DE102021103323A1 - Pannini lens and imaging optical device - Google Patents

Abstract

An objective 1 is specified, which has image-forming optical elements 2 with non-rotationally symmetrical curved surfaces, which are arranged along an optical axis z. The image-forming optical elements 2 are designed and arranged in such a way that an image to be formed on an image area 3 is a Pannini image with the following features: equal image scale in the center of the image along each direction in the image area 3, straight vertical imaging of straight vertical lines across the entire image field and barrel distortion of the object space in the horizontal direction. In addition, an imaging optical device 10, a mobile terminal device 100 and a set comprising a lens 1 or an imaging optical device 10 and instructions for use 201 are specified.

Description

The invention relates to a lens, an imaging optical device, a mobile terminal device and a set comprising a lens or an imaging optical device and instructions for use.

The ideal mapping is the rectilinear projection onto a plane, also known as gnomonic projection. A rectilinear projection is a central projection where the projection center is at the center of the body to be imaged. It corresponds to the imaging of a pinhole camera, i. H. each point of the 3-dimensional object space is mapped onto a plane called the image plane along a straight line through the hole forming the perspective center PC of the projection. The normal of the image plane passing through the perspective center PC is defined as the optical axis z. The angle d between the optical axis z and the light ray from the object point Ob through the perspective center PC is called the object angle.

The distance between the perspective center PC and the image plane along the optical axis z is denoted by f. If the hole is replaced by a simple, limited lens, then f corresponds to the focal length of the lens if these object points are sharply imaged onto the image plane at an infinitely large distance. The image height, i.e. the distance that the light beam emanating from the object point Ob has in the image plane from the optical axis z, is denoted by h'. Then the rectilinear or gnomonic mapping is defined as: $$H\text{'}=ƒ\cdot \text{tan}\vartheta$$

The rectilinear projection depicts straight lines as a straight line in the image plane, regardless of their position and orientation in the object space. It leads to the well-known laws of central perspective: parallel lines all meet at the horizon on the line of the eye in a single vanishing point. In a plane positioned perpendicular to the projection axis, all straight lines are imaged identically, in particular horizontal and vertical lines remain horizontal and vertical. Deviations from this projection are referred to as distortion. Camera lenses should generally be distortion-free.

In order to achieve an extensive freedom from distortion, the field of view increases. field-of-view, FOV, the technological effort is considerable. With lenses from 90° FOV, especially with lenses over 110° FOV, a considerable barrel distortion is usually permitted, which is referred to as the fish-eye look. Fisheye lenses form a FOV of up to 180° or more, e.g. B. also 230 °, from where lines at the edge of the field are very strongly curved. These lenses have a much simpler structure than the rectilinear lenses mentioned.

For the majority of photographic situations captured with wide-angle lenses, a rectilinear perspective is very disadvantageous. In group shots, people appear fatter at the edges of the image, and their heads are shown as egg-shaped deformed. This distortion of three-dimensional objects is referred to as rectilinear perspective distortion, sometimes referred to as elliptical distortion to match the depiction of eggheads. The reason for the rectilinear perspective distortion is that the rectilinear projection of 3-dimensional objects becomes progressively wider towards the edge of the field in the radial direction.

Apart from photographing people, there are many other motifs for which a rectilinear perspective is very unfavorable in terms of image design. For example, in interior photography, rectilinear lenses lead to a very large spatial depth, which is often very unfavorable. Areas at the end of the room often appear very small, while lateral areas, e.g. B. side walls, oversized and unnaturally stretched.

The topic of "perspective representation of three-dimensional space" has been dealt with in the field of computer vision in recent years, e.g. B. Carroll R., Agrawal M., Agrawala A. Optimizing content-preserving projections for wide-angle images. In SIGGRAPH '09: ACM SIGGRAPH 2009 papers (New York, NY, USA, 2009), ACM, pp. 1-9.

Alternative representations of the projection of 3D space have been partially implemented in CAD programs, 3D renderers or computer games. The development of some projective representations in computer graphics arose from analysis of paintings by artists over the past centuries.

Important painters used perspective as a stylistic device for composition, such as Leonardo da Vinci in “The Last Supper”. Important Italian vedute painters, such as Piranesi or Canaletto, deliberately deviated from the central perspective in some paintings. Canaletto used a camera obscura to realistically depict spatial perspective, but deliberately modified the proportions to emphasize central image content. In his painting of St. Mark's Square in Venice, the side buildings follow separate vanishing points. Perspectively correct, they run towards the same eye line as the vanishing lines (floor slabs) of the central St. Mark's Square, so that the violation of the central perspective remains unnoticeable to the viewer. The side buildings are more pointed towards the vanishing point than St. Mark's Square, so that a greater spatial depth is suggested to the viewer. In this way, the long facade of the buildings is rendered compressed and the buildings in the far center, St. Mark's Basilica and St. Mark's Tower, appear relatively enlarged.

The veduta painter Giovanni Paolo Pannini (1691-1765) was strongly influenced by Canaletto, but chose a different path. An analysis of his paintings by computer graphics scientists in Sharpless, T.K., Postle, B., German, D.G. Pannini: A New Projection for Rendering Wide Angle Perspective Images, Computational Aesthetics in Graphics, Visualization and Imaging. The Eurographics Association (2010), hereinafter referred to as Sharpless et al. called, revealed that their perspective is not gnomonic, although it initially appears so. All the columns and vertical lines in the painting The Interior of Saint Peter's with the Visit of the Duc de Choiseul, 1756-57. Giovanni Paolo Panini. oil on canvas; 164.3x223.5 cm. The Boston Athenaeum are shown as straight lines. The vanishing lines into the depths of the cathedral also appear straight. Unlike the Canaletto painting discussed above, there is only a single vanishing point. However, on closer analysis of the painting, it turned out that the side wings are compressed, so that despite the wide-angle perspective, the central corridor with Bernini's ciborium still appears relatively large despite the great distance and the side areas do not appear too dominant. This compression towards the horizontal edge is achieved by a non-gnomonic projection in the horizontal direction. It is barrel-shaped horizontally compared to the gnomonic figure.

Because many of the buildings Pannini depicted survive, Sharpless et al. determine the selected perspective projection via reverse engineering, with the result that Pannini generally chose the rectilinear projection in the vertical direction and another one in the horizontal direction, which compresses the image content towards the edge of the image. The analysis showed that the projection in the horizontal direction was not chosen uniformly in different paintings, in some approximately stereographic, in others more cylindrical.

Sharpless et al. have proposed a parameterization of the "Pannini projection". 1 shows the definition of the coordinates used for this, whereby the coordinates have been adapted to the general conventions of geometric optics, i.e. the optical axis is denoted by z, the horizontal direction by x and the vertical direction by y. In addition, a distinction is made between an object point Ob = (x,y,z) ∈ ℝ ³ of the object space "without a line" and the projected image point in the image plane Im = (x',y') ∈ ℝ ² "with a line". In addition, parameters of the optical system can be introduced: If the light comes from infinity, the parameter f', the image-side focal length, is sufficient.

The object point Ob is located in a y-tangential plane that runs through the y-axis and encloses an angle ϕ in the x-z plane between the y-tangential plane and the z-axis, ie the optical axis. From Ob, a ray, the object ray s, meets the coordinate origin O at the angle d to the optical axis. It should be noted that the object-side angle or object angle d of the ray is defined “skew” to the optical axis z and not in the y-tangent plane lies. The three-dimensional space is mapped onto a sphere, the panosphere, along the rays to the perspective center PC. The meeting point of the object ray s with the panosphere is P. Since all object rays run normal to the panosphere through the origin O, there is a clear assignment of Ob to P.

