p}}_{\mathrm{wm}}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)=\left(\left({\mathrm{z}}_{\mathrm{m}}\mathrm{x}-{\mathrm{x}}_{\mathrm{m}}\mathrm{z}\right)/\left({\mathrm{z}}_{\mathrm{m}}-\mathrm{z}\right).\left({\mathrm{z}}_{\mathrm{m}}\mathrm{y}-{\mathrm{y}}_{\mathrm{m}}\mathrm{z}\right)/\left({\mathrm{z}}_{\mathrm{m}}-\mathrm{z}\right).\left({\mathrm{z}}_{\mathrm{m}}\mathrm{z}\right)/\left({\mathrm{z}}_{\mathrm{m}}-\mathrm{z}\right)\right)$$

Each grid point w ∈ W is assigned a subset M ( w ) of the drawing plane. Here are the different subsets disjoint for different halftone dots.

$${\mathit{f}}_{\mathrm{2}}\left(\mathit{x},\mathit{y},\mathit{z}\right)=\mathrm{Helligkeit\; des\; K}\ddot{\mathrm{o}}\mathrm{rpers\; K\; an\; der\; Stelle}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)$$

ist.The body K to be modeled is defined by the function f = (f ₁ , f ₂ ): R ³ → R ² , where $$f_1\left(x,y,z\right)=\{\begin{array}{cc}1& \mathit{if\; x}\in \mathrm{<figure-callout\; id="0"\; label="K"\; filenames="imgf0002.png"\; state="\{\{state\}\}">K</figure-callout>}\hfill \\ 0& \mathit{otherwise}\hfill \end{array}$$

$${\mathit{f}}_{\mathrm{2}}\left(\mathit{x},\mathit{y},\mathit{z}\right)=\mathrm{Brightness\; of\; the\; K}\ddot{\mathrm{O}}\mathrm{K\; at\; the\; point}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)$$

is.

Then the above formula can be understood as follows: $$$$

Claims (21)

1. A depiction arrangement for security papers, value documents, electronic display devices or other data carriers, having a raster image arrangement for depicting a specified three-dimensional solid that is given by a solid function f(x,y,z), which indicates a characteristic property of the solid at the position (x,y,z), such as a brightness distribution, a color distribution, a binary distribution or another solid property, such as transparency, reflectivity, density or the like, having

   - a motif image that is subdivided into a plurality of cells, in each of which are arranged imaged regions of the specified solid,

   - a viewing grid composed of a plurality of viewing elements for depicting the specified solid when the motif image is viewed with the aid of the viewing grid,

   - the motif image exhibiting, with its subdivision into a plurality of cells, an image function m(x,y) that is given by $$\mathrm{m}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{f}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{K}}\\ {\mathrm{y}}_{\mathrm{K}}\\ {\mathrm{z}}_{\mathrm{K}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)\end{array}\right)\cdot \mathit{g}\left(\mathit{x},\mathit{y}\right),$$

   

   where $$\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{K}}\\ {\mathrm{y}}_{\mathrm{K}}\end{array}\right)=\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\mathit{V}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)\cdot \left(\left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)\right)\mathrm{modW}\right)-{\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)-{\mathrm{w}}_{\mathrm{c}}\left(\mathrm{x},\mathrm{y}\right)\right)$$

   

   $${\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{d}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{d}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$$

   

   and $${\mathrm{w}}_{\mathrm{c}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{c}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right),$$

   

   wherein

   - the unit cell of the viewing grid is described by lattice cell vectors ${\mathrm{w}}_1=$

   

   $\left(\begin{array}{c}{w}_{11}\\ w_{21}\end{array}\right)$

   

   and ${\mathrm{w}}_2=\left(\begin{array}{c}{w}_{12}\\ w_{22}\end{array}\right)$

   

   and combined in the matrix $\mathrm{W}=\left(\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& w_{22}\end{array}\right),$

   

   and x_m and y_m indicate the lattice points of the W-lattice,

   - the magnification term V(x,y, x_m,y_m) is either a scalar $\mathrm{V}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)=\left(\frac{{\mathrm{z}}_{\mathrm{K}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)}{\mathrm{e}}-\mathrm{1}\right),$

   

   where e is the effective distance of the viewing grid from the motif image, or a matrix V(x,y, x_m,y_m) =(A(x,y, x_m,y_m) - I), the matrix $\mathrm{A}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{11}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)& {\mathrm{a}}_{\mathrm{12}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)\\ {\mathrm{a}}_{\mathrm{21}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)& {\mathrm{a}}_{\mathrm{22}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)\end{array}\right)$

   

   describing a desired magnification and movement behavior of the specified solid and I being the identity matrix,

   - the vector (c₁(x,y), c₂(x,y)), where 0 ≤ c₁ (x, y), c₂ (x, y) < 1, indicates the relative position of the center of the viewing elements within the cells of the motif image,

   - the vector (d₁(x,y), d₂(x,y)), where 0 ≤ d, (x, y), d₂ (x, y) < 1, represents a displacement of the cell boundaries in the motif image, and

