RU2794281C2 - Method of manufacturing light redirecting surface of caustic layer, optical protective element containing manufactured light redirecting surface of caustic layer, marked object, application and method of object authentication - Google Patents

Abstract

FIELD: imaging.

SUBSTANCE: method includes a discrete representation of the input target image, calculating a piecewise linear representation of the light-redirecting surface of the caustic layer by means of intersecting surface pieces. For a given set of vertex heights of the surface pieces, the corresponding set of light intensities that are focused by the caustic layer is calculated, and the vertex heights of the corresponding surface pieces that minimize the differences between the values of the calculated light intensities and the corresponding values of the target light intensities are calculated, a refractive transparent, or a partially transparent, or reflective light redirecting surface of the caustic layer having a relief pattern for redirecting incident light from the light source and for forming an image containing the caustic pattern is obtained.

EFFECT: creation of a fast, scalable, reliable and accurate manufacturing method and the possibility of visual object authentication using publicly available tools.

13 cl, 8 dwg

Description

The field of technology to which the invention belongs

The present invention relates to the field of manufacturing caustic optical elements, in particular to the manufacture of a refractive transparent or partially transparent light redirecting surface (or reflective light redirecting surface) of the caustic layer, as well as refractive or reflective optical security elements capable of projecting caustic patterns under acceptable illumination.

Background of the invention

There is a need for security features on objects that can be authenticated by the so-called "man on the street" using public means. These tools include the use of the five senses - mainly sight and touch - plus the use of widely used tools such as the mobile phone.

Some common examples of security features are expert fibers, filaments or foils (embedded in a substrate such as paper), watermarks, gravure or microprinting features (possibly printing on the substrate with optically variable inks) that can be found on banknotes, credit cards , identity cards, tickets, certificates, documents, passports, etc. These security features may include optically variable inks, invisible inks or luminescent inks (fluorescent or phosphorescent under suitable illumination from a particular excitation light source), holograms, and/or tactile features. The main aspect of a security feature is that it has some physical property (optical effect, magnetic effect, material structure or chemical composition) that is very difficult to fake, so an object marked with such a security feature can be reliably recognized as genuine if the specified property can be see or detect (visually or with a specific device).

However, when the object is transparent or partially transparent, these features may not be suitable. In fact, transparent objects often require that a security element having the desired security properties does not change its transparency or appearance for either aesthetic or functional reasons. Notable examples may include blisters and vials for pharmaceutical products. For example, a transparent window has recently been incorporated into the design of polymer and hybrid banknotes, thereby creating a need for security features that are compatible with it. While the transparency argument does not apply to reflective caustics, the approach naturally extends to reflective caustics, where, for example, the use case requires the appearance of polished, mirror-like surfaces to be preserved.

Most of the existing security features for documents, banknotes, security tickets, passports, etc. have not been specifically designed for transparent objects/areas and as such are not suitable for this application. Other features, such as those obtained with invisible and fluorescent paints, require special excitation and/or detection tools that may not be available to the "man on the street".

Translucent optically variable features (eg, liquid crystal coatings or latent images from surface structures) are known to provide such functionality. Unfortunately, markings incorporating such security features can usually be seen against a dark/uniform background to make the effect more noticeable.

Other known features are diffractive optical elements such as non-metallized surface holograms. The disadvantage of these features is that they show a very low contrast visual effect when viewed directly. Also, when used in conjunction with a monochromatic light source to project a pattern, they typically require a laser to produce a satisfactory result. In addition, a fairly accurate relative spatial arrangement of the light source, the diffractive optical element, and the user's eyes is required to provide a clearly visible optical effect.

For example, laser-engraved microtext and/or microcodes have been used for, for example, glass vials. However, they require expensive tools to implement them and a special magnifying tool to detect them.

Accordingly, in order to solve the problems of the prior art, many further attempts have been made to develop optical security elements suitable for transparent or partially transparent objects.

One possible approach is to implement a fabrication methodology that uses a caustic layer having a refractive transparent or partially transparent light redirecting surface or a reflective light redirecting surface, wherein the caustic layer has a relief pattern suitable for redirecting incident light received from a light source, and to form a projected image containing a caustic pattern. This approach naturally extends to the use of a reflective light-redirecting surface in the case of opaque objects with a specular surface.

This approach allows you to control the caustic pattern by forming the surface of the caustic layer. Computational tools based on light transfer have been developed to form almost any desired shape by optimizing (calculating) the geometry of the refractive or reflective surface of caustic optical elements, starting from the target image.

In an ideal workflow, the calculation of a suitable surface, starting from the target image, should be fast and applicable to a wide range of target images, should not be overly computationally intensive, and should not require user intervention beyond selecting and providing the target image.

Methods for calculating the caustic surface from a target image have been disclosed in the prior art.

Document EP2711745 A2 discloses sampling the generated surface into a mesh, which is then deformed to adjust the brightness of the corresponding area of the image. The normal field associated with the grid is then determined and integrated to find the corresponding caustic surface. However, for an arbitrary image, there is no guarantee that the corresponding normal field will be integrable, and additional measures must be taken to ensure the same.

EP2963464 A1 uses a similar approach to determine the optimal transport map (OTM) and similarly requires the calculation and integration of a normal field whose integrability is not guaranteed.

US9188783B2 and US2016041398 divide the generated surface into a set of micropatches, each of which is responsible for projecting the Gaussian caustic kernel, with the superposition of the kernels zooming in on the desired image. However, as also noted in EP2711745 A2, the disadvantages of the method are sampling artifacts and difficulty in resolving low intensity areas. The normal field also needs to be integrated.

In all these cases, the actual image projected by the calculated caustic surface is ultimately modeled using ray tracing; target figure image adjustment may be required when the caustic pattern does not approximate the target image with sufficient accuracy. This requires additional time and effort, and yet cannot guarantee the complete accuracy of the resulting caustic pattern.

Therefore, it is an object of the present invention to provide a method for manufacturing a refractive transparent or partially transparent light redirecting surface or a reflective light redirecting surface of a caustic layer that is fast, scalable, reliable and accurate. This allows for a significant reduction in the overall time required to transition from the target image to the corresponding surface, since fewer iterations are required due to corrections or adjustments, and the iterations are faster. It also reduces the overall time required for manufacturing.

Another goal of the present invention is to eliminate the step of calculating and integrating the normal field. Eliminating the need to integrate the normal field eliminates one of the major limitations and sources of inaccuracies of previously known methods.

Yet another object of the present invention is to reduce or eliminate user intervention beyond specifying the target image and accepting the resulting surface. Eliminating the need for user intervention greatly simplifies the implementation of the method in the context of a manufacturing workflow where specialist skills are not always available.

Yet another object of the present invention is to provide optical security elements capable of projecting caustic patterns under reasonable illumination, which are suitable for transparent or partially transparent objects and for reflective objects.

Another object of the present invention is to provide a marked object selected from the group consisting of consumer goods, documents of value and banknotes, which contains an optical security element.

Yet another object of the present invention is to provide a method for visually authenticating an object marked with an optical security element using public means.

Yet another object of the present invention is the use of an optical security element for authentication or anti-counterfeiting of an object selected from the group consisting of consumer goods, documents of value and banknotes.