This panosphere in object space can be thought of as transformed (mirrored, i.e. of opposite sign) directly in front of the image plane (x',y'), which is usually an image plane. Now the perspective transformation defines along which path each point on the panosphere is mapped onto the image surface: From the perspective center PC the chief ray t runs through P into the image surface. The position of the perspective center PC can be parameterized with d, the distance between the perspective center PC and the coordinate origin O. The perspective center PC is the center of the entrance pupil, i. H. the intersection of the optical axis z with the entrance pupil.

For example, gnomonic perspective (d = 0) maps along the normal to the panosphere, orthographic perspective (d → ∞) along the normal to the image plane, or stereographic perspective (d = 1) along the direction from the opposite side of the panosphere im Image space, at the intersection with the optical axis z. Values between 1.3 and 715365 were determined for the parameter d for various paintings by Pannini.

$$\frac{y\text{'}}{ƒ\text{'}}=v\text{'}=tan\vartheta \frac{d+1}{d+cos\phi }$$

The Sharpless et al. proposed parameterization of the general Pannini transform in modified notation for optical imaging is: $$\frac{x\text{'}}{ƒ\text{'}}=H\text{'}=sin\phi \frac{\mathrm{i.e}+1}{\mathrm{i.e}+cOs\phi }$$

$$\frac{y\text{'}}{ƒ\text{'}}=v\text{'}=tan\vartheta \frac{\mathrm{i.e}+1}{\mathrm{i.e}+cOs\phi }$$

$$\frac{y\text{'}}{ƒ\text{'}}=\frac{tan\vartheta }{cos\phi }$$

Here v' is the vertical image coordinate and h' is the horizontal image coordinate. The image coordinates are normalized with the focal length f'. The horizontal image coordinate h' is in the value range [-h'max, +h'max]. The entire image field has the "image width 2*h'max. The parameter d ≥ 0 ∈ R characterizes an infinite family of possible transformations, where d = 0 results in a rectilinear or gnomonic transformation, both in the vertical and in the horizontal direction. For d = 0, Equations I and II yield: $$\frac{x\text{'}}{ƒ\text{'}}=tan\phi$$

$$\frac{y\text{'}}{ƒ\text{'}}=\frac{tan\vartheta }{cOs\phi }$$

1. 1. Die Mitte des Bildes ist nicht verzerrt. Die Brennweite bzw. der Abbildungsmaßstab sind in der Bildmitte gleich.
2. 2. Vertikale gerade Linien werden über das gesamte Bildfeld als gerade vertikale Linien abgebildet.
3. 3. Der Objektraum ist in horizontaler Richtung gegenüber der rektilinearen Abbildung tonnenförmig verzeichnet, d. h. die Abweichung zum gnomonischen Bildort ist horizontaler Richtung monoton steigend.

The parameterization by Sharpless et al. can be generalized. A projection of the object space, referred to below as "Pannini mapping" or "Pannini perspective", is characterized by the following features:

1. 1. The center of the image is not distorted. The focal length or image scale are the same in the center of the image.
2. 2. Vertical straight lines are displayed as straight vertical lines over the entire field of view.
3. 3. The object space is barrel-shaped in the horizontal direction compared to the rectilinear image, ie the deviation from the gnomonic image location increases monotonically in the horizontal direction.

In contrast to the parameterization given above, this definition also contains, for example, the cylindrical perspective, according to which x is proportional to d in the horizontal direction (“f-ϑ mapping”) and rectilinear in the vertical direction.

The Pannini perspective is now established on real-time platforms in computer graphics. It has recently been implemented in the main developer platforms for real-time 3D visualization, Unreal Engine or Unity.

In photography or cinematography, on the other hand, the Pannini perspective is unknown. The transformations implemented in Unity or Unreal Engine for animation or computer games can in principle be transferred to the software for image processing, especially in the post-processing of professional film productions. However, there is no optimized digital imaging chain: This means that the image quality (resolution, contrast) sometimes suffers considerably and also that the image information that is available during the recording is lost during the display.

It should also be noted that the implementation of the imaging proposed here is clearly distinguished from anamorphic imaging, which is widespread in professional film: with anamorphic imaging, similar to Pannini imaging, an object field of different sizes is used in the horizontal and vertical direction on the image sensor (same geometry). However, this is achieved by choosing different focal lengths or imaging scales in the horizontal and vertical directions. In contrast to optical Pannini imaging, this necessitates a two-stage process in order to compensate for the deformation of the image during reproduction.

In addition, the Pannini mapping characterizes the compression of the image information in the horizontal direction (in the case of conformal mapping of straight lines in the vertical direction). This is not the case with anamorphic imaging. The barrel distortion, mostly in both horizontal and vertical directions, in some patented and commercially available wide-angle anamorphic lenses is a result of a simple design principle.

Horizontally, this is close to Pannini perspective, but crooked vertical lines are undesirable. With integration of the required afocal components as described in U.S. 2013/0 022 345 A1 , severe distortion can be avoided even with wide-angle lenses.

The anamorphic image does not meet the above mentioned. Features 1st and 3rd of the Pannini illustration. Nevertheless, there is a similarity between Pannini imaging and anamorphic imaging, which should be mentioned here because it is very popular in cinematography with anamorphic lenses and is partly the reason for its use, despite the considerable additional costs. Defocused highlights in the image background appear oval in anamorphic imaging. For example, lights have Streetlights, for example, have a “candle-like” shape, which many filmmakers appreciate, as well as the fact that changing the focal plane has the dynamic effect of vertically “pulling” the sharpness.

In the Pannini image there is a round bokeh in the center of the image, which becomes increasingly oval towards the horizontal edges of the image and also longer in the vertical direction, i.e. "candle-shaped". This results from the fact that the local image scale decreases towards the horizontal edge of the image. Since in the present case it is essentially wide-angle images with a generally large depth of field that are assumed, this effect is not as effective as with lenses of a longer focal length, but it is still clearly visible with larger image formats.

In addition, there is a second type of lens whose image is clearly different from the Pannini image, so-called panomorph lenses, particularly as an alternative to circular fish-eye lenses that image the entire hemisphere onto the image plane. These enable extremely wide-angle shots with an aspect ratio that is not equal to 1:1 in the horizontal and vertical directions. Since image sensors are usually rectangular, they can therefore be better utilized. These lenses use cylindrical or toric lenses. However, the figure differs fundamentally from the Pannini figure - vertical lines neither remain straight nor is the horizontal figure barrel-shaped. Rather the opposite is the case. The local image scale increases towards the horizontal edge of the image, so that the unfavorable perspective image becomes even stronger. Consequently, the above Characteristics 2. and 3. of the Pannini illustration not fulfilled.

As already mentioned, the generation of Pannini images by means of digital transformation is known from the prior art. However, such a digital transformation has several disadvantages, which become more and more important with increasing field of view angles. On the one hand, after the digital transformation of an image recorded with a fisheye lens, the image section has to be restricted, with a large part of the recorded image information being lost.

On the other hand, in fisheye imaging, information is "squeezed" at the edge of the object field on the equidistant grid of the image sensor. In this process, resolution of object details at the edge of the image is inevitably lost. In addition, the contrast decreases. The lack of resolution and the reduction in contrast are of course lost after the transformation to rectilinear or Pannini perspective, so that both resolution and contrast vary greatly in the image field.