   - g(x,y) is a mask function for adjusting the visibility of the solid, which is either identical to 1, or which is zero in subregions, especially in edge regions of the cells of the motif image, and in this way describes an angle limit when the depicted solid is viewed, and
2. The depiction arrangement according to claim 1, characterized in that the magnification term is given by a matrix V(x,y, x_m,y_m) = (A(x,y, x_m,y_m) - I), where a₁₁(x,y, x_m,y_m) = z_K(x,y, x_m,y_m)/e, such that the raster image arrangement depicts the specified solid when the motif image is viewed with the eye separation being in the x-direction, or that the magnification term is given by a matrix V(x,y, x_m,y_m) = (A(x,y, x_m,y_m) - I), where (a₁₁ cos²ψ +(a₁₂+ a₂₁) cosψ sinψ + a₂₂ sin²ψ) = z_K(x,y, x_m,y_m) / e, such that the raster image arrangement depicts the specified solid when the motif image is viewed with the eye separation being in the direction ψ to the x-axis.
3. The depiction arrangement according to at least one of claims 1 to 3, characterized in that, in addition to the solid function f(x,y,z), a transparency step function t(x,y,z) is given, wherein t(x,y,z) is equal to 1 if, at the position (x,y,z), the solid f(x,y,z) covers the background, and otherwise is equal to 0, and wherein, for a viewing direction substantially in the direction of the z-axis, for z_K(x,y,x_m,y_m), the smallest value is to be taken for which t(x,y,z_K) is not equal to zero in order to view the solid front from the outside, and wherein, for a viewing direction substantially in the direction of the z-axis, for z_K(x,y,x_m,y_m), the largest value is to be taken for which t(x,y,z_K) is not equal to zero in order to view the solid back from the inside.
4. A depiction arrangement for security papers, value documents, electronic display devices or other data carriers, having a raster image arrangement for depicting a specified three-dimensional solid that is given by a height profile having a two-dimensional depiction of the solid f(x,y) and a height function z(x,y), where the two-dimensional depiction of the solid f(x,y) indicates a brightness distribution, a color distribution, a binary distribution or another image property, such as transparency, reflectivity, density or the like, and the height function z(x,y) includes, for every point (x,y) of the specified solid, height/depth information, having

   - a motif image that is subdivided into a plurality of cells, in each of which are arranged imaged regions of the specified solid,

   - a viewing grid composed of a plurality of viewing elements for depicting the specified solid when the motif image is viewed with the aid of the viewing grid,

   - the motif image exhibiting, with its subdivision into a plurality of cells, an image function m(x,y) that is given by $$\mathrm{m}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{f}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{K}}\\ {\mathrm{y}}_{\mathrm{K}}\end{array}\right)\cdot \mathit{g}\left(\mathit{x},\mathit{y}\right),$$

   

   where $$\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{K}}\\ {\mathrm{y}}_{\mathrm{K}}\end{array}\right)=\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+\mathit{V}\left(\mathrm{x},\mathrm{y}\right)\cdot \left(\left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)\right)\mathrm{modW}\right)-{\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)-{\mathrm{w}}_{\mathrm{c}}\left(\mathrm{x},\mathrm{y}\right)\right),$$

   

   $${\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{d}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{d}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$$

   

   and $${\mathrm{w}}_{\mathrm{c}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{c}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right),$$

   

   wherein

   - the unit cell of the viewing grid is described by lattice cell vectors w₁ = $\left(\begin{array}{c}{w}_{11}\\ w_{21}\end{array}\right)$

   

   and ${\mathrm{w}}_2=\left(\begin{array}{c}{w}_{12}\\ w_{22}\end{array}\right)$

   

   and combined in the matrix $\mathrm{w}=\left(\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& w_{22}\end{array}\right),$

   

   - the magnification term V(x,y) is either a scalar $\mathrm{V}\left(\mathrm{x},\mathrm{y}\right)=\left(\frac{z\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}-\mathrm{1}\right),$

   

   where e is the effective distance of the viewing grid from the motif image, or a matrix $$\mathrm{V}\left(\mathrm{x},\mathrm{y}\right)=\left(\mathrm{A}\left(\mathrm{x},\mathrm{y}\right)-\mathrm{I}\right),\mathrm{the\; matrix\; A}\left(\mathrm{x},\mathrm{y}\right)=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{11}}\left(\mathrm{x},\mathrm{y}\right)& {\mathrm{a}}_{\mathrm{12}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{a}}_{\mathrm{21}}\left(\mathrm{x},\mathrm{y}\right)& {\mathrm{a}}_{\mathrm{22}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$$

   

   describing a desired magnification and movement behavior of the specified solid and I being the identity matrix,

   - the vector (c₁(x,y), c₂(x,y)), where 0 ≤ c₁(x, y), c₂ (x, y) < 1, indicates the relative position of the center of the viewing elements within the cells of the motif image,

   - the vector (d₁(x,y), d₂(x,y)), where 0 ≤ d₁ (x, y), d₂ (x, y) < 1, represents a displacement of the cell boundaries in the motif image, and