Brief description of the invention

According to one aspect, the present invention relates to a method for manufacturing a refractive transparent or partially transparent light redirecting surface or a reflective light redirecting surface of a caustic layer suitable for redirecting incident light received from a light source and for forming a projected image comprising a caustic pattern, the method comprising computerized stages, in which:

в плоскости изображения, со связанными целевыми ненулевыми интенсивностями света {I_i}, i=1,…,N, распределенными в заданной области целевого изображения и соответствующими целевому каустическому узору целевого изображения;- provide a discrete representation of the input target image containing P a set of N image pixels p _i coordinates

in the image plane, with associated target non-zero light intensities {I _i }, i=1,...,N, distributed in a given area of the target image and corresponding to the target caustic pattern of the target image;

- calculate a piecewise linear representation of the light-redirecting surface z = F(x,y) of the caustic layer with a height z above the coordinate plane (x,y) based on the representation of the light-redirecting surface by intersecting surface pieces z = f _i (x,y), i=1,…,N, respectively obtained from the stationarity of the length of the optical path of the rays refracted or reflected by the caustic layer and focused at the points P(i) of the image plane of the coordinates (x _i ,y _i ), i=1,…,N, at each piece of the surface z = f _i (x,y) is a surface of revolution around the axis passing through the point P(i) with a vertex at the point (x _i ,y _i ,z _i ), height z _i = f _i (x _i ,y _i ), i=1,…,N, moreover, the piecewise linear representation of the light-redirecting surface associated with the respective values of the heights of the vertices N is formed by the envelope of the intersections of the corresponding N pieces of surfaces z = f _i (x, y), i= 1,…,N;

- for a given set of corresponding values of heights z ₁ ,…,z of _N vertices of N pieces of surfaces, a corresponding set of values of light intensities I(1),…,I(N) is calculated, which are respectively focused at points P(1),…,P( N) a caustic layer that redirects incident light through an associated piecewise linear light redirecting surface; And

- calculate the respective values of N heights z ₁ ,…,z _N N of the vertices of the respective N pieces of surfaces, which minimize the differences between the respective values of the calculated light intensities I(1),…,I(N) focused at the points P(1),… ,P(N) through the associated light-redirecting surface, and the corresponding corresponding values of the target light intensities I ₁ ,…,I _N ,

thereby obtaining a light redirecting surface having a relief pattern suitable for redirecting incident light received from the light source and forming a projected image containing the target caustic pattern of the target image.

According to the present invention, each surface piece z=f _i (x,y), i=1,…,N, can be approximated paraxially by expanding into a Taylor series of order k greater than or equal to two of the surface piece expression obtained from the stationarity of the optical length way.

In the above method, the step of calculating the heights z _i minimizing the differences between the calculated light intensities I(i) and the corresponding target light intensities I _i , for i=1,...,N, can be performed by a non-gradient optimization method. The advantage of these methods is the simplification of high computational costs.

Preferably, the step of calculating the heights z _i minimizing the differences between the calculated light intensities I(i) and the corresponding target light intensities I _i , for i=1,...,N, can be performed by the (limited capacity) power diagram method associated with the method optimization, which may be without a gradient, or preferably based on a gradient (in order to reduce computational costs).

In a further aspect of the present invention, a fabricated light redirecting surface is used to generate a machine-compatible representation for processing purposes, for example, using industry standard formats such as stereolithography (STL) or native graphics information exchange specification (IGES). In particular, the machine-compatible representation can be used to control a machining tool to process a light-redirecting surface of an optical material substrate or an intermediate substrate further used for mass production of caustic optical elements by copying.

In a further aspect of the present invention, the fabricated light-redirecting surface is the original light-redirecting surface to be used to create a replica of the light-redirecting surface. In this case, the method may further include copying the light-redirecting surface onto the substrate. Such copying may include one of the following: roll-to-roll printing, foil-to-foil printing, UV casting, and embossing.

In another aspect, the present invention provides an optical security element comprising at least one of a refractive or reflective light redirecting surface of the caustic layer, which is manufactured according to the method as described above. Said optical security element may mark an object selected from the group consisting of consumer goods, documents of value, identity cards, tax stamps and banknotes.

In another aspect, the present invention provides a marked object selected from the group consisting of consumer goods, documents of value and banknotes, which contains an optical security element as described above.

In another aspect, the present invention provides a method for visually authenticating an object marked with an optical security element by a user, the method comprising the steps of:

- illuminate the light-redirecting surface of the optical security element with a point light source at a distance d _s from the light-redirecting surface;

- visually observe the caustic pattern projected on the projection surface, at a distance d _i from the optical protective element; And

- decide whether the object is genuine, when assessing by the user whether the projected caustic pattern is visually similar to the reference image.

In yet another aspect, the present invention provides for the use of an optical security element, as described above, to authenticate or protect against forgery an object selected from the group including consumer goods, documents of value, identity cards, tax stamps, and banknotes.

Hereinafter, the present invention will be described more fully with reference to the accompanying drawings, which illustrate the main aspects and features of the present invention.

Brief description of the drawings

In FIG. 1 is a schematic illustration of a conventional optical configuration of a refractive optical security element for projecting a caustic image.

In FIG. 2 illustrates the caustic surface for one image pixel (image point).

In FIG. 3 illustrates a caustic surface for a plurality of image pixels (image points).

In FIG. 4 illustrates the target image.

In FIG. 5 illustrates a generic power diagram.

In FIG. 6 illustrates a view of the caustic surface.

In FIG. 7 illustrates a side view of an object in front of the detector (ray tracing geometry).

In FIG. Figure 8 illustrates the ray tracing simulation of a gray scale image at a distance of 40 mm from the object.

Detailed description

In optics, the term "caustic" refers to the envelope of light rays refracted or reflected by one or more surfaces, at least one of which is curved, as well as the projection of such light rays onto another surface. More specifically, a caustic is a curve or surface tangent to each light ray that defines the envelope boundary of the rays as a curve of concentrated light. For example, the light pattern formed by the sun's rays on the bottom of a pool is a caustic "image" or pattern formed by a single light-redirecting surface (the wavy air-water interface), while light passing through the curved surface of the water glass creates a spiky a pattern on a table on which liquid glass rests when it crosses two or more surfaces (eg air-glass, glass-water, air-water, etc.) that redirect its path.

In the following, the most common configuration will be used as an example, in which the (refractive) caustic layer of an optical security element is bound by a single curved surface or light redirecting surface, and a single flat surface will be used as an example, without limiting the more general cases. In this document, reference will be made to the more general term "caustic pattern" (or "caustic image") as a light pattern formed on a screen (projection surface) when the optical surface is suitably shaped (with a light redirecting surface having an acceptable relief pattern) The caustic layer redirects light from a source to divert it from some areas of the screen and concentrate it on other areas of the screen in a predetermined light pattern (ie, thus forming said "caustic pattern"). Redirection refers to the change in the path of light rays from the source in the presence of a caustic layer in relation to the path from the source to the screen in the absence of a caustic layer. A caustic layer (refractive or reflective) is thus a piece of optical material having a light redirecting surface with a relief pattern suitable for redirecting light received from a light source to form a caustic image. The optical security element according to the present invention comprises a caustic layer and may further comprise additional optical element(s) (eg a lens or support substrate) involved in light redirection.

In turn, a curved optical surface will mean a "relief pattern", and an optical element associated with this surface will mean a caustic layer. It should be noted that a caustic pattern can result from more than one curved surface and more than one object redirecting light, although perhaps at the cost of increased complexity. In addition, the relief pattern to create a caustic pattern should not be confused with a diffractive pattern (such as in security holograms).

The concept of the present invention can, for example, be applied to common objects such as consumer goods, ID/credit cards, banknotes, and so on. This requires a sharp reduction in the size of the optical security element and, in particular, bringing the relief depth of the relief pattern below acceptable values. An efficient workflow is especially useful for this purpose, as it allows for multiple design iterations until all operational constraints are met.

In this description, the term "relief" should be understood as the existence of a height difference (measured along the optical axis of the optical security element) between the highest point and the lowest point of the surface, by analogy with the difference in absolute height between the bottom of the valley and the top of the mountain (i.e. in peak-to-valley scale). Since the method of the present invention is not limited to a particular relief, for many applications it is assumed that the maximum depth of the relief pattern of an optical security element is generally less than or equal to 250 μm, or more preferably less than or equal to 30 μm, and exceeds the limit imposed by ultra-precise mechanical processing (UPM) and playback process, i.e. approximately 0.2 µm.