Against this background, it is the object of the present invention to find ways in which Pannini images can be recorded and/or reproduced without the disadvantages explained above.

This problem is solved by the subject matter of the independent claims. The dependent claims relate to special configurations of these solutions according to the invention.

A basic idea of the invention is to implement the recording and playback in Pannini perspective with purely optical means, ie without the need for digital transformation.

A first aspect of the invention relates to an objective. The lens can be used, for example, as a lens for a camera, e.g. B. a film camera or a photo camera, or as a lens for a projection device be trained. The lens has image-forming optical elements with one or more non-rotationally symmetrical curved surfaces, which are arranged along an optical axis. The lens can be designed in particular as a wide-angle lens, z. B. with a diagonal angle of view of 60 ° and larger, preferably 75 ° and larger, more preferably 85 ° and larger. The lens can also be designed as a zoom lens.

According to the invention, it is provided that the image-forming optical elements are designed and arranged in such a way that an image to be formed on an image surface is a Pannini image. Such a Pannini map is characterized by the following features: equal magnification at image center along each direction in the image plane (feature 1), straight vertical mapping of straight vertical lines across the entire image field (feature 2), and barrel distortion of object space in the horizontal direction ( Feature 3).

The "centre of the image" means the area on the image surface in the immediate vicinity of the optical axis.

Feature 1 requires that the focal length of the lens in the area of the rays that generate the center of the image is also the same in the x and y directions. Due to the same magnification in the center of the image, the center of the image is not distorted. Minor deviations, i. H. Deviations that are imperceptible to the viewer are permissible. Such minor deviations can, for example, amount to a maximum of 15%, i. H. the compression or stretching factor in the center of the image may deviate from the ideal “no distortion” by a maximum of 15%.

Feature 2 means that vertical straight lines are imaged as straight vertical lines over the entire field of view. Here, too, minor deviations, i. H. Deviations that are imperceptible to the viewer are permissible. Such minor deviations can amount to a maximum of 5%, for example. The deviation of 5% refers to the so-called "TV distortion", which describes the crookedness of a desired straight line. TV distortion is defined as the change in distance from the center of the picture to the top of the picture divided by the distance from the bottom of the picture to the top of the picture. It is therefore measured how much the image is bent by a straight line at the edge of the image, in particular at the long edge of a rectangular format. This deflection is related to the total image height and expressed as a percentage.

Feature 3 means that the image space is barrel-wrapped in the horizontal direction compared to the rectilinear mapping, i. H. the deviation from the gnomonic image location increases monotonically in the horizontal direction. Since the deviation can be expressed as a 3rd order polynomial, this means that the deviations increase towards the vertical edge of the image. In other words, the image information is increasingly squeezed in the direction of the vertical edge of the image, i. H. wide areas are mapped narrower.

The Pannini image can be placed on an arbitrarily shaped image surface, for example a flat image surface, e.g. B. on a flat surface of an image sensor in the case of a film or photo camera or a flat projection surface in the case of a projection lens. Other forms of the picture surface, e.g. B. cylindrical or toric image surfaces can also be used. The design and arrangement of the image-forming optical elements is carried out in such a way that a Pannini image can be generated on the selected image area.

The lens according to the invention is characterized in that a Pannini image with features 1 to 3 can be generated directly using exclusively optical means. Such a lens is also referred to below as a Pannini lens. Thus, no digital transformation of the image is required, nor are multiple optical systems required to generate the Pannini image.

Concrete and generally applicable information on the optical elements to be used and their arrangement - with the exception of the presence of at least one optical element with a non-rotationally symmetrical curved surface - cannot be given, since these are e.g. depend on the other lens properties, the selected image area and the intended purpose. For example, a certain optical effect can be generated by means of a single optical element or by a plurality of optical elements.

In addition, the specific purpose of use defines the framework for the minimum properties to be met by the lens, including image quality requirements, technological requirements and costs. In other words, the construction of Pannini lenses can differ significantly depending on the specific intended application, e.g. B. Use of the Pannini lens in a smartphone camera, an action video camera (action cam), as a camera lens for a reflex camera or a mirrorless system camera, as a camera lens in professional cinematography, as a projection lens, etc. There is also the possibility of using existing lenses with To modify attachment or back adapter optics in such a way that a Pannini lens is obtained, so that consequently the selection and arrangement of the additional image-forming optical elements is dependent on the existing lens.

Another field of application includes afocal systems, ie binoculars, spotting scopes, telescopes, etc. In many of these systems, particularly in terrestrial systems, it is advantageous to increase the object-side visual field angle in the horizontal direction. It doesn't matter if the information at the edge of the field of view is compressed and therefore has poorer resolution, it's just a matter of roughly seeing peripheral information at the edge of the field of view without having to take your eye off the eyepiece (e.g. wildlife or bird watching with binoculars/spotting scope; in the case of peripheral detection, swivel to enlarged main field of view). It follows that a Pannini lens for such an afocal system has a significantly different structure than a Pannini lens for a photo or film camera. Consequently, the specific structure of the Pannini lens is highly dependent on the desired application.

Using the general procedure explained below and using the attached exemplary embodiments, those skilled in the art will be able to find an objective with optical elements that are suitably designed and arranged for generating a Pannini image.

For an optical realization of the Pannini image without digital transformation of the image coordinates, one or more non-rotationally symmetrical acting optical elements are necessary in order to be able to influence the beam path differently in different areas and thereby, among other things. to achieve the desired distortion. In addition, of course, optical elements that act in a rotationally symmetrical manner can also be present.

Optical elements that act non-rotationally symmetrical can either actually be non-rotationally symmetrical optical elements that are positioned in such a way along the optical axis, e.g. B. are arranged perpendicular to the optical axis, that the non-rotational symmetry is not z. B. is canceled by tilting, or it can be rotationally symmetrical optical elements that are arranged along the optical axis in such a way that they still do not appear rotationally symmetrical, for example by tilting relative to the optical axis.

For example, cylindrically curved, toric surfaces or free-form surfaces can be used as non-rotationally symmetrical curved surfaces.

The optical elements of the lens can have optically effective surfaces that act as lenses (refraction) or mirrors (reflection). In addition, optical elements can also be implemented as diffractive elements (diffraction). The latter also includes optical meta surfaces.

Optically active surfaces can preferably be used which are symmetrical to the x-axis and to the y-axis of the lens, e.g. B. cylindrically curved surfaces, toroidal surfaces or correspondingly designed free-form surfaces.

The “x-axis” here means the axis aligned in the horizontal direction at a right angle to the z-axis, which corresponds to the optical axis. The "y-axis" means the axis oriented in the vertical direction at right angles to the z-axis.

Cylindrically curved surfaces are characterized by a radius (either rx or ry), which can be positive or negative, convex or concave. The other radius is infinite. A cylindrically curved surface corresponds to a section of a lateral surface of a vertical circular cylinder.

A "toric surface" is a surface that has two mutually perpendicular principal sections of different curvature and where the cross-sections in both principal sections are nominal are circular. It is therefore a matter of symmetrical surfaces whose two surface radii can be finitely large and have any sign. In addition, the toric surface shape also includes a rotationally symmetrical asphere on the toric surface. This is usually given by a conic constant k and a set of polynomial coefficients a4, a6,...

In the case of free-form surfaces, the asphere is generally not rotationally symmetrical.