   - g(x,y) is a mask function for adjusting the visibility of the solid, which is either identical to 1, or which is zero in subregions, especially in edge regions of the cells of the motif image, and in this way describes an angle limit when the depicted solid is viewed, and
5. The depiction arrangement according to claim 4, characterized in that two height functions z₁(x,y) and z₂(x,y) and two angles φ₁ (x, y) and φ₂ (x, y) are specified, and in that the magnification term is given by a matrix V(x,y) = (A(x,y) -1), where $$\mathrm{A}\left(\mathrm{x},\mathrm{y}\right)=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{11}}\left(\mathrm{x},\mathrm{y}\right)& {\mathrm{a}}_{\mathrm{12}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{a}}_{\mathrm{21}}\left(\mathrm{x},\mathrm{y}\right)& {\mathrm{a}}_{\mathrm{22}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)=\left(\begin{array}{cc}\frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}& \frac{{\mathrm{z}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\cdot {\cot \mathit{\phi }}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\\ \frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\cdot {\tan \mathit{\phi }}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)& \frac{{\mathrm{z}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\end{array}\right),$$

   

   or that two height functions z₁(x,y) and z₂(x,y) are specified, and in that the magnification term is given by a matrix V(x,y) = (A(x,y) - I), where $$\mathrm{A}\left(\mathrm{x},\mathrm{y}\right)=\left(\begin{array}{cc}\frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}& \mathrm{0}\\ \mathrm{0}& \frac{{\mathrm{z}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\end{array}\right),$$

   

   or that a height function z(x,y) and an angle φ₁ are specified, and in that the magnification term is given by a matrix V(x,y) = (A(x,y) - I), where $$\mathrm{A}\left(\mathrm{x},\mathrm{y}\right)=\left(\begin{array}{cc}\frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}& \mathrm{0}\\ \frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\cdot {\tan \mathit{\phi }}_{\mathrm{1}}& \mathrm{1}\end{array}\right)$$

   

   such that the depicted solid, upon viewing with the eye separation being in the x-direction and tilting the arrangement in the x-direction, moves in the direction φ₁ to the x-axis, and upon tilting in the y-direction, no movement occurs, especially that the viewing grid is a slot grid, cylindrical lens grid or cylindrical concave reflector grid whose unit cell is given by $$\mathrm{w}=\left(\begin{array}{cc}\mathrm{d}& \mathrm{0}\\ \mathrm{0}& \mathrm{\infty }\end{array}\right)$$

   

   where d is the slot or cylinder axis distance, or that a height function z(x,y), an angle φ₁ and a direction, by an angle γ, are specified, and in that the magnification term is given by a matrix V(x,y) = (A(x,y) - I), where $$\mathrm{A}=\left(\begin{array}{cc}\mathrm{cos\gamma }& -\mathrm{sin\gamma }\\ \mathrm{sin\gamma }& \mathrm{cos\gamma }\end{array}\right)\cdot \left(\begin{array}{cc}\frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}& \mathrm{0}\\ \frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}& \mathrm{1}\end{array}\right)\cdot \left(\begin{array}{cc}\mathrm{cos\gamma }& \mathrm{sin\gamma }\\ -\mathrm{sin\gamma }& \mathrm{cos\gamma }\end{array}\right),$$

   

   especially that the viewing grid is a slot grid, cylindrical lens grid or cylindrical concave reflector grid whose unit cell is given by $$\mathrm{W}=\left(\begin{array}{cc}\mathrm{cos\gamma }& -\mathrm{sin\gamma }\\ \mathrm{sin\gamma }& \mathrm{cos\gamma }\end{array}\right)\cdot \left(\begin{array}{cc}\mathrm{d}& \mathrm{0}\\ \mathrm{0}& \mathrm{\infty }\end{array}\right)$$

   

   wherein d indicates the slot or cylinder axis distance and γ the direction of the slot or cylinder axis, or that two height functions z₁(x,y) and z₂(x,y) and an angle φ₂ are specified, and in that the magnification term is given by a matrix V(x,y) = (A(x,y) - I), where $$\mathrm{A}\left(\mathrm{x},\mathrm{y}\right)=\left(\begin{array}{cc}\mathrm{0}& \frac{{\mathrm{z}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\cdot {\cot \mathit{\phi }}_{\mathrm{2}}\\ \frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}& \frac{{\mathrm{z}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\end{array}\right),\mathrm{A}\left(\mathrm{x},\mathrm{y}\right)=\left(\begin{array}{cc}\mathrm{0}& \frac{{\mathrm{z}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}\\ \frac{{\mathrm{z}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}& \mathrm{0}\end{array}\right){\mathrm{if\; \phi }}_{\mathrm{2}}=\mathrm{0}$$

   

   such that the depicted solid, upon viewing with the eye separation being in the x-direction and tilting the arrangement in the x-direction, moves normal to the x-axis, and upon viewing with the eye separation being in the y-direction and tilting the arrangement in the y-direction, the depicted solid moves in the direction φ₂ to the x-axis.
6. A depiction arrangement for security papers, value documents, electronic display devices or other data carriers, having a raster image arrangement for depicting a specified three-dimensional solid that is given by n sections f_j (x,y) and n transparency step functions t_j (x,y), where j = 1,...n, wherein, upon viewing with the eye separation being in the x-direction, the sections each lie at a depth z_j, z_j > z_j-1, and wherein f_j(x,y) is the image function of the j-th section, and indicates a brightness distribution, a color distribution, a binary distribution or another image properties, such as transparency, reflectivity, density or the like, and the transparency step function t_j(x,y) is equal to 1 if, at the position (x,y), the section j covers objects lying behind it, and otherwise is equal to 0, having