According to this description, the height difference between the highest and the lowest point in the relief pattern on the light-redirecting surface refers to the relief depth ε.

Under the caustic pattern (image), forming the approximation of a digital image, should be understood as a light pattern projected by the optical security element when illuminated by a suitable point source. As mentioned above, an optical security element should be understood as a plate of refractive material responsible for creating a caustic image.

The light redirecting surface(s) is(are) the surface (or surfaces) of the caustic layer (optical security element) responsible for redirecting incoming light from the source to the screen or (pre-flat) projection surface where the caustic pattern is formed .

An optical material substrate used for manufacturing an optical (protective) member is a raw material substrate whose surface is specially formed to have a relief pattern and thus form a light redirecting surface. In the case of a reflective light-redirecting surface, the optical material substrate is not necessarily homogeneous or transparent; the same applies in the case of an original surface used only for further copying. For example, the material may be opaque to visible light, and the reflectivity may then be obtained by conventional plating of the formed surface. In the case of a refractive light-redirecting surface, the source material substrate is transparent (or partially transparent) and homogeneous with a refractive index n (for photons of the spectrum visible to the human eye), and by the corresponding light redirecting surface is meant "a refractive transparent or partially transparent light redirecting surface with refractive index n".

The original light-redirecting surface according to this description is the first physical implementation of the light-redirecting surface from the computed surface. It can be copied into multiple copies (stamps), which are then used for serial copying.

The point source used in this description is a light source whose angular size (from the point of view of the optical security element) is small enough that the light can be considered to come from a single point at a distance d _s from the light redirecting surface. Experience has shown that this means that the amount of: (source diameter) xd _i /d _s , is less than the desired resolution (e.g. 0.05-0.1 mm) of the target caustic pattern in the projected image on the projection surface at a distance d _i from redirecting light surface (see Fig. 1). The screen should be understood as the surface onto which the caustic pattern is projected. The distance between the source and the light-directing surface also refers to the source distance d _s , and the distance between the light-directing surface and the screen refers to the image distance d _i .

The term "stamp" (or copy stamp when necessary to disambiguate) is primarily used to refer to the physical object bearing the profile of the light redirecting surface that is used for serial copying. This can be used, for example, to create a copy of the original light-redirecting surface (original relief reproduced by embossing or injecting from an original bearing the corresponding inverted relief). For the tool used to machine the relief pattern of the light-redirecting surface, the term "machining tool" is used for disambiguation.

In FIG. 1 is a schematic illustration of a conventional optical configuration of a refractive optical security element for projecting a caustic image. An optical security element (1) containing a caustic layer having a refractive surface redirects light from a point source S and projects it onto a suitable screen (3), which can be any surface of any object, etc., where a meaningful image is formed, as shown in FIG. 1. The particular design of the light-redirecting surface may allow a (recognizable) caustic pattern to be projected onto a curved surface. The image may be, for example, a logo, picture, number, or any other information that may be relevant to a particular context. Preferably, the screen is a flat projection surface or a flat part of any object.

The configuration in Fig. 1 shows that light from a source S is redirected by an optical surface of a suitable shape having a relief pattern (2). This general idea is known, for example, from reflective surfaces for car headlights, reflectors and lenses for LED lighting, optical systems in laser optics, projectors and cameras. However, in general, the goal is to convert a non-uniform distribution of light into a uniform distribution. On the contrary, the aim of the present invention is to obtain a non-uniform light pattern, i.e. a caustic pattern, which (approximately) reproduces some areas of relative brightness of the reference image (as represented in the (digital) reference image). If the illuminated relief pattern (2) of the optical element makes it possible to form a caustic pattern (4) on the screen (3), reproducing with sufficient quality (possibly differing in the overall intensity scaling factor) a known reference image, then a person visually observing the caustic pattern on the screen will easily see whether it is a true reproduction of the reference image, and if the caustic pattern is sufficiently similar to the reference image, the object marked by the optical security element is considered to be (most likely) genuine.

According to the embodiment shown in FIG. 1, light rays from a light source S, which is a point source according to this example, are propagated onto a (refractive) optical security element 1 (caustic layer) at a source distance _ds with a light redirecting surface having a relief pattern (2). The optical protective element in this case is made of a transparent or partially transparent homogeneous material with a refractive index n. The caustic pattern (4) is projected on the screen (3) at an image distance d _i from the light-redirecting surface of the optical security element (1). The authenticity of an optical security element (and hence of an object marked by that security element) can be directly assessed by visually checking the degree of similarity between the projected caustic pattern and the reference image.

Preferably, the relief pattern (2) is calculated based on the particular target digital image. From this computed relief pattern, a corresponding physical relief pattern can be created on the surface of a suitable optical material substrate (e.g., a transparent or partially transparent material with a refractive index n or a reflective surface of an opaque material) using UPM. In the case of mechanical processing of a relief on the surface of a substrate of an opaque optical material to form a reflective surface, good reflectivity will be obtained either by suitable properties of the material itself or by an additional conventional operation of depositing a thin layer of metal (metallization) on the relief. UPM uses diamond machining tools and nanotechnology to achieve very high precision so that tolerances can reach the “submicron” level or even the “nanoscale” level. In contrast, "high precision" in conventional machining means micron tolerances in single digits. Other potentially suitable technologies for creating a physical relief pattern on a surface are laser ablation and grayscale lithography. As is known in the field of microfabrication, each of these technologies has its own strengths and weaknesses in terms of cost, accuracy, speed, resolution, etc.

A suitable substrate of optical material for the refractive light-redirecting optical element should be optically transparent, transparent or at least partially transparent and mechanically stable. Generally, a transmittance T ≥ 50% is preferred and T ≥ 90% is most preferred. Alternatively, low haze H ≤ 10% can be used, but H ≤ 3% is preferred and H ≤ 1% is most preferred. The optical material must also behave correctly during the machining process to ensure a smooth and defect-free surface. An example of a suitable substrate is an optically transparent PMMA plate (also known under the commercial names Plexiglas, Lucite, Perspex, etc.). For reflective caustic redirecting optical elements, a suitable substrate of optical material must be mechanically stable and must be able to be given a mirror surface. An example of a suitable substrate is a metal, such as those used to make original rifled gratings and laser mirrors, or a non-reflective substrate, which can be further metallized.

For large-scale production, further steps are required to create a stamp and serially copy the optical security element on the target object. A suitable process for creating a stamp from an original is, for example, electroforming. Suitable processes for serial copying are, for example, hot stamping of a polymer film or molding of a photopolymer under the influence of UV radiation, and these can be carried out in the printing process either from roll to roll or from foil to foil. For serial copying purposes, neither the original nor the resulting stamp need be optically transparent, so opaque materials (particularly metals) can also be used, even when the final product is a refractive optical element. However, in some cases it may be advantageous for the original to be transparent as it allows the quality of the caustic image to be checked before proceeding with embossing and serial copying.

A critical aspect of using optical elements (with a light redirecting surface having an embossed pattern) as security features is their physical scale, which must be compatible with the target object, and the optical configuration required to project a caustic image.

As a rule, the maximum lateral dimension is limited by the overall size of the object and can usually vary from a few cm to less than 1 cm in less favorable cases. For certain applications, such as, for example, banknotes, the target overall thickness may be extremely thin (on the order of 100 μm or less). In addition, allowable thickness variations (embossing) are even smaller for a number of reasons, including mechanical limitations (weak points associated with thinner areas) and operational considerations (e.g. when stacking banknotes, the stack will be convex due to the thicker portion of the paper, which complicates banknote handling and storage). Typically, for a banknote with an overall thickness of approximately 100 µm, the target thickness for the embossed pattern of the optical security element to be included in that banknote may be approximately 30 µm. For a credit card or ID card with a thickness of approximately 1 mm, the target thickness for the optical security element relief pattern to be included in that credit card/ID card will be less than about 400 μm, and preferably no more than about 250 μm.