For the latter there is a large number of surface representations, e.g. B. "anamorphic asphere" (in each case in x- and y-direction independent radius, conical constant and polynomial coefficients), XY-polynomial (radius, conical constant and mixed polynomial coefficients x ^m y ⁿ ) or corresponding representations with Zernike polynomials or other orthogonal ones functional systems.

A “freeform surface” is to be understood in a broader sense as a complex surface that can be represented in particular by means of functions defined in certain areas, in particular functions defined in different areas that can be continuously differentiated twice. Examples of suitable region-wise defined functions are (particularly piecewise) polynomial functions (particularly polynomial splines, such as bicubic splines, fourth-degree or higher degree splines, or polynomial non-uniform rational B-splines (NURBS)). A distinction must be made between simple surfaces such as e.g. B. spherical surfaces, aspherical surfaces, cylindrical surfaces, toric surfaces, which are described at least along a main meridian as a circle. In particular, a free-form surface does not need to have axial symmetry and point symmetry and can have different values for the mean surface refractive index in different areas of the surface. A free-form surface is usually produced on an optical element by machining the optical element, for example by milling, as part of a CNC process in which the free-form surface is produced under numerical control on the basis of a mathematical description of the surface. However, it is also possible to blank press the free form. For this purpose, the negative press mold must be processed with the appropriate allowances for temperature-dependent shrinkage using CNC processes.

With these surfaces designed symmetrically to the x-axis and to the y-axis of the lens, the optical system, that is to say all of the image-forming optical elements, can be arranged along a single optical axis. The surface geometry applies in particular to the surface shape, ie the deformation of the surface, but can also have an additional advantageous effect if the optical surface has a correspondingly symmetrical boundary. Likewise, separate field diaphragms with symmetry about the x and y axes can have an advantageous effect. Boundaries of optical surfaces or field stops cut off the light beam of an object point. This is called "vignetting" and reduces optical artifacts. This enables lenses with a larger aperture to be realized with sufficiently good sharpness in the corners of the image. However, as a result of the clipping of the light beam, less light reaches the edge of the image field, so that it becomes darker there. The presence of symmetry axes considerably simplifies both the design of the optical system and its assembly and adjustment.

The desired barrel distortion can be achieved, for example, using the lens configuration "-, aperture stop, +", i. H. with a configuration in which the object-side lens part has a negative refractive power and the image-side lens part has a positive refractive power, the object-side and image-side lens parts being separated from one another by the aperture stop.

mit $$B_p=\frac{n\left(n\text{'}-n\right)w_p\left(u{\text{'}}_p+i_p\right)}{2n\text{'}hn{\text{'}}_{k}u{\text{'}}_k},$$

mit h Objekt(Bild)-Höhe, n, n' Brechungsindices der Medien an der Grenzfläche, i Strahleinfallswinkel an der Grenzfläche, u Strahlwinkel zur optischen Achse, n_k' den Brechungsindex zwischen letzter optischer Fläche und Bildfläche und u_k' den entsprechenden Strahlwinkel zur optischen Achse zwischen letzter optischer Fläche und Bildfläche und w Strahlhöhe an der Grenzfläche zur optischen Achse. Der Index „p“ kennzeichnet dabei den Hauptstrahl, der durch die Pupillenmitte verläuft; die entsprechenden nicht indizierten Größen den Marginalstrahl des Achsbüschels (siehe Warren Smith (1993), Modern Lens Design, McGraw-Hill, Seite 447 und 448). Aus diesem Ausdruck folgt, dass eine einzelne positive Linse (beliebiger Form) hinter der Blende zu einer tonnenförmigen Verzeichnung führt, genauso wie eine negative Linse vor der Blende. Folglich führt eine Konfiguration mit negativer Linse, dann Blende, dann positive Linse auch zu einer tonnenförmigen Verzeichnung.According to Seidel's theory of aberrations, the distortion coefficient is of the 3rd order, $$DC=H\left[B_{p}i{i}_p+\left(\mathrm{and}{\text{'}}_{\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="DE102021103323A1\_0033.png,DE102021103323A1\_0038.png"\; state="\{\{state\}\}">p</figure-callout>}}^2-{\mathrm{and}}_p^2\right)\text{/}2\right]$$

With $$B_p=\frac{n\left(n\text{'}-n\right)w_p\left(\mathrm{and}{\text{'}}_p+i_p\right)}{2n\text{'}Hn{\text{'}}_k\mathrm{and}{\text{'}}_k},$$

with h object (image) height, n, n' refractive indices of the media at the interface, i ray incidence angle at the interface, u ray angle to the optical axis, n _k ' the refractive index between the last optical surface and the image surface and _uk ' the corresponding ray angle to the optical axis between the last optical surface and the image surface and w ray height at the interface to the optical axis. The index “p” designates the principal ray that runs through the center of the pupil; the corresponding non-indexed sizes den Axial tuft marginal ray (see Warren Smith (1993), Modern Lens Design, McGraw-Hill, pp. 447 and 448). From this expression it follows that a single positive lens (of any shape) behind the stop will produce barrel distortion, as will a negative lens in front of the stop. Consequently, a negative lens, then aperture, then positive lens configuration also results in barrel distortion.

bzw. für endliche Objektentfernungen die Abbildungsmaßstäbe, $${\beta }_x={\beta }_y.$$

According to the first Pannini condition, the focal lengths in x and y are equal, $$f_x=f_y,$$

or for finite object distances the image scales, $${\beta }_x={\beta }_y.$$

Only the surface shapes (radii and higher-order deformations) and not the surface distances are available as lines of freedom for realizing the desired perspective projection in the horizontal and vertical direction. Starting from a symmetrical lens arrangement for rectilinear projection in the vertical direction, the refractive power of the arrangement in the horizontal direction must be more negative at the front, i.e. close to the object, and more positive at the back, i.e. close to the image. The near-field lenses in front and behind the rectilinear direction become more front-diverging and rear-converging in the barrel deformed direction.

A Pannini lens with at least two non-rotationally symmetrical optical components, e.g. B. cylindrical or toric lenses required. Otherwise, the focal length at the center of the image would be unequal in the horizontal and vertical directions, because changing a single component power without changing distances between the components (which are the same in both slices x and y) also changes the total power.

ändert und damit auch der Petzvalradius im Bild.This cannot be avoided, at least for moderate and large numerical apertures, by choosing the same refractive power locally in both directions close to the axis, but changing the refractive power at a single point off-axis, for example by using an asphere close to the field that is locally the same in the axial direction has refractive power, but which assumes a different curvature towards the edge. The reason for this is that the position of the image then also changes locally, ie an image shell would be introduced because the Petzval sum $${\displaystyle \sum \frac{{\varphi }_j}{n_j}}$$

changes and with it the Petzval radius in the image.

gleich sein. Der Abstand der Hauptebenen der beiden Teilsysteme ist mit d_HE bezeichnet. Dieser Abstand d_HE ist beim Pannini-Objektiv für die x- und y-Komponente gleich, so dass gelten muss: $${\varphi }_{x\mathrm{,1}}+{\varphi }_{x\mathrm{,2}}-d_{HE}{\varphi }_{x\mathrm{,1}}{\varphi }_{x\mathrm{,2}}={\varphi }_{y\mathrm{,1}}+{\varphi }_{y\mathrm{,2}}-d_{HE}{\varphi }_{y\mathrm{,1}}{\varphi }_{y\mathrm{,2}}$$