   - a motif image that is subdivided into a plurality of cells, in each of which are arranged imaged regions of the specified solid,

   - a viewing grid composed of a plurality of viewing elements for depicting the specified solid when the motif image is viewed with the aid of the viewing grid,

   - the motif image exhibiting, with its subdivision into a plurality of cells, an image function m(x,y) that is given by $$\mathrm{m}\left(\mathrm{x},\mathrm{y}\right)={\mathrm{f}}_{\mathrm{j}}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{K}}\\ {\mathrm{y}}_{\mathrm{K}}\end{array}\right)\cdot \mathit{g}\left(\mathit{x},\mathit{y}\right),$$

   

   where $$\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{K}}\\ {\mathrm{y}}_{\mathrm{K}}\end{array}\right)=\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathit{V}}_{\mathit{j}}\cdot \left(\left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)\right)\mathrm{modW}\right)-{\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)-{\mathrm{w}}_{\mathrm{c}}\left(\mathrm{x},\mathrm{y}\right)\right),$$

   

   ${\mathrm{w}}_{\mathrm{d}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{d}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{d}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$

   

   and ${\mathrm{w}}_{\mathrm{c}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{c}}_{\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right),$

   

   wherein, for j, the smallest or the largest index is to be taken for which ${\mathrm{t}}_{\mathrm{j}}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{K}}\\ {\mathrm{y}}_{\mathrm{K}}\end{array}\right)$

   

   is not equal to zero, and wherein

   - the unit cell of the viewing grid is described by lattice cell vectors ${\mathrm{w}}_{\mathrm{1}}=$

   

   $\left(\begin{array}{c}{w}_{11}\\ w_{21}\end{array}\right)$

   

   and ${\mathrm{w}}_2=\left(\begin{array}{c}{w}_{12}\\ w_{22}\end{array}\right)$

   

   and combined in the matrix $\mathrm{W}=\left(\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& w_{22}\end{array}\right),$

   

   - the magnification term V_j is either a scalar ${\mathrm{V}}_{\mathrm{j}}=\left(\frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}-\mathrm{1}\right),$

   

   where e is the effective distance of the viewing grid from the motif image, or a matrix V_j = (A_j - I), the matrix ${\mathrm{A}}_{\mathrm{j}}=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{j}\mathrm{11}}& {\mathrm{a}}_{\mathrm{j}\mathrm{12}}\\ {\mathrm{a}}_{\mathrm{j}\mathrm{21}}& {\mathrm{a}}_{\mathrm{j}\mathrm{22}}\end{array}\right)$

   

   describing a desired magnification and movement behavior of the specified solid and I being the identity matrix,

   - the vector (c₁(x,y), c₂(x,y)), where 0 ≤ c₁ (x, y), c₂ (x, y) < 1, indicates the relative position of the center of the viewing elements within the cells of the motif image,

   - the vector (d₁(x,y), d₂(x,y)), where 0 ≤ d₁ (x, y), d₂ (x, y) < 1, represents a displacement of the cell boundaries in the motif image, and

   - g(x,y) is a mask function for adjusting the visibility of the solid, which is either identical to 1, or which is zero in subregions, especially in edge regions of the cells of the motif image, and in this way describes an angle limit when the depicted solid is viewed.
7. The depiction arrangement according to claim 6, characterized in that a change factor k not equal to 0 is specified and the magnification term is given by a matrix V_j = (A_j - I), where $${\mathrm{A}}_{\mathrm{j}}=\left(\begin{array}{cc}\frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}& \mathrm{0}\\ \mathrm{0}& \mathrm{k}\cdot \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\end{array}\right)$$

   

   such that, upon rotating the arrangement, the depth impression of the depicted solid changes by the change factor k, or that a change factor k not equal to 0 and two angles φ₁ and φ₂ are specified, and the magnification term is given by a matrix V_j = (A_j - I), where $${\mathrm{A}}_{\mathrm{j}}=\left(\begin{array}{cc}\frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}& \mathrm{k}\cdot \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}{\cot \mathit{\phi }}_{\mathrm{2}}\\ \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\cdot {\tan \mathit{\phi }}_{\mathrm{1}}& \mathrm{k}\cdot \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\end{array}\right)$$

   

   such that the depicted solid, upon viewing with the eye separation being in the x-direction and tilting the arrangement in the x-direction, moves in the direction φ to the x-axis, and upon viewing with the eye separation being in the y-direction and tilting the arrangement in the y-direction, moves in the direction φ₂ to the x-axis and is stretched by the change factor k in the depth dimension, or that an angle φ₁ is specified, and in that the magnification term is given by a matrix V_j =(A_j - I), where $${\mathrm{A}}_{\mathrm{j}}=\left(\begin{array}{cc}\frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}& \mathrm{0}\\ \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\cdot {\tan \mathit{\phi }}_{\mathrm{1}}& \mathrm{1}\end{array}\right)$$