In addition, the distance between the source and the image is usually limited by user convenience to a few tens of centimeters. Notable exceptions are the sun or a spotlight mounted on the ceiling, which are, however, less accessible under certain circumstances. In addition, the ratio d _s /d _i between the two distances is typically greater than 5-10 in order to obtain a sharper image (and good contrast) that is easier to recognize. In addition, the ratio d _s /d _i ≥ 5, together with a light source S that is preferably a point light (for example, the lighting LED of a conventional mobile phone), makes it possible to consider that the light source is actually approximately "infinitely distant" and thus the projection surface at only approximately a focal length from the optical security element will be suitable for clearly viewing the projected caustic pattern. As a consequence, conditions of good visual observation by the user do not require too strict relative spatial arrangement of the light source, the optical security element, and the user's eyes.

In general, thickness and relief are among the most important parameters. Given an arbitrary target image (reference image) and an optical geometry configuration (i.e. geometric lighting/observation conditions of the projected caustic pattern), there is no guarantee that the computed optical surface will have a relief pattern below a given limit. In fact, in general, the opposite is likely to happen: this is especially true for the severe restrictions placed on the optical security elements described above. The proposed method does not automatically take into account these additional restrictions. However, since it can quickly calculate the caustic surface corresponding to a given input image, it allows multiple iterations in the image design until the rendered terrain is constrained compatible. On the contrary, traditional methods, which require large computational resources, usually not only do not automatically take into account the limitations, but also impose serious restrictions on the iterative manufacturing process.

Although only the configuration of a transmissive caustic optical element is described in this document, the same considerations can be applied to a reflective configuration with slight modifications (in particular with respect to the application of Fermat's principle).

осью, выровненной с оптической осью (перпендикулярно каустическому оптическому элементу) и направленной от источника к изображению. Для иллюстрации концепции настоящего изобретения рассмотрен простой каустический оптический элемент «плоско-выпуклого» типа (фиг. 2-3), который освещается пучком (по существу) параллельных лучей. Распространение на случай источника света на конечном расстоянии является простым путем добавления линзообразного оптического элемента, преобразующего источник на конечном расстоянии в виртуальный источник на бесконечности. Функция линзообразного элемента в конечном итоге может быть реализована непосредственно в каустическом оптическом элементе. Оси

и

, таким образом, лежат на плоскости каустического оптического защитного элемента (которая параллельна входной грани каустического оптического элемента). Каустическая поверхность математически описана скалярной функцией

, определяющей расстояние

поверхности от эталонной плоскости

в точке координат

каустического оптического элемента. Для удобства дальнейшего описания эта плоскость может быть расположена на задней поверхности каустического оптического элемента, и в этом случае

равна толщине каустического оптического элемента (см. фиг. 2). В примере, показанном на фиг. 2, эта плоскость параллельна плоскости каустического изображения.For purposes of description, it is convenient to define a Cartesian frame of reference with

an axis aligned with the optical axis (perpendicular to the caustic optical element) and directed from the source to the image. To illustrate the concept of the present invention, a simple "plano-convex" type caustic optical element (FIGS. 2-3) is considered, which is illuminated by a beam of (essentially) parallel beams. Extending to the case of a light source at a finite distance is simple by adding a lenticular optical element that converts the source at a finite distance into a virtual source at infinity. The function of the lenticular element can eventually be implemented directly in the caustic optical element. axes

And

, thus lie on the plane of the caustic optical security element (which is parallel to the input face of the caustic optical element). The caustic surface is mathematically described by the scalar function

, which determines the distance

surface from the reference plane

at the point of coordinates

caustic optical element. For the convenience of further description, this plane can be located on the rear surface of the caustic optical element, and in this case

equal to the thickness of the caustic optical element (see Fig. 2). In the example shown in FIG. 2, this plane is parallel to the plane of the caustic image.

, определяющей интенсивность света в точке (или пикселе) координат

на плоскости изображения.Similarly, the caustic image is described by the scalar function

, which determines the intensity of light at a point (or pixel) of coordinates

on the image plane.

It should be noted that the use of Cartesian coordinates is a matter of convenience, and other systems may also be used instead (for example, in the case where the caustic surface is part of or supported by a curved object). Similarly, the back surface of a caustic optical element does not have to be flat, although obviously this must be taken into account in the calculations.

каустического изображения пучок лучей малого поперечного сечения, сходящийся на ней, прошел пути одинаковой оптической длины. Обычно рельефный узор перенаправляющей свет поверхности каустического слоя имеет глубину рельефа ε, очень малую по сравнению с расстоянием d между каустическим слоем и плоскостью изображения, на которой формируется каустическое изображение (см. фиг. 2): действительно, как правило, полученное в результате значение ε меньше 300 мкм, в то время как d больше 5 см (таким образом, ε/d <6 10^-3), глубина рельефа ε определяется как разность высот между самой высокой и самой низкой точкой рельефного узора. Общая толщина каустического слоя равна (e+ε), где e - толщина однородной части оптического материала каустического слоя. Как правило, толщина e также очень мала по сравнению с расстоянием наблюдения d, т.е. обычно e меньше миллиметра (таким образом, e/d ≤ 2 10^-2, и (e+ε)/d ≤ 2.6 10^-2). Однако слой толщиной е, соответствующий распространению входящих световых лучей внутри каустического слоя в виде простых параллельных лучей, не влияет на разность оптического пути и, таким образом, не учитывается. Рассматривая каустический слой, проиллюстрированный на фиг. 2, для источника света, расположенного в бесконечности (для простоты, получают параллельные входящие лучи),

, учитывают разность длины оптического пути

между (i) длиной оптического пути

прямого луча, входящего в грань плоскости (на уровне z=0) каустического слоя, в точке (x₀,y₀), проходящего через каустический слой (каустический оптический элемент) преломляющего индекса n до точки (x₀,y₀) на уровне z₀ перенаправляющей свет поверхности уравнения z = F(x,y), и достигающего фокусной точки (x₀,y₀) плоскости изображения, и (ii) длиной оптического пути

луча, входящего в грань плоскости (на уровне z=0) каустического слоя, в точке (x,y), ближайшей к точке (x₀,y₀), проходящего через каустический слой до точки (x,y) на уровне z перенаправляющей свет поверхности, и отклоняемого в точку (x₀,y₀) плоскости изображения. Если r является расстоянием между точками (x₀,y₀) и (x,y), т.е.

, то имеют:The present invention exploits the property that light propagates along paths of constant optical length, where the optical path length is a local extremum with respect to any small path change (Fermat's principle). For any given point

caustic image, a beam of rays of small cross section, converging on it, has traveled paths of the same optical length. Typically, the relief pattern of the light-redirecting surface of the caustic layer has a relief depth ε very small compared to the distance d between the caustic layer and the image plane on which the caustic image is formed (see FIG. 2): indeed, as a rule, the resulting value of ε less than 300 μm, while d is greater than 5 cm (thus ε/d <6 10 ^-3 ), the relief depth ε is defined as the height difference between the highest and lowest point of the relief pattern. The total thickness of the caustic layer is equal to (e+ε), where e is the thickness of the homogeneous part of the optical material of the caustic layer. As a rule, the thickness e is also very small compared to the observation distance d, i.e. usually e is less than a millimeter (thus e/d ≤ 2 10 ^-2 , and (e+ε)/d ≤ 2.6 10 ^-2 ). However, a layer of thickness e, corresponding to the propagation of incoming light rays inside the caustic layer as simple parallel rays, does not affect the optical path difference and is thus not taken into account. Considering the caustic layer illustrated in FIG. 2, for a light source located at infinity (for simplicity, get parallel incoming beams),

, take into account the difference in the length of the optical path

between (i) optical path length

of a direct beam entering the face of the plane (at the level z=0) of the caustic layer, at the point (x ₀ ,y ₀ ), passing through the caustic layer (caustic optical element) of refractive index n to the point (x ₀ ,y ₀ ) at the level z ₀ of the light redirecting surface of the equation z = F(x,y), and reaching the focal point (x ₀ ,y ₀ ) of the image plane, and (ii) the length of the optical path

of the ray entering the face of the plane (at the level z=0) of the caustic layer, at the point (x,y), closest to the point (x ₀ ,y ₀ ), passing through the caustic layer to the point (x,y) at the level z redirecting surface light, and deflected to a point (x ₀ ,y ₀ ) of the image plane. If r is the distance between the points (x ₀ ,y ₀ ) and (x,y), i.e.