In a lens with two non-rotationally symmetrical groups with the refractive powers φ _x,1 , φ _x,2 in the x-direction and φ _y,1 , φ _y,2 in the y-direction, the total refractive power must be φ = 1/f' for the entire lens, $$\varphi ={\mathrm{<figure-callout\; id="1"\; label="\phi "\; filenames="DE102021103323A1\_0024.png,DE102021103323A1\_0026.png"\; state="\{\{state\}\}">\varphi </figure-callout>}}_1+{\mathrm{<figure-callout\; id="2"\; label="\phi "\; filenames="DE102021103323A1\_0033.png,DE102021103323A1\_0038.png"\; state="\{\{state\}\}">\varphi </figure-callout>}}_2-{\mathrm{i.e}}_{HE}{\varphi }_1{\mathrm{<figure-callout\; id="2"\; label="\phi "\; filenames="DE102021103323A1\_0033.png,DE102021103323A1\_0038.png"\; state="\{\{state\}\}">\varphi </figure-callout>}}_2$$

be equal. The distance between the main planes of the two subsystems is denoted by d _HE . With the Pannini lens, this distance d _HE is the same for the x and y components, so that the following must apply: $${\varphi }_{x\mathrm{,1}}+{\varphi }_{x\mathrm{,2}}-{\mathrm{i.e}}_{HE}{\varphi }_{x\mathrm{,1}}{\varphi }_{x\mathrm{,2}}={\varphi }_{y\mathrm{,1}}+{\varphi }_{y\mathrm{,2}}-{\mathrm{i.e}}_{HE}{\varphi }_{y\mathrm{,1}}{\varphi }_{y\mathrm{,2}}$$

In the case of an objective given in the y-direction, assumed as a symmetrical system without loss of generality (characterized more generally by a specific ratio of the partial refractive powers), ϕ _x,1 = ϕ _x,2 ≡ ϕ _x must apply $$2{\varphi }_x-{\mathrm{i.e}}_{HE}{\varphi }_x^2={\varphi }_{y\mathrm{,1}}+{\varphi }_{y\mathrm{,2}}-{\mathrm{i.e}}_{HE}{\varphi }_{y\mathrm{,1}}{\varphi }_{y\mathrm{,2}}.$$

In the vertical direction, the distortion is corrected by the quasi-symmetrical structure of the lens, in the horizontal direction, on the other hand, the desired distortion of the near stereographic perspective can be achieved by a retrofocus structure.

If a specific perspective transformation is required, for example stereographic or equiangular, then there is no guarantee that the Petzval condition can also be strictly satisfied. However, a strictly fulfilled Petzval condition only means that a complete image field flattening is possible for any aperture. With moderate apertures, on the other hand, it is quite possible to achieve good Pannini imaging with just two toric or cylindrical lenses.

Another way of realizing a Pannini image with a simple lens structure is as follows: It is known that the Petzval condition for the flattening of the image shell reduces the complexity of the lens, i. H. Number of lenses, size, length, weight etc. essentially determined. This applies in particular to lenses with a large aperture and larger image formats (i.e. small depth of field and thus increased requirements for image field flattening), especially for wide-angle lenses, for which image flattening must be guaranteed over a very large range of solid angles.

It is also known that a spherically curved field of view, with the center of curvature facing the lens, can significantly reduce complexity, particularly for the wider aperture lenses mentioned. For example, with very simply constructed, almost concentric lenses made up of a few spherical elements, an image that is as good as that achieved with lenses with more than a dozen lenses or aspheric lenses for imaging on one plane.

A comparatively simple lens structure can also be used when using non-flat image areas such. B. cylindrically curved image surfaces can be achieved. It is to be expected that cylindrically curved image surfaces, e.g. B. image sensors, are much easier to implement than spherically curved. A problem with spherically curved image sensors is that the uniform curvature on a square pixel grid causes different shearing forces on the grid elements, since the orientation of the individual pixels to the surface curvature varies. This is different with a cylindrically curved image sensor, since the same forces act homogeneously in each x or y direction.

It may therefore be possible to realize Pannini images on cylindrically curved image surfaces with a very large field of view angle on the order of the entire hemisphere, in contrast to rotationally symmetrical fisheye images on rotationally symmetrically curved image surfaces. As already mentioned, a Pannini image can already be generated with two toric elements as long as the aperture is not very large (focal ratio f/8 or smaller). It is also possible to realize a Pannini image with only two toric elements if the aperture is relatively large but the field of view angle is not very large.

If, instead of two toric elements, three or more toric elements are used to realize the Pannini image, significantly more degrees of freedom are opened up in order to obtain Pannini lenses both for large field of view angles and for large apertures. In Pannini lenses with three toric elements, the light paths in the lens pass through the lens elements at significantly different heights. In addition to sufficient degrees of freedom for the simultaneous correction of distortion and image shell, this also enables the correction of other image errors, such as e.g. B. the transverse chromatic aberration. For example, by choosing an entrance pupil position for the stereographic imaging direction with a strongly curved front lens closer to the objective entrance, the light can pass through the lens at a lower height and produce a transverse chromatic aberration comparable to that in the vertical direction.

A general procedure for achieving the desired barrel-shaped distortion in the horizontal direction is based on the knowledge that bundles of rays must be able to be influenced independently of one another. Elements close to the field are primarily influenced here, while elements close to the pupil are of little importance.

Optical surfaces that have a subaperture ratio between greater than 0.5 and 1, in particular between 0.7 and 1, are referred to as “near the pupil”. In contrast, optical surfaces that are “near the field” have a subaperture ratio of between 0 and 0.5, in particular between 0 and 0.3. The subaperture ratio SAR of an optical surface of an optical element in the beam path is defined as the quotient between the subaperture diameter DSA and the optically free diameter DCA, ie SAR = DSA / DCA. The subaperture diameter DSA is the maximum diameter of a partial area of the optical element that is illuminated with rays of a beam bundle emanating from a given field point. The optically free diameter DCA is the diameter of the smallest circle around a reference axis of the optical element, the circle enclosing that area of the surface of the optical element that is illuminated by all rays coming from the object field. In a field level, e.g. B. object plane or image plane, SAR = 0 applies accordingly, in a pupil plane SAR = 1.

Starting from rotationally symmetrical elements, the geometry and arrangement of elements close to the field in particular can be varied in order to obtain a Pannini image. The effect of free-form surfaces is all the more effective the smaller the associated sub-aperture diameter DSA is.

mit $$H={\left(\frac{h\text{'}}{d+1}\right)}^2$$

For the above transformations of the object angles (ϑ, ϕ) into image coordinates (x′, y′) or scaled image coordinates (h′, v′) according to Equations I and II, the numerical assignments can be specified using exemplary data for focal length and image length where the object angles are specified in degrees. Since the image coordinates, e.g. B. positions on the image sensor, are given in the calculation, the object angles are required as a function of the image coordinates. For the horizontal coordinate we get $$cOs\phi =\frac{-H\mathrm{i.e}\pm \sqrt{{\left(H\mathrm{i.e}\right)}^2-\left(H+1\right)\left(H{\mathrm{i.e}}^2-1\right)}}{\mathrm{<figure-callout\; id="1"\; label="H\; +"\; filenames="DE102021103323A1\_0024.png,DE102021103323A1\_0026.png"\; state="\{\{state\}\}">H</figure-callout>}+1}$$