   

   such that the depicted solid, upon viewing with the eye separation being in the x-direction and tilting the arrangement in the x-direction, moves in the direction φ₁ to the x-axis, and no movement occurs upon tilting in the y-direction, especially that the viewing grid is a slot grid, cylindrical lens grid or cylindrical concave reflector grid whose unit cell is given by $$\mathrm{W}=\left(\begin{array}{cc}\mathrm{d}& \mathrm{0}\\ \mathrm{0}& \mathrm{\infty }\end{array}\right)$$

   

   where d is the slot or cylinder axis distance, or that an angle φ₁ and a direction, by an angle γ, are specified and that the magnification term is given by a matrix V_j =(A_j - I), where $${\mathrm{A}}_{\mathrm{j}}=\left(\begin{array}{cc}\mathrm{cos\gamma }& -\mathrm{sin\gamma }\\ \mathrm{sin\gamma }& \mathrm{cos\gamma }\end{array}\right)\cdot \left(\begin{array}{cc}\frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}& \mathrm{0}\\ \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\cdot {\tan \mathit{\phi }}_{\mathrm{1}}& \mathrm{1}\end{array}\right)\cdot \left(\begin{array}{cc}\mathrm{cos\gamma }& \mathrm{sin\gamma }\\ -\mathrm{sin\gamma }& \mathrm{cos\gamma }\end{array}\right),$$

   

   especially the viewing grid is a slot grid, cylindrical lens grid or cylindrical concave reflector grid whose unit cell is given by $$\mathrm{W}=\left(\begin{array}{cc}\mathrm{cos\gamma }& -\mathrm{sin\gamma }\\ \mathrm{sin\gamma }& \mathrm{cos\gamma }\end{array}\right)\cdot \left(\begin{array}{cc}\mathrm{d}& \mathrm{0}\\ \mathrm{0}& \mathrm{\infty }\end{array}\right)$$

   

   wherein d indicates the slot or cylinder axis distance and γ the direction of the slot or cylinder axis, or that a change factor k not equal to 0 and an angle φ are specified, and in that the magnification term is given by a matrix V_j =(A_j - I), where $${\mathrm{A}}_{\mathrm{j}}=\left(\begin{array}{cc}\mathrm{0}& \mathrm{k}\cdot \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\cdot \cot \mathit{\phi }\\ \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}& \mathrm{k}\cdot \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\end{array}\right),{\mathrm{A}}_{\mathrm{j}}=\left(\begin{array}{cc}\mathrm{0}& \mathrm{k}\cdot \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\\ \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}& \mathrm{0}\end{array}\right)\mathrm{if\; \phi }=\mathrm{0}$$

   

   such that the depicted solid, upon horizontal tilting, moves normal to the tilt direction, and upon vertical tilting, in the direction φ to the x-axis, or that a change factor k not equal to 0 and an angle φ₁ are specified, and in that the magnification term is given by a matrix V_j =(A_j - I), where $${\mathrm{A}}_{\mathrm{j}}=\left(\begin{array}{cc}\frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}& \mathrm{k}\cdot \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\cdot {\cot \mathit{\phi }}_{\mathrm{1}}\\ \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\cdot {\tan \mathit{\phi }}_{\mathrm{1}}& \mathrm{k}\cdot \frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{e}}\end{array}\right)$$

   

   such that the depicted solid always moves, independently of the tilt direction, in the direction φ₁ to the x-axis.
8. The depiction arrangement according to at least one of claims 1 to 7, characterized in that the cell boundaries in the motif image are location-dependently displaced, preferably in that the motif image exhibits two or more subregions having a different, in each case constant, cell grid.
9. The depiction arrangement according to at least one of claims 1 to 8, characterized in that the relative position of the center of the viewing elements is location independent within the cells of the motif image, in other words the vector (c₁, c₂) is constant, or that the relative position of the center of the viewing elements is location dependent within the cells of the motif image.
10. A depiction arrangement for security papers, value documents, electronic display devices or other data carriers, having a raster image arrangement for depicting a plurality of specified three-dimensional solids that are given by solid functions f_i(x,y,z), i=1,2,...N, where N≥1, each of which indicates a characteristic property of the i-th solid at the position (x,y,z), such as a brightness distribution, a color distribution, a binary distribution or another solid property, such as transparency, reflectivity, density or the like, having

    - a motif image that is subdivided into a plurality of cells, in each of which are arranged imaged regions of the specified solids,

    - a viewing grid composed of a plurality of viewing elements for depicting the specified solids when the motif image is viewed with the aid of the viewing grid,

    - the motif image exhibiting, with its subdivision into a plurality of cells, an image function m(x,y) that is given by $$m\left(x,y\right)=F\left(h_1,h_2,\dots ,h_N\right),$$

    

    having the describing functions $${\mathrm{h}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y}\right)={\mathrm{f}}_{\mathrm{i}}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{iK}}\\ {\mathrm{y}}_{\mathrm{iK}}\\ {\mathrm{z}}_{\mathrm{iK}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)\end{array}\right)\cdot {\mathit{g}}_{\mathit{i}}\left(\mathit{x},\mathit{y}\right),$$