, then they have:

, и, таким образом, при решении квадратного уравнения в z, получают:According to Fermat's principle, they must have

, and thus, when solving the quadratic equation in z, one gets:

имеют

Таким образом, указанное z = f₀(x,y), обозначающее локальное представление поверхности z = F(x,y) (т.е. вокруг точки (x₀,y₀)), и z₀ = f₀(x₀,y₀), представляющее собой расчет на вершине, можно написать следующее:where, due to

have

Thus, the specified z = f ₀ (x,y), denoting a local representation of the surface z = F(x,y) (i.e. around the point (x ₀ ,y ₀ )), and z ₀ = f ₀ (x ₀ ,y ₀ ), which is the calculation at the top, you can write the following:

which is a surface of revolution around the z-axis with a vertex at (x ₀ ,y ₀ ,z ₀ ).

Therefore, if instead of the focal point (x ₀ ,y ₀ ) we consider any of the focal points (x _i ,y _i ) on the image plane (i=1,…,N), we can determine the local one (i.e. with the vertex at the point (x _i ,y _i )) approximation F(x,y) as follows:

и

. Следовательно, функция F(x, y), задающая общую форму перенаправляющей свет поверхности каустического слоя, может быть локально представлена в соответствии с вышеупомянутой стационарностью оптического пути кусочно-линейной поверхностью, являющейся огибающей, полученной в результате пересечения кусков поверхностей с «функциями элементарной формы» z = f_i(x,y) вокруг вершин (x_i,y_i), соответствующих заданным точкам (x_i,y_i), i=1,…,N, на плоскости изображения.Where

And

. Therefore, the function F(x, y), which defines the general shape of the light-redirecting surface of the caustic layer, can be locally represented in accordance with the aforementioned stationarity of the optical path by a piecewise linear surface, which is an envelope obtained as a result of the intersection of pieces of surfaces with "elementary shape functions" z = f _i (x,y) around the vertices (x _i ,y _i ) corresponding to the given points (x _i ,y _i ), i=1,…,N, on the image plane.

d, и, таким образом,

, это локальное представление каустической поверхности может дополнительно приближаться вблизи от (x_i,y_i) с помощью первых нескольких ненулевых членов разложения в ряд Тейлора выражения в пределах квадратных скобках:The present invention further follows from the observation that, in the paraxial approximation, i. where r

d, and thus

, this local representation of the caustic surface can be further approximated near (x _i ,y _i ) using the first few non-zero terms of the Taylor series expansion of the expression within square brackets:

For example, if we consider the local approximation f _i (x,y) F(x,y) around the point (x _i ,y _i ) and take into account only the first nonzero term of the Taylor series expansion, we obtain a simplified approximation of the local representation:

, как показано на фиг. 3, с «высотой»

относительно плоскости (x,y) в z = 0 и соответствующей вершине параболоида (пространственных координат (x_i,y_i,z_i)).which describes a paraboloid of revolution with an axis centered at

, as shown in FIG. 3, with "height"

relative to the plane (x,y) at z = 0 and the corresponding vertex of the paraboloid (spatial coordinates (x _i ,y _i ,z _i )).

For the Taylor series expansion up to the next non-zero order (k = 4), we obtain an approximation of the local representation around the point (x _i ,y _i ) as follows:

When considering the piecewise linear approximation F(x,y) given by the approximation up to the first non-zero order of the local representation f _i (x, y), the intersection of two such (circular) paraboloids having respectively a height above the point (x _i ,y _i ) and the height z _j above the adjacent point (x _j ,y _j ), usually defines a parabola in the plane perpendicular to the straight line connecting the two points (x _i ,y _i ) and (x _j ,y _j ). Thus, for a set of points {(x _i ,y _i ), i=1,…,N} of the image plane and a corresponding set of heights {z _i , i=1,…,N} of paraboloid vertices, respectively associated with the specified points, the resulting (outer) envelope of the intersection of these paraboloids (defining the piecewise linear light-redirecting surface) is formed by portions of the paraboloids bounded by sharp parabolic curves. These curves can be computed by solving simple second order algebraic equations. In the case of expansion into a Taylor series of order k = 4 or higher, the corresponding “elementary form functions” z = f _i (x, y) are more complex than simple paraboloids, and the calculation of lines of intersection of pieces of surfaces (when setting different heights of their vertices) becomes more laborious.

In the example shown in FIG. 3, the incident parallel rays illuminate the flat (inlet) face z = 0 of the caustic layer with a uniform intensity I ₀ , and thus, for a given piecewise linear approximation of the light redirecting surface z = F(x,y), i.e. for a given set of N vertices (x _i ,y _i ,z _i ) and the corresponding functions of the elementary form f _i (x, y), i=1,…,N, the contribution to the intensity I(j) at the point (x _j ,y _j ) the image plane from the envelope of the intersections of elementary pieces of the surface can be mathematically described as follows:

using the "trace function" (i,j belong to {1,…,N}):

, и где интегралы взяты по опорному домену каустического элемента (т.е. «окно» или область сбора света). Стоит обратить внимание, что в принципе нет никаких конкретных ограничений на форму и/или размер окна. Однако простые геометрические формы, компактные формы и выпуклые формы являются преимущественными для вычислительных и практических целей.the function H[X] is the usual Heaviside step function defined by

, and where the integrals are taken over the reference domain of the caustic element (i.e., the "window" or light collection area). It is worth noting that in principle there are no specific restrictions on the shape and/or size of the window. However, simple geometric shapes, compact shapes, and convex shapes are advantageous for computational and practical purposes.

Thus, the expression for the piecewise linear approximation (for a given number N of image points (x _i ,y _i ), i=1,…,N) of the representation of the caustic surface z = F(x,y) has the following form:

² минимизируется. After obtaining a piecewise linear approximation of the light-redirecting surface z = F(x,y) (for a given set of N vertices), it is necessary to estimate the corresponding light intensity distribution I(i), i=1,…,N at the selected corresponding points (x _i ,y _i ),i=1,…,N, of the image plane, and evaluate the difference for each target point (x _i ,y _i ) between I(i) and the given (target) intensity I _i at the same point corresponding to the target caustic pattern to be reproduced. Thus, the heights z _i , i=1,…,N of the vertices are iteratively specified in such a way that the sum

² is minimized.