With $$H={\left(\frac{H\text{'}}{\mathrm{i.e}+1}\right)}^2$$

This results in the vertical component with $$tan\vartheta =\frac{v\text{'}\left(\mathrm{i.e}+cOs\phi \right)}{\mathrm{<figure-callout\; id="1"\; label="i.e\; +"\; filenames="DE102021103323A1\_0024.png,DE102021103323A1\_0026.png"\; state="\{\{state\}\}">i.e</figure-callout>}+1}.$$

ergeben sich damit die expliziten Winkel als Funktion der Bildkoordinaten (h', v') nach Ersetzen von Gleichung V $$\phi \left(h\text{'}\right)=atan2{\left(h\text{'},S\left(cos\phi \right)cos\phi \right)|}_{cos\phi =c\left(h\text{'}\right)}$$

$$\vartheta \left(h\text{'},v\text{'}\right)=atan2{\left(v\text{'},S\left(cos\phi \right)\right)|}_{cos\phi =c\left(h\text{'}\right)}$$

With the abbreviation $$S\left(cOs\phi \right)=\frac{\mathrm{i.e}+1}{\mathrm{i.e}+cOs\phi }$$

the explicit angles result as a function of the image coordinates (h', v') after replacing Equation V $$\phi \left(H\text{'}\right)=atan2{\left(H\text{'},S\left(cOs\phi \right)cOs\phi \right)|}_{cOs\phi =c\left(H\text{'}\right)}$$

$$\vartheta \left(H\text{'},v\text{'}\right)=atan2{\left(v\text{'},S\left(cOs\phi \right)\right)|}_{cOs\phi =c\left(H\text{'}\right)}$$

Tables 1 and 2 below list the resulting angles for various h-values and v-values. Table 1a: Horizontal angles for rectilinear projection, rotationally symmetrical, 36 × 24 mm ² , focal length f = 15 mm v' ↓ 12 0.0 11.3 21:8 31.0 38.7 45.0 50.2 9 0.0 11.3 21:8 31.0 38.7 45.0 50.2 6 0.0 11.3 21:8 31.0 38.7 45.0 50.2 3 0.0 11.3 21:8 31.0 38.7 45.0 50.2 0 0.0 11.3 21:8 31.0 38.7 45.0 50.2 h' → 0 3 6 9 12 15 18 Table 1b: Vertical angles in rectilinear projection, rotationally symmetrical, 36 × 24 mm ² , focal length f = 15 mm v' ↓ 12 38.7 38.1 36.6 34.4 32.0 29.5 27.1 9 31.0 30.5 29.1 27.2 25.1 23.0 21.0 6 21:8 21:4 20.4 18.9 17.3 15.8 14.4 3 11.3 11.1 10.5 9.7 8.9 8.0 7.3 0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 h' → 0 3 6 9 12 15 18 Table 2a: Horizontal angles in Pannini projection, d = 1, h - stereographic, v - rectilinear, 36 × 24 mm ² , focal length f = 15 mm v' ↓ 12 0.0 11.4 22.6 33.4 43.6 53.1 61.9 9 0.0 11.4 22.6 33.4 43.6 53.1 61.9 6 0.0 11.4 22.6 33.4 43.6 53.1 61.9 3 0.0 11.4 22.6 33.4 43.6 53.1 61.9 0 0.0 11.4 22.6 33.4 43.6 53.1 61.9 h' → 0 3 6 9 12 15 18 Table 2b: Vertical angles in Pannini projection, d = 1, h - stereographic, v - rectilinear, 36 × 24 mm ² , focal length f = 15 mm v' ↓ 12 38.7 38.4 37.6 36.3 34.6 32.6 30.5 9 31.0 30.7 30.0 28.8 27.3 25.6 23.8 6 21:8 21:6 21.0 20.2 19.0 17.7 16.4 3 11.3 11.2 10.9 10.4 9.8 9.1 8.4 0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 h' → 0 3 6 9 12 15 18

In both cases, along the field bisector in the vertical direction (h = 0), a rectilinear perspective results, i.e. the same allocation of object angle to image height. For example, the same object angle of 38.7° to the optical axis results at the maximum height.

In the horizontal direction along the field bisecting line (v = 0), the maximum angle is rectilinear 50.2°, while according to Pannini (d = 1) or stereographically 61.9° are still imaged. Near the center of the image, the assignment of object angles to image heights is almost identical, since the focal length is the same in both directions.

The resulting angles can be calculated accordingly for other focal lengths and image widths.

These determined assignments must be fulfilled for the conditions of the optical design. This can be done, for example, in a plane directly in front of the lens. For this it is useful to express the direction of the main ray in the object space by the direction cosine, ie the components (L,M,N)=(sinϕ cosϑ, sinϑ, cosϕ cosϑ). This vector is normalized so that L ² + M ² + N ² = 1 and the direction are already clearly defined by two components.

It should be noted that object angles and image positions are assigned, but the object position (e.g. beam passage on the plane in front of the lens entrance) is free. In particular, it is not necessary for all object rays to meet at one and the same point of intersection (“entrance pupil”) on the object side. A (generally slight) parallax of the principal rays is thus permissible, which means that the entrance pupil is not a point but is spatially distributed. This is particularly common with fisheye lenses. Table 3a: Values for L = sin(ϕ)*cos(ϑ) for rectilinear projection, rotationally symmetrical, 36 × 24 mm ² , focal length f = 15 mm v' ↓ 12 0.000 0.154 0.298 0.424 0.530 0.615 0.684 9 0.000 0.169 0.324 0.457 0.566 0.651 0.717 6 0.000 0.183 0.348 0.487 0.596 0.680 0.744 3 0.000 0.192 0.365 0.507 0.617 0.700 0.762 0 0.000 0.196 0.371 0.514 0.625 0.707 0.768 h' → 0 3 6 9 12 15 18 Table 3b: Values for M = sin(ϑ) for rectilinear projection, rotationally symmetrical, 36 × 24 mm ² , focal length f = 15 mm v' ↓ 12 0.625 0.617 0.596 0.566 0.530 0.492 0.456 9 0.514 0.507 0.487 0.457 0.424 0.391 0.359 6 0.371 0.365 0.348 0.324 0.298 0.272 0.248 3 0.196 0.192 0.183 0.169 0.154 0.140 0.127 0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 h' → 0 3 6 9 12 15 18 Table 3c: Values for N = cos(ϕ)*cos(ϑ) for rectilinear projection, rotationally symmetrical, 36 × 24 mm ² , focal length f = 15 mm v' ↓ 12 0.781 0.772 0.745 0.707 0.662 0.615 0.570 9 0.857 0.845 0.811 0.762 0.707 0.651 0.598 6 0.928 0.913 0.870 0.811 0.745 0.680 0.620 3 0.981 0.962 0.913 0.845 0.772 0.700 0.635 0 1,000 0.981 0.928 0.857 0.781 0.707 0.640 h' → 0 3 6 9 12 15 18 Table 3d: Values for L ² +M ² +N ² with rectilinear projection, rotationally symmetrical, 36 × 24 mm ² , focal length f = 15 mm v' ↓ 12 1 1 1 1 1 1 1 9 1 1 1 1 1 1 1 6 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 h' → 0 3 6 9 12 15 18 Table 4a: Values for L = sin(ϕ)*cos(ϑ) for Pannini projection, d = 1, 36 × 24 mm ² , focal length f = 15 mm v' ↓ 12 0.000 0.155 0.305 0.444 0.568 0.674 0.761 9 0.000 0.170 0.333 0.482 0.613 0.721 0.807 6 0.000 0.184 0.359 0.517 0.652 0.762 0.846 3 0.000 0.194 0.378 0.541 0.680 0.790 0.873 0 0.000 0.198 0.385 0.550 0.690 0.800 0.882 h' → 0 3 6 9 12 15 18 Table 4b: Values for M = sin(ϑ) for Pannini projection, d = 1, 36 × 24 mm ² , focal length f = 15 mm v' ↓ 12 0.625 0.621 0.610 0.592 0.568 0.539 0.507 9 0.514 0.511 0.500 0.482 0.459 0.433 0.404 6 0.371 0.368 0.359 0.345 0.326 0.305 0.282 3 0.196 0.194 0.189 0.180 0.170 0.158 0.145 0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 h' → 0 3 6 9 12 15 18 Table 4c: Values for N = cos(ϕ)*cos(ϑ) for Pannini projection, d = 1, 36 × 24 mm ² , focal length f = 15 mm v' ↓ 12 0.781 0.768 0.732 0.673 0.596 0.505 0.406 9 0.857 0.843 0.800 0.731 0.643 0.541 0.431 6 0.928 0.911 0.862 0.784 0.685 0.571 0.451 3 0.981 0.962 0.906 0.821 0.714 0.592 0.466 0 1,000 0.980 0.923 0.835 0.724 0.600 0.471 h' → 0 3 6 9 12 15 18 Table 4d: Values for L ² +M ² +N ² in Pannini projection, d = 1, 36 × 24 mm ² , focal length f = 15 mm v' ↓ 12 1 1 1 1 1 1 1 9 1 1 1 1 1 1 1 6 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 h' → 0 3 6 9 12 15 18