    

    where $$\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{iK}}\\ {\mathrm{y}}_{\mathrm{iK}}\end{array}\right)=\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathrm{V}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)\cdot \left(\left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)\right)\mathrm{modW}\right)-{\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)-{\mathrm{w}}_{\mathrm{ci}}\left(\mathrm{x},\mathrm{y}\right)\right)$$

    

    $${\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{d}}_{\mathrm{i}\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{d}}_{\mathrm{i}\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$$

    

    and $${\mathrm{w}}_{\mathrm{ci}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{i}\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{c}}_{\mathrm{i}\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right),$$

    

    - wherein F(h ₁, , h ₂ , ... h_N ) is a master function that indicates an operation on the N describing functions h_i(x,y), and wherein

    - the unit cell of the viewing grid is described by lattice cell vectors w₁ = $\left(\begin{array}{c}{w}_{11}\\ w_{21}\end{array}\right)$

    

    and ${\mathrm{w}}_2=\left(\begin{array}{c}{w}_{12}\\ w_{22}\end{array}\right)$

    

    and combined in the matrix $\mathrm{W}=\left(\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& w_{22}\end{array}\right),$

    

    and x_m and y_m indicate the lattice points of the W-lattice,

    - the magnification terms V_i(x,y, x_m,y_m) are either scalars ${\mathrm{V}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)=\left(\frac{{\mathrm{z}}_{\mathrm{iK}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)}{\mathrm{e}}-\mathrm{1}\right),$

    

    where e is the effective distance of the viewing grid from the motif image, or matrices V_i(x,y, x_m,y_m) =(A_i(x,y, x_m,y_m) - I), the matrices ${\mathrm{A}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{i}\mathrm{11}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)& {\mathrm{a}}_{\mathrm{i}\mathrm{12}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)\\ {\mathrm{a}}_{\mathrm{i}\mathrm{21}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)& {\mathrm{a}}_{\mathrm{i}\mathrm{22}}\left(\mathrm{x},\mathrm{y},{\mathrm{x}}_{\mathrm{m}},{\mathrm{y}}_{\mathrm{m}}\right)\end{array}\right)$

    

    each describing a desired magnification and movement behavior of the specified solid f_i and I being the identity matrix,

    - the vectors (c_i1(x,y), c_i2(x,y)), where 0 ≤ c_i1 (x, y),c_i2 (x, y) < 1, indicate in each case, for the solid f_i, the relative position of the center of the viewing elements within the cells i of the motif image,

    - the vectors (d_i1(x,y), d_i2(x,y)), where 0 ≤ d_i1(x, y), d_i2(x, y) < 1, each represent a displacement of the cell boundaries in the motif image, and

    - g_i(x,y) are mask functions for adjusting the visibility of the solid f_i. which are either identical to 1, or which define a strip-like or checkerboard-like alternation of the visibility of the solids f_i.
11. The depiction arrangement according to claim 10, characterized in that, in addition to the solid functions f_i(x,y,z), transparency step functions t_i(x,y,z) are given, wherein t_i(x,y,z) is equal to 1 if, at the position (x,y,z), the solid f_i(x,y,z) covers the background, and otherwise is equal to 0, and wherein, for a viewing direction substantially in the direction of the z-axis, for Z_iK(x,y,x_m,y_m), the smallest value is to be taken for which t_i(x,y,z_K) is not equal to zero in order to view the solid front of the solid f_i from the outside, and wherein, for a viewing direction substantially in the direction of the z-axis, for Z_iK(x,y,x_m,y_m), the largest value is to be taken for which t_i(x,y,z_K) is not equal to zero in order to view the solid back of the solid f_i from the inside.
12. A depiction arrangement for security papers, value documents, electronic display devices or other data carriers, having a raster image arrangement for depicting a plurality of specified three-dimensional solids that are given by height profiles having two-dimensional depictions of the solids f_i(x,y), i=1,2,...N, where N≥1, and by height functions z_i(x,y), where the two-dimensional depictions of the solid f_i(x,y) each indicates a brightness distribution, a color distribution, a binary distribution or another image property, such as transparency, reflectivity, density or the like, and the height functions z_i(x,y) each includes height/ depth information for every point (x,y) of the specified solid f_i, having

    - a motif image that is subdivided into a plurality of cells, in each of which are arranged imaged regions of the specified solids,

    - a viewing grid composed of a plurality of viewing elements for depicting the specified solids when the motif image is viewed with the aid of the viewing grid,

    - the motif image exhibiting, with its subdivision into a plurality of cells, an image function m(x,y) that is given by m(x, y) = F(h ₁,h ₂,...h_N ), having the describing functions $${\mathrm{h}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y}\right)={\mathrm{f}}_{\mathrm{i}}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{iK}}\\ {\mathrm{y}}_{\mathrm{iK}}\end{array}\right)\cdot {\mathit{g}}_{\mathit{i}}\left(\mathit{x},\mathit{y}\right),$$