(и возможно с границей окна каустического слоя). В случае, если параболоид (j) полностью маскируется по меньшей мере одним параболоидом (i) (т.е. если z_i является достаточно большим относительно z_j), интенсивность I(j) равна нулю. Как упомянуто выше, контур пересечения двух параболоидов (i) и (j) представляет собой параболу в плоскости, перпендикулярной прямой, соединяющей две точки (x_i,y_i) и (x_j,y_j), при этом данная плоскость параллельна оптической оси вдоль z: пересечение этой плоскости с плоскостью (x, y) при z = 0 определяет прямой отрезок. При рассмотрении пересечений параболоида (j) с соседними параболоидами (i) соответствующие прямые отрезки на плоскости z=0 очерчивают выпуклую многоугольную ячейку Ω_j. Ясно, что интенсивность света I(j), доставляемая в точку (x_j,y_j) плоскости изображения, является только результатом входящих (однородных) параллельных лучей, собранных ячейкой Ω_j, и, таким образом, доставляемая интенсивность I(j) пропорциональна области a(j) ячейки Ω_j. Конечно, сумма всех областей ячеек, связанных с огибающей всех пересекающихся параболоидов, должна быть равна полной области A (на плоскости z=0) сбора входящих световых лучей (окна):

Это ограничение учитывается путем выбора приемлемой нормализации при (итеративной) минимизации суммы

². Каждый раз при изменении относительных разностей между высотами вершин параболоидов (путем увеличения или уменьшения хотя бы одной из N высот), области ячеек изменяются соответствующим образом: изменение высоты вершин, таким образом, эквивалентно изменению областей ячеек. При изменении высот z_i и z_j соответственных вершин двух параболоидов, соответствующих двум смежным точкам (x_i,y_i) и (x_j,y_j), например, путем изменения z_j в z_j + δz_j (другие высоты остаются неизменными), отрезок границы между ячейкой Ω_i (относящейся к параболоиду (i)) и ячейкой Ω_j (относящейся к параболоиду (j)) будет перемещаться в сторону ячейки Ω_i, если δz_j является положительным (т.е. область a(i) уменьшается), и будет перемещаться в сторону ячейки Ω_j, если δz_j является отрицательным (т.е. область a(i) увеличивается). Более того, поскольку интенсивности пропорциональны областям ячеек, минимизация суммы S эквивалентна минимизации суммы

², где a_i является значением области, соответствующим целевой интенсивности I_i, i=1,…,N. Область a(i) можно рассматривать как параметр, связанный с ячейкой Ω_i, и изменение высот вершин параболоидов эквивалентно изменению параметров ячеек, формирующих разбиение области A. Область a(j) является результатом пересечения параболоидов и может быть рассчитана с помощью вышеупомянутой функции трассировки как (интегрирование осуществляют по плоскости (x,y) области A):For example, if local pieces of surfaces f _j (x, y) are approached by the main term of the expansion in a Taylor series, i.e. paraboloids, non-zero intensity I(j) at the point (x _j ,y _j ) on the image, the plane arises only from what is left of the vertex paraboloid (x _j ,y _j ,z _j ), i.e. paraboloid (j), after crossing with the rest of the paraboloids forming a piecewise linear surface F having the corresponding vertices (x _i ,y _i ,z _i ),

(and possibly with the border of the caustic layer window). In case the paraboloid (j) is completely masked by at least one paraboloid (i) (ie if z _i is sufficiently large relative to z _j ), the intensity I(j) is zero. As mentioned above, the contour of the intersection of two paraboloids (i) and (j) is a parabola in a plane perpendicular to the line connecting the two points (x _i ,y _i ) and (x _j ,y _j ), while this plane is parallel to the optical axis along z: the intersection of this plane with the plane (x, y) at z = 0 defines a line segment. When considering the intersections of the paraboloid (j) with neighboring paraboloids (i), the corresponding straight segments on the plane z=0 outline a convex polygonal cell Ω _j . It is clear that the light intensity I(j) delivered to the point (x _j ,y _j ) of the image plane is only the result of the incoming (homogeneous) parallel rays collected by the cell Ω _j , and thus the delivered intensity I(j) is proportional to area a(j) of the cell Ω _j . Of course, the sum of all cell areas associated with the envelope of all intersecting paraboloids must be equal to the total area A (on the z=0 plane) of collecting incoming light rays (windows):

This limitation is taken into account by choosing an acceptable normalization in the (iterative) minimization of the sum

² . Each time the relative differences between the heights of the vertices of the paraboloids change (by increasing or decreasing at least one of the N heights), the cell regions change accordingly: changing the height of the vertices is thus equivalent to changing the cell regions. When changing the heights z _i and z _j of the corresponding vertices of the two paraboloids corresponding to two adjacent points (x _i ,y _i ) and (x _j ,y _j ), for example, by changing z _j to z _j + δz _j (other heights remain unchanged ), the segment of the boundary between the cell Ω _i (related to the paraboloid (i)) and the cell Ω _j (related to the paraboloid (j)) will move towards the cell Ω _i if δz _j is positive (i.e., the area a(i ) decreases), and will move towards the cell Ω _j if δz _j is negative (i.e., the area a(i) increases). Moreover, since the intensities are proportional to the cell areas, minimizing the sum S is equivalent to minimizing the sum

² , where a _i is the area value corresponding to the target intensity I _i , i=1,…,N. The region a(i) can be considered as a parameter associated with the cell Ω _i , and changing the heights of the vertices of the paraboloids is equivalent to changing the parameters of the cells that form the partition of the region A. The region a(j) is the result of the intersection of the paraboloids and can be calculated using the above tracing function as (integration is carried out over the plane (x, y) of region A):

ограничение является

, а функция потерь является

². Приближение перенаправляющей свет поверхности описано следующим образом.The above reasoning on the example of paraboloid surfaces remains valid even if the expression for the surface piece, directly derived from the stationarity of the optical path length, does not approach or approximates by the Taylor expression to any (even) order k > 2 (since the resulting expression still describes the surface of revolution): at step n of the iteration of the minimization operation, the set of values {z _i ⁽ⁿ⁾ , i=1,…,N} defines the set of cells {Ω _i ⁽ⁿ⁾ , i=1,…,N} representing the intersections of the surfaces of N pieces {z = f _i ⁽ⁿ⁾ (x,y), i=1,…,N} and the corresponding set of cell areas {a ⁽ⁿ⁾ (i), i=1,…,N}, where

limitation is

, and the loss function is

² . The approximation of the light redirecting surface is described as follows.

² можно осуществлять согласно любому известному методу минимизации, такому как, например, симплекс-метод Недлера-Мида (без использования производных) (J.A. Nelder and R. Mead, “A simplex method for function minimization”, The Computer Journal, vol.7 (4), 1965 г., стр. 308-313). Конечно, можно использовать другие методы оптимизации без использования производных, например, метод покоординатного спуска (см. Stephen J. Wright, “Coordinate Descent Algorithms”, Mathematical Programming, vol.151 (1), июнь 2015 г., стр. 3-34) или метод многоуровневого координатного поиска («MCS») (см. W. Huyer and A. Neumaier, “Global Optimization by Multilevel Coordinate Search”, Journal of Global Optimization, vol.14 (4), июнь 1999 г., стр. 331-355).Functional minimization process (i.e. loss function)

² can be carried out according to any known minimization method, such as, for example, the Nedler-Mead simplex method (without the use of derivatives) (JA Nelder and R. Mead, “A simplex method for function minimization”, The Computer Journal, vol.7 ( 4), 1965, pp. 308-313). Of course, other nonderivative optimization methods can be used, such as the coordinate descent method (see Stephen J. Wright, “Coordinate Descent Algorithms”, Mathematical Programming, vol.151 (1), June 2015, pp. 3-34 ) or Multilevel Coordinate Search (“MCS”) method (see W. Huyer and A. Neumaier, “Global Optimization by Multilevel Coordinate Search”, Journal of Global Optimization, vol.14 (4), June 1999, p. 331-355).