The optimization of the concrete lens design can be carried out with standard optical design programs, such as e.g. B. Code-V, Zemax Optic Studio, OSLO, where the direction cosines as well as other equivalent ray angles can be entered as indicated and set as boundary conditions.

The Pannini lens can be used in a variety of applications, e.g. B. in cinematography, DSLR photography (DSLR, English digital single lens reflex), CSC photography (CSC, English compact system camera), in action cams, in photography with mobile devices and the corresponding projection applications.

Of course, the Pannini lens can be combined with a suitable digital transformation, for example for distortion correction, warping, etc., so that specific image quality requirements, e.g. B. in terms of resolution, contrast, chromatic aberrations, etc., can be maintained in the image field.

Despite the need to use non-rotationally symmetrical optical elements, the purely optical implementation of the Pannini perspective in the form of the lens described offers advantages over a digital-optical solution, which become greater the wider the recorded area is. In comparison to a fisheye image after transformation into a rectilinear representation, in which the initially rectangular image section is distributed over a large star-shaped area, so that when it is cropped rectangularly, e.g. B. for viewing on a monitor, a large part of the recorded image information is lost, the entire image information is retained. In addition, a high resolution of object details can also be achieved at the edge of the image, with the contrast being retained. Consequently, an image field with little variation in resolution as well as contrast can be obtained.

Advantageously, the lens can have a field stop, the shape of which deviates from a circular shape and a rectangular shape.

As is well known, the use of field stops is essential for photographic lenses, especially for interchangeable lenses for larger image formats, e.g. B. 36 × 24 mm ² . The beams of rays towards the edge of the field are increasingly cut off by the field stop. This makes it much easier to correct the image errors.

Particularly with high-aperture wide-angle lenses, large gradients of the wavefront towards the edge of the pupil easily form, which considerably reduce the image contrast. If these large wavefront errors are cut out with the help of diaphragms, which are arranged in the front and/or rear of the lens in the area close to the field, the contrast can also be significantly increased at the edges of the image. These field stops are also called "coma stops" because coma is a typical, dominant image defect that is reduced by the field stops.

However, field pruning is accompanied by a reduction in irradiance. With camera lenses, the irradiance should be checked for each pixel. Typical specifications of permitted variations in irradiance in the image with wide-angle lenses are 2 to 3 f-stops (EV), i.e. a factor of 25% or slightly less compared to the center of the image. If the differences in brightness are greater, this is often perceived as annoying, especially with evenly bright subjects. The edge light drop is often digitally compensated. But then the noise at the edge of the field is higher than in the middle of the image.

For a given rectangular image field, there are "footprints" on the optical surfaces, i.e. areas of the light used for imaging for each object point. The totality of these footprints defines the necessary surface area and thus the boundary of the optical component. 13 shows footprint distributions of a Pannini lens in different planes. In the area close to the field, the area is angular, but generally not rectangular, while in the area close to the pupil it is circular (if a circular pupil). 13 shows the footprints in specific planes and makes it clear that the field stops would have the shape indicated.

Such more complex shapes can be advantageous for minimal design sizes as well as a defined influence on the light fall-off in the image field. They deviate from the rectangular or circular shape. In addition, it is advantageous with Pannini lenses if the field diaphragm is not flat but shaped three-dimensionally, further advantageous with four sectors of the same shape, i.e. x-y-symmetrical. This means that the optimal position of the field stop in the z-direction can deviate in the x-direction from the y-direction. A corresponding three-dimensional shape is therefore often advantageous for a uniform irradiance, but not absolutely necessary.

According to various embodiment variants, the image-forming optical elements can be designed and arranged in such a way that the image to be formed can be formed on a flat image surface, i. H. the Pannini image is generated on a flat image surface.

This allows the use of known planar image surfaces, e.g. B. flat image sensors in the generation of the Pannini image. The Pannini lens can thus be used advantageously together with already existing components of an imaging optical device.

Alternatively, the image-forming optical elements can be designed and arranged in such a way that the image to be formed can be formed on a cylindrically curved image surface, i. H. the Pannini image is generated on a cylindrically curved image surface.

As already mentioned above, a simplification of the structure of the Pannini lens can be achieved in this way, so that it i.a. can be produced inexpensively.

According to further embodiment variants, the lens can be a camera lens or a projection lens.

The camera lens can be used, for example, for a photo camera, ie as a photo lens, or for a film camera, ie as a film lens. Combined use is also possible. Other applications, e.g. B. in the areas of observation, measurement, qualification, quantification are possible.

The projection lens can be used accordingly for a projection device. In comparison to a camera lens, the light path is reversed. The object to be imaged, e.g. B. a self-illuminating OLED display, an illuminated slide, etc. The projection surface takes the place of the object plane. Analogously to a curved image surface, the object to be imaged or projected can be curved or the object can be flat, but the projection surface can be curved. The projection surface can be curved towards the viewer, for example, and can therefore have the advantages of a domed cinema.

The formation of the lens as a camera or projection lens advantageously enables the recording or reproduction of Pannini images.

According to further embodiment variants, the lens can be an attachment lens or a back adapter lens.

In other words, the lens can be designed as an optical system that is used before (attachment lens) or after (back adapter lens) another lens and z. B. changed the focal length or imaging properties of this other lens. In this case, the Pannini image is generated by combining both lenses. In this case, the specific design and arrangement of the image-forming optical elements depends on the structure and the properties of the further objective.

Due to the design of the lens as an attachment lens or back adapter lens, existing lenses can be used flexibly, i. H. with and without an attachment lens or back adapter lens to create Pannini images or other images.

A further aspect of the invention relates to an imaging optical device with an objective as described above.

All statements regarding the lens can be transferred accordingly to the imaging optical device. The advantages of the lens are correspondingly connected with the imaging optical device.

The imaging optical device can be used, for example, as a camera, e.g. B. as a photo camera or film camera, be designed, the camera can have, for example, a flat or cylindrically curved image surface. In addition to the lens, the camera has an image sensor, preferably a full-frame image sensor.

Alternatively, the imaging optical device can also be a projection device, a telescope, binoculars, a microscope, etc. A projection device has a light source in addition to the projection objective.