    

    where $$\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{iK}}\\ {\mathrm{y}}_{\mathrm{iK}}\end{array}\right)=\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathit{V}}_{\mathit{i}}\left(\mathrm{x},\mathrm{y}\right)\cdot \left(\left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)\right)\mathrm{modW}\right)-{\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)-{\mathrm{w}}_{\mathrm{ci}}\left(\mathrm{x},\mathrm{y}\right)\right)$$

    

    $${\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{d}}_{\mathrm{i}\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{d}}_{\mathrm{i}\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$$

    

    and $${\mathrm{w}}_{\mathrm{ci}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{i}\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{c}}_{\mathrm{i}\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right),$$

    

    - wherein F(h ₁, h ₂,... h_N ) is a master function that indicates an operation on the N describing functions h_i(x,y), and wherein

    - the unit cell of the viewing grid is described by lattice cell vectors ${\mathrm{w}}_1=$

    

    $\left(\begin{array}{c}{w}_{11}\\ w_{21}\end{array}\right)$

    

    and ${\mathrm{w}}_2=\left(\begin{array}{c}{w}_{12}\\ w_{22}\end{array}\right)$

    

    and combined in the matrix $\mathrm{W}=\left(\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& w_{22}\end{array}\right),$

    

    - the magnification terms V_i(x,y) are either scalars ${\mathrm{V}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y}\right)=\left(\frac{{\mathrm{z}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y}\right)}{\mathrm{e}}-\mathrm{1}\right),$

    

    where e is the effective distance of the viewing grid from the motif image, or matrices V_i(x,y) = (Ai(x,y) - I), the matrices ${\mathrm{A}}_{\mathrm{i}}\left(\mathrm{x},\mathrm{y}\right)=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{i}\mathrm{11}}\left(\mathrm{x},\mathrm{y}\right)& {\mathrm{a}}_{\mathrm{i}\mathrm{12}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{a}}_{\mathrm{i}\mathrm{21}}\left(\mathrm{x},\mathrm{y}\right)& {\mathrm{a}}_{\mathrm{i}\mathrm{22}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$

    

    each describing a desired magnification and movement behavior of the specified solid f_i and I being the identity matrix,

    - the vectors (c_i1(x,y), c_i2(x,y)), where 0 ≤ c _i1 (x, y), c_i2 (x, y) < 1, indicate in each case, for the solid f_i, the relative position of the center of the viewing elements within the cells i of the motif image,

    - the vectors (d_i1(x,y), d_i2(x,y)), where 0 ≤ d_i1 (x, y), d_i2 (x, y) < 1, each represent a displacement of the cell boundaries in the motif image, and

    - g_i(x,y) are mask functions for adjusting the visibility of the solid f_i. which are either identical to 1, or which define a strip-like or checkerboard-like alternation of the visibility of the solids f_i.
13. A depiction arrangement for security papers, value documents, electronic display devices or other data carriers, having a raster image arrangement for depicting a plurality (N≥1) of specified three-dimensional solids that are each given by n_i sections f_ij(x,y) and n_i transparency step functions t_ij(x,y), where i=1,2,...N and j = 1,2,... n_i, wherein, upon viewing with the eye separation being in the x-direction, the sections of the solid i each lie at a depth z_ij and wherein f_ij(x,y) is the image function of the j-th section of the i-th solid and indicates a brightness distribution, a color distribution, a binary distribution or another image properties, such as transparency, reflectivity, density or the like, and the transparency step function t_ij(x,y) is equal to 1 if, at the position (x,y), the section j of the solid i covers objects lying behind it, and otherwise is equal to 0, having

    - a motif image that is subdivided into a plurality of cells, in each of which are arranged imaged regions of the specified solids,

    - a viewing grid composed of a plurality of viewing elements for depicting the specified solids when the motif image is viewed with the aid of the viewing grid,

    - the motif image exhibiting, with its subdivision into a plurality of cells, an image function m(x,y) that is given by $$m\left(x,y\right)=F\left(h_{11},h_{12},\dots ,h_{1n_1},h_{21},h_{22},\dots ,h_{2n_2},\dots ,h_{N1},h_{N2},\dots ,h_{{\mathit{Nn}}_N}\right),$$

    

    having the describing functions $${\mathrm{h}}_{\mathrm{ij}}={\mathrm{f}}_{\mathrm{ij}}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{iK}}\\ {\mathrm{y}}_{\mathrm{iK}}\end{array}\right)\cdot g_{\mathrm{ij}}\left(x,y\right),$$

    

    where $$\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{iK}}\\ {\mathrm{y}}_{\mathrm{iK}}\end{array}\right)=\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathit{V}}_{\mathrm{ij}}\cdot \left(\left(\left(\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)+{\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)\right)\mathrm{modW}\right)-{\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)-{\mathrm{w}}_{\mathrm{ci}}\left(\mathrm{x},\mathrm{y}\right)\right)$$

    

    $${\mathrm{w}}_{\mathrm{di}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{d}}_{\mathrm{i}\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{d}}_{\mathrm{i}\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right)$$