According to the present invention, and with the aforementioned piecewise linear representation of the light-redirecting surface, the technical problem is to calculate the light-redirecting surface of a caustic layer that is suitable for redirecting incident light received from a light source to form a projected image containing a given caustic pattern (i.e., a given distribution of non-zero light intensity) of the target image is solved as follows:

, i = 1,…N, в плоскости изображения, со связанными ненулевыми целевыми интенсивностями света {I_i}, распределенными в заданной области целевого изображения и соответствующими целевому каустическому узору целевого изображения;- provide a discrete representation of the input target image containing P a set of N image pixels p _i coordinates

, i = 1,...N, in the image plane, with associated non-zero target light intensities {I _i } distributed in a given region of the target image and corresponding to the target caustic pattern of the target image;

- calculate a piecewise linear light-redirecting surface z = F(x,y) of the caustic layer with a height z above the coordinate plane (x,y) based on the representation of the light-redirecting surface by intersecting pieces of surfaces f _i (x, y), i=1 ,…,N, respectively obtained from the stationarity of the length of the optical path of the rays refracted or reflected by the caustic layer and focused at the points P(i) of the image plane of the coordinates (x _i ,y _i ), i=1,…,N, with each piece surface z = f _i (x,y) is a surface of revolution around an axis passing through the point P(i) with a vertex at the point (x _i ,y _i ,z _i ), height z _i = f _i (x _i ,y _i ), i=1,…,N, wherein the piecewise-linear light-directing surface associated with the respective values of the heights of the vertices N is formed by the envelope of the intersections of the corresponding N pieces of surfaces;

- for a given set of corresponding values of heights z ₁ ,…,z of _N vertices of N pieces of surfaces, a corresponding set of values of light intensities I(1),…,I(N) is calculated, which are respectively focused at points P(1),…,P( N) a caustic layer that redirects incident light through an associated piecewise linear light redirecting surface; And

- calculate the respective values of N heights z ₁ ,…,z _N N of the vertices of the respective N pieces of surfaces, which minimize the differences between the respective values of the calculated light intensities I(1),…,I(N) focused at the points P(1),… ,P(N) through the associated light-directing surface, and the corresponding corresponding values of the target light intensities I ₁ ,…,I _N .

. Затем метод осуществляет последовательность преобразований рабочего симплекса S, направленную на уменьшение значений функции потерь в его вершинах. На каждом этапе преобразование определяется путем вычисления одной или более контрольных точек вместе с их значениями функции потерь и сравнения этих значений функции потерь со значениями в текущих вершинах с целью замены наихудшей вершины, т. е. одной с наибольшим значением функции потерь, лучшей вершиной. Контрольные точки можно выбрать в соответствии с одной из четырех эвристик: (1) отражение или (2) расширение от худшей вершины; или (3) сжатие или (4) сжатие в сторону лучшей вершины (вершин). Минимизация завершается, когда рабочий симплекс S становится достаточно малым или когда значения функции потерь в вершинах становятся достаточно близкими. Посредством четырех эвристических преобразований алгоритм Нелдера-Мида обычно требует только одного или двух вычислений функции на каждом шаге, в то время как многие другие методы прямого поиска используют как минимум N вычислений функции потерь. Интуитивное объяснение алгоритма Нелдера-Мида приведено в (Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 10.5. Downhill Simplex Method in Multidimensions". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8): «Симплекс-метод спуска с вершины теперь включает серию шагов, большинство из которых просто перемещают точку симплекса, где функция наибольшая («самая высокая точка»), через противоположную сторону симплекса в нижнюю точку. Эти шаги называются отражениями, и они построены так, чтобы сохранить объем симплекса (и, следовательно, сохранить его невырожденность). Когда это возможно, метод расширяет симплекс в том или ином направлении, чтобы сделать более крупные шаги. Когда он достигает «дна долины», метод сжимается в поперечном направлении и пытается просочиться вниз по долине. Если возникает ситуация, когда симплекс пытается «пройти сквозь игольное ушко», он сжимается во всех направлениях, втягиваясь вокруг своей самой нижней (лучшей) точки».For example, when minimizing the loss function Σ using the Nelder and Mead simplex method, optimization begins with a set of N+1 points Q(1),…,Q(N+1) located at the vertices of the nondegenerate simplex S in the N-dimensional space optimization (i.e. N heights z ₁₁ ,…,z _N ), and the corresponding set of loss function values

. Then the method performs a sequence of transformations of the working simplex S aimed at reducing the values of the loss function at its vertices. At each step, the transformation is determined by computing one or more control points along with their loss function values and comparing these loss function values with the values at the current vertices in order to replace the worst vertex, i.e. the one with the largest loss function value, with the best vertex. Control points can be chosen according to one of four heuristics: (1) reflection or (2) extension from the worst vertex; or (3) contraction or (4) contraction towards the best vertex(s). The minimization ends when the working simplex S becomes small enough or when the values of the loss function at the vertices become close enough. Through four heuristic transformations, the Nelder-Mead algorithm usually requires only one or two function evaluations at each step, while many other direct search methods use at least N loss function evaluations. An intuitive explanation of the Nelder-Mead algorithm is given in (Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 10.5. Downhill Simplex Method in Multidimensions". Numerical Recipes: The Art of Scientific Computing (3rd ed. .), New York: Cambridge University Press, ISBN 978-0-521-88068-8): "The vertex descent simplex method now involves a series of steps, most of which simply move the point of the simplex where the function is greatest ("the highest point ”), through the opposite side of the simplex to the lower point. These steps are called reflections, and they are constructed in such a way as to preserve the volume of the simplex (and therefore preserve its non-degeneracy). When possible, the method expands the simplex in one direction or another to make larger steps. When it reaches the "bottom of the valley", the method shrinks laterally and tries to seep down the valley. If a situation arises where the simplex tries to "pass through the eye of a needle," it contracts in all directions, retracting around its lowest (best) point.

According to a preferred embodiment of the present invention, the optimal light redirecting surface is advantageously obtained by the (universal) power diagram method (also known as the Voronoi diagram method or the Laguerre/Voronoi diagram method (see F. de Goes et al., “Blue Noise through Optimal Transport” , CAN Transactions on Graphics, vol. 31(6), (SIGGRAPH Asia) 2012) (also see http://www.geometry.caltech.edu/BlueNoise/, source code available). the method is powerful, and it has been proven that in the case corresponding to the optimization problem of the present invention, the power diagram method as a unique solution "...for any given capacitance constraints" minimizes the concave function of the values, values w _i , corresponding in this case to the heights z _i , and capacitances m _i , corresponding in this case to the areas of the cell a(i) (see, in particular, the appendix to the article cited above by de Goes et al.)

(см. фиг. 4) задача расчета каустической поверхности, которая его генерирует, сводится к нахождению приблизительного набора значений

для заданного набора приближающихся точек

Since any image can be approximated by a finite set of pixels, the caustic surface can be approximated by the composition of the corresponding pieces of surfaces (eg, paraboloids). Therefore, given the target image

(see Fig. 4) the task of calculating the caustic surface that generates it is reduced to finding an approximate set of values

for a given set of approaching points

(в данном случае высот {z_i}) для диаграммы мощности сайтов

, так что емкости

(в данном случае области ячейки {a(i)}) пропорциональны целевым интенсивностям изображения

. После получения оптимального набора высот {z_i, i=1,...,n} и соответствующих границ ячеек

(ячеек

областей a(i)) с помощью метода диаграммы мощности, кусочно-линейная поверхность восстанавливается с учетом пересечений цилиндров, построенных вдоль оси z, основания которых сформированы границами ячеек, с соответственными кусками поверхностей с вершинами на указанных полученных высотах. В предпочтительном варианте осуществления куски поверхностей приближаются параболоидами: в этом случае граница

ячейки

является многоугольной, и расчеты расстояний точки до границы и градиентов значительно упрощаются. В более общем случае (т.е. куски поверхностей не приближаются или приближаются путем разложения в ряд Тейлора порядка более 2), граница

ячейки

по-прежнему является замкнутой кривой, но состоит из изогнутых линий, и вышеупомянутые расчеты расстояний точки до границы и градиентов более сложные. According to the optimal transfer hypothesis (see the aforementioned paper by de Goes et al.), this is equivalent to finding the values

(in this case heights {z _i }) for site power diagram

, so the capacitance

(in this case, cell areas {a(i)}) are proportional to the target image intensities

. After obtaining the optimal set of heights {z _i , i=1,...,n} and the corresponding cell boundaries

(cells

regions a(i)) using the power diagram method, the piecewise linear surface is restored taking into account the intersections of cylinders built along the z axis, the bases of which are formed by the cell boundaries, with the corresponding pieces of surfaces with vertices at the indicated obtained heights. In the preferred embodiment, the surface pieces are approached by paraboloids: in this case, the boundary

cells

is polygonal, and calculations of point-to-border distances and gradients are greatly simplified. In a more general case (i.e., pieces of surfaces do not approach, or approach by expansion into a Taylor series of order greater than 2), the boundary

cells

is still a closed curve, but is made up of curved lines, and the aforementioned calculations for point-to-border distances and gradients are more complex.

The results obtained for the target image depicted in FIG. 4 are shown in FIG. 5–8. The target image has 100x100 pixels, of which 900 are non-zero (i.e., represent areas of the caustic image with non-zero light intensity). In FIG. 5 shows the calculated power diagram; in fig. 6 illustrates the corresponding caustic surface, FIG. 7 shows the ray tracing geometry, and FIG. 8 shows the results of ray tracing.

Minimization _of the functional Σ _i │m _i -I _i │ ² with respect to w i can be solved by a simple gradient descent algorithm. The process starts from the initial set {wi _i } (most often, when all values are taken equal), and then converges to the optimal set {wi _i } of the corresponding tiling Ω _i of capacities m _i . Then, from the resulting optimal set {w _i }, a set of heights of elements of the paraboloid {z _i } is obtained, and from the boundaries ∂Ωi of the resulting polygonal cells Ω _i , by crossing the vertical (along z) cylinders of the base ∂Ωi with the help of paraboloids, the final piecewise linear caustic surface.

A caustic layer having a light redirecting surface calculated and manufactured according to the present invention forms a projected image that contains a caustic pattern that reproduces a reference image that is easily recognized by a person without the use of additional means (for example, the naked eye) or typical and freely available means. , so that an object marked with this optical security element can be easily visually authenticated by a person. The transparent aspect of the refractive optical security element makes it particularly suitable for marking at least partially transparent substrates (eg glass or plastic bottles, bottle caps, watch glasses, jewelry, precious stones, etc.).

The disclosed method for making a refractive transparent or partially transparent light-redirecting surface or a reflective light-redirecting surface of a caustic layer is fast, scalable, reliable and accurate. This greatly reduces the number of iterations required to move from the target image to the corresponding surface, as there is no need for corrections or adjustments. It also reduces the overall time required for manufacturing.

In addition, the normal field calculation and integration step is eliminated, and an efficient optimization method is provided by minimizing capacitance constraints.

In addition, user intervention other than specifying the target image and accepting the resulting surface is completely eliminated. Eliminating the need for user intervention greatly simplifies the implementation of the method in a manufacturing context where specialist skills are not always available.

The foregoing subject matter is to be considered illustrative and not restrictive and serves to better understand the present invention as defined by the independent claims.

Claims (21)

1. A method of manufacturing a refractive transparent or partially transparent light redirecting surface or a reflective light redirecting surface of a caustic layer having a relief pattern formed on a lenticular element and suitable for redirecting incident light received from a light source and for forming a projected image containing a caustic pattern, characterized in that the method includes computerized steps in which: provide a discrete representation of the input target image containing P a set of N image pixels p _i coordinates

in the image plane, with associated target non-zero light intensities {I _i }, i=1,...,N, distributed in a given area of the target image and corresponding to the target caustic pattern of the target image; compute a piecewise linear representation of the light-redirecting surface z = F(x,y) of the caustic layer with a height z above the coordinate plane (x,y) based on the representation of the light-redirecting surface by intersecting surface pieces z = f _i (x,y), i =1,…,N, respectively obtained from the stationarity of the length of the optical path of the rays refracted or reflected by the caustic layer and focused at the points P(i) of the image plane of the coordinates (x _i ,y _i ), i=1,…,N, while each piece of the surface z = f _i (x,y) is a surface of revolution around the axis passing through the point P(i) with a vertex at the point (x _i ,y _i ,z _i ), height z _i = f _i (x _i ,y _i ), i=1,…,N, and the piecewise-linear representation of the light-redirecting surface associated with the corresponding values of the heights of the vertices N is formed by the envelope of the intersections of the corresponding N pieces of surfaces z = f _i (x,y), i=1 ,…,N; for a given set of corresponding values of heights z ₁ ,…,z of _N vertices of N pieces of surfaces, a corresponding set of values of light intensities I(1),…,I(N) is calculated, which are respectively focused at points P(1),…,P(N ) a caustic layer that redirects incident light through a coupled piecewise-linear light-redirecting surface; And calculate the respective values of N heights z ₁ ,…,z _N N of the vertices of the respective N pieces of surfaces, which minimize the differences between the respective values of the calculated light intensities I(1),…,I(N) focused at the points P(1),…, P(N) through the associated light-redirecting surface, and the respective respective values of the target light intensities I ₁ ,…,I _N , thereby obtaining a light redirecting surface having a relief pattern suitable for redirecting incident light received from the light source and forming a projected image containing the target caustic pattern of the target image. 2. The method according to claim 1, characterized in that each piece of the surface z=f _i (x,y), i=1,…,N is approximated in paraxial approximation by expanding in a Taylor series of order k, greater than or equal to two of the expression of the piece surface obtained from the stationarity of the optical path length. 3. The method according to any one of paragraphs. 1 and 2, characterized in that the stage of calculating the heights z _i , minimizing the difference between the calculated light intensities I(i) and the corresponding target light intensities I _i , for i=1,...,N, is carried out by means of an optimization method without a gradient. 4. The method according to any one of paragraphs. 1 and 2, characterized in that the stage of calculating the heights z _i , minimizing the difference between the calculated light intensities I(i) and the corresponding target light intensities I _i , for i=1,...,N, is carried out by means of an optimization method using a power diagram for calculation of the associated loss function and its derivatives. 5. The method according to any one of paragraphs. 1-4, characterized in that the fabricated light redirecting surface is used to generate a machine-matchable representation for controlling a machining tool. 6. The method according to any one of paragraphs. 1-5, characterized in that the manufactured light-redirecting surface is the original light-redirecting surface used to create a copy of the light-redirecting surface. 7. The method of claim 5, wherein the method further comprises copying the light-redirecting surface onto the substrate. 8. The method according to claim 6, characterized in that the copying comprises one of roll-to-roll printing, foil-to-foil printing, UV casting and embossing. 9. An optical security element containing at least one refractive or reflective light redirecting surface of the caustic layer, characterized in that the element is manufactured according to the method according to any one of paragraphs. 1-7. 10. Optical security element according to claim. 9, designed to mark an object selected from the group including consumer goods, documents of value, identity cards, excise stamps and banknotes. 11. Marked object selected from the group including consumer goods, documents of value and banknotes, characterized in that the object contains an optical security element according to any one of paragraphs. 9 and 10. 12. The method of visual authentication of an object marked with an optical security element according to any one of paragraphs. 8 and 9 by a user, characterized in that the method includes the steps of: illuminating the light-directing surface of the optical security element with a point light source at a distance d _s from the light-directing surface; visually observe the caustic pattern projected on the projection surface at a distance d _i from the optical security element; And decide whether the object is genuine, when the user evaluates whether the projected caustic pattern is visually similar to the reference image. 13. The use of an optical security element according to any one of paragraphs. 9 and 10 to authenticate or protect against forgery an object selected from the group including consumer goods, documents of value, identity cards, tax stamps and banknotes.

Images