    

    and $${\mathrm{w}}_{\mathrm{ci}}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{W}\cdot \left(\begin{array}{c}{\mathrm{c}}_{\mathrm{i}\mathrm{1}}\left(\mathrm{x},\mathrm{y}\right)\\ {\mathrm{c}}_{\mathrm{i}\mathrm{2}}\left(\mathrm{x},\mathrm{y}\right)\end{array}\right),$$

    

    wherein, for ij in each case, the index pair is to be taken for which ${\mathrm{t}}_{\mathrm{ij}}\left(\begin{array}{c}{\mathrm{x}}_{\mathrm{iK}}\\ {\mathrm{y}}_{\mathrm{iK}}\end{array}\right)$

    

    is not equal to zero and z_ij is minimal or maximal, and

    - wherein F(h ₁₁ , h ₁₂ ,...,h _{1n 1} , h ₂₁ ,h ₂₂ ,...,h _{2n 2} ,...,h _N1 ,h _N2 ,...,h_NnN ) is a master function that indicates an operation on the describing functions h_ij(x,y), and wherein

    - the unit cell of the viewing grid is described by lattice cell vectors ${\mathrm{w}}_1=$

    

    $\left(\begin{array}{c}{w}_{11}\\ w_{21}\end{array}\right)$

    

    and ${\mathrm{w}}_2=\left(\begin{array}{c}{w}_{12}\\ w_{22}\end{array}\right)$

    

    and combined in the matrix $\mathrm{W}=\left(\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& w_{22}\end{array}\right),$

    

    - the magnification terms V_ij are either scalars ${\mathrm{V}}_{\mathrm{ij}}=\left(\frac{{\mathrm{z}}_{\mathrm{ij}}}{\mathrm{e}}-\mathrm{1}\right),$

    

    where e is the effective distance of the viewing grid from the motif image, or matrices V_ij =(A_ij - I), the matrices ${\mathrm{A}}_{\mathrm{ij}}=\left(\begin{array}{cc}{\mathrm{a}}_{\mathrm{ij}\mathrm{11}}& {\mathrm{a}}_{\mathrm{ij}\mathrm{12}}\\ {\mathrm{a}}_{\mathrm{ij}\mathrm{21}}& {\mathrm{a}}_{\mathrm{ij}\mathrm{22}}\end{array}\right)$

    

    each describing a desired magnification and movement behavior of the specified solid f_i and I being the identity matrix,

    - the vectors (c_i1(x,y), c_i2(x,y)), where 0 ≤ c _i1 (x, y), c_i2 (x, y) < 1, indicate in each case, for the solid f_i, the relative position of the center of the viewing elements within the cells i of the motif image,

    - the vectors (d_i1(x,y), d_i2(x,y)), where 0 ≤ d_i1 (x, y), d_i2 (x, y) < 1, each represent a displacement of the cell boundaries in the motif image, and

    - g_i(x,y) are mask functions for adjusting the visibility of the solid f_i. which are either identical to 1, or which define a strip-like or checkerboard-like alternation of the visibility of the solids f_i.
14. The depiction arrangement according to one of claims 10 to 13, characterized in that at least one of the describing functions h_i(x,y) or h_ij(x,y) is designed as specified in claims 1 to 7 for the image function m(x,y), and/or that the raster image arrangement depicts an alternating image, a motion image or a morph image, and/or that the master function F constitutes the sum function, and/or that two or more three-dimensional solids f_i are visible simultaneously.
15. The depiction arrangement according to at least one of claims 1 to 14, characterized in that the viewing grid and the motif image are firmly joined together to form a security element having a stacked, spaced-apart viewing grid and motif image, or that the viewing grid and the motif image are arranged at different positions of a data carrier such that the viewing grid and the motif image are stackable for self-authentication and form a security element in the stacked state, especially that the viewing grid and the motif image are stackable by bending, creasing, buckling or folding the data carrier.
16. The depiction arrangement according to at least one of claims 1 to 15, characterized in that, to amplify the three-dimensional visual impression, the motif image is filled with Fresnel patterns, blaze lattices or other optically effective patterns, such as subwavelength patterns.
17. The depiction arrangement according to at least one of claims 1 to 16, characterized in that after the determination of the image function m(x,y), the image contents of individual cells of the motif image are interchanged.
18. The depiction arrangement according to at least one of claims 1 to 14 or 17, characterized in that the motif image is displayed by an electronic display device, and the viewing grid for viewing the displayed motif image is firmly joined with the electronic display device, or that the motif image is displayed by an electronic display device, and in that the viewing grid, as a separate viewing grid for viewing the displayed motif image, is bringable onto or in front of the electronic display device.
19. A security paper for manufacturing security or value documents, such as banknotes, checks, identification cards, certificates or the like, having a depiction arrangement according to at least one of claims 1 to 17.
20. A data carrier, especially a branded article, value document, decorative article or the like, having a depiction arrangement according to at least one of claims 1 to 17, wherein the viewing grid and/or the motif image of the depiction arrangement is preferably arranged in a window region of the data carrier.
21. An electronic display arrangement having an electronic display device, especially a computer or television screen, a control device and a depiction arrangement according to at least one of claims 1 to 14 or 17 to 18, the control device being designed and adjusted to display the motif image of the depiction arrangement on the electronic display device.

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