A further aspect of the invention relates to a mobile terminal device with an imaging optical device as described above.

The mobile end device can be in the form of a smartphone, tablet, smartwatch, data glasses, etc., for example. In particular, the mobile terminal device can have an imaging optical device designed as a camera.

All statements regarding the imaging optical device and the associated lens can be transferred accordingly to the mobile terminal device. The advantages of the imaging optical device and the associated objective are correspondingly connected to the mobile terminal device. Another advantage is that by implementing the lens in a mobile device, Pannini images can also be generated on the go.

A further aspect of the invention relates to a so-called kit-of-parts, namely a set which, firstly, includes a lens as described above or an imaging optical device as described above, and, secondly, an instruction manual with instructions for using the Specify the lens or the imaging optical device with a flat or curved image surface, includes.

The instructions for use can be designed, for example, in the form of a pictogram or instructions in text form. It defines the use of the lens or the imaging optical device with either a flat or a curved image surface, e.g. B. a cylindrically curved image surface.

All statements regarding the lens and the imaging optical device can be transferred to the set accordingly. With the set, the advantages of the lens or the imaging optical device are connected accordingly.

• 1 Darstellung zur Veranschaulichung der Koordinaten zur Beschreibung der Pannini-Projektion;
• 2a ein erstes beispielhaftes Objektiv (X-Schnitt);
• 2b das erste beispielhafte Objektiv (Y-Schnitt);
• 3a ein zweites beispielhaftes Objektiv (X-Schnitt);
• 3b das zweite beispielhafte Objektiv (Y-Schnitt);
• 4a ein drittes beispielhaftes Objektiv (X-Schnitt);
• 4b das dritte beispielhafte Objektiv (Y-Schnitt);
• 5a ein viertes beispielhaftes Objektiv (X-Schnitt);
• 5b das vierte beispielhafte Objektiv (Y-Schnitt);
• 6a ein fünftes beispielhaftes Objektiv (X-Schnitt);
• 6b das fünfte beispielhafte Objektiv (Y-Schnitt);
• 6c Verzeichnungsgitter des fünften beispielhaften Objektivs;
• 6d Verlauf der Modulationstransferfunktion des fünften beispielhaften Objektivs;
• 7a ein sechstes beispielhaftes Objektiv (X-Schnitt);
• 7b das sechste beispielhafte Objektiv (Y-Schnitt);
• 7c Verzeichnungsgitter des sechsten beispielhaften Objektivs;
• 8a ein siebentes beispielhaftes Objektiv (X-Schnitt);
• 8b das siebente beispielhafte Objektiv (Y-Schnitt);
• 8c-e Queraberrationsdiagramme des siebenten beispielhaften Objektivs;
• 8f Verlauf der Modulationstransferfunktion des siebenten beispielhaften Objektivs;
• 9a ein achtes beispielhaftes Objektiv (X-Schnitt);
• 9b das achte beispielhafte Objektiv (Y-Schnitt);
• 10 eine schematische Darstellung eines beispielhaften abbildenden optischen Geräts;
• 11 eine schematische Darstellung eines beispielhaften mobilen Endgeräts;
• 12 eine schematische Darstellung eine beispielhaften Sets; und
• 13 Footprints über das Bildfeld in verschiedenen Objektivebenen.

In the following, the invention is explained by way of example with reference to the enclosed figures using preferred embodiments, with the features presented below being able to represent an aspect of the invention, both individually and in various combinations with one another. Show it:

• 1 Diagram to illustrate the coordinates to describe the Pannini projection;
• 2a a first exemplary lens (X-cut);
• 2 B the first exemplary lens (Y-cut);
• 3a a second exemplary lens (X-cut);
• 3b the second exemplary lens (Y-cut);
• 4a a third exemplary lens (X-cut);
• 4b the third exemplary lens (Y-cut);
• 5a a fourth exemplary lens (X-cut);
• 5b the fourth exemplary lens (Y-cut);
• 6a a fifth exemplary lens (X-cut);
• 6b the fifth exemplary lens (Y-cut);
• 6c Distortion grating of the fifth example lens;
• 6d Progression of the modulation transfer function of the fifth exemplary lens;
• 7a a sixth example lens (X-cut);
• 7b the sixth exemplary lens (Y-cut);
• 7c Sixth Example Lens Distortion Grating;
• 8a a seventh exemplary lens (X-cut);
• 8b the seventh exemplary lens (Y-cut);
• 8c-e transverse aberration diagrams of the seventh example lens;
• 8f Course of the modulation transfer function of the seventh exemplary lens;
• 9a an eighth exemplary lens (X-cut);
• 9b the eighth exemplary lens (Y-cut);
• 10 a schematic representation of an exemplary imaging optical device;
• 11 a schematic representation of an exemplary mobile terminal device;
• 12 a schematic representation of an exemplary set; and
• 13 Footprints across the field of view in different lens planes.

1 serves to illustrate the coordinates for describing the Pannini projection. In this regard, reference is made to the explanations in the introductory description.

the 2a and 2 B show the lens 1 of a first embodiment with a focus setting at infinity. This first exemplary embodiment is a lens 1 designed as a compact wide-angle lens for mirrorless cameras for forming a rectilinear-stereographic Pannini image in a format of 36×24 mm ² .

Lens 1 with an opening of f/7.5 and a focal length of 15 mm forms an object angle of (hFOV, vFOV) = (121°, 78°) and is therefore almost 25% more horizontally than a rectilinear lens of the same focal length (hFOV_rect = 98°) and without perspective distortion. The lens 1 is very short with a length of 33.4 mm. The ratio of overall length to sensor diagonal is just 0.77.

The objective 1 has five toric lenses (without aspheric orders) as image-forming optical elements 2 . The focal length on the image side is 11.6 mm. This makes it possible to use this lens 1 as an interchangeable lens on certain mirrorless cameras.

The basic construction data of the lens 1 of the first exemplary embodiment are given in Table 5 below, with the tolerance range of the given values being able to be ±5%. The curved surfaces of the image-forming optical elements 2 of the first embodiment are denoted by A1 to A12. Table 5: Basic construction data of the first embodiment Surface surface type Crest radius of surface curvature in y-direction [mm] Crest radius of surface curvature in x-direction [mm] Distance to the following surface (air distance or lens thickness) [mm] refractive index n _d Abbe number v _d Half the diameter of the optically used area in the y-direction [mm] Half the diameter of the optically used area in the y-direction [mm] object spherical infinite infinite infinite A1 spherical infinite infinite 0 22.74 22.74 A2 X toroidal 357.956738 49.405612 1.00 1.804602 34.64 17.70 17.70 A3 X toroidal 23.879334 40.436790 4:15 15:41 15:41 A4 X toroidal 168.330868 -178.098135 3.00 1.804742 31.82 12.00 15:41 A5 X toroidal -60.849834 45.240455 14:13 13.81 13.81 A6 X toroidal 6.487694 6.095405 0.94 1.623215 46.00 3.01 3.01 A7 X toroidal 54.659318 42.141089 1.34 2.86 2.86 aperture stop spherical infinite infinite 0.72 1.20 1.20 A9 X toroidal -14.523181 -21.662830 5.39 1.804200 46.50 1.72 1.72 A10 X toroidal -6.122898 -6.024103 0.78 3.62 3.62 A11 X toroidal -4.256482 -4.382202 1.98 1.805180 25.36 3.62 3.62 A12 X toroidal -12.314677 -19.228587 11.58 5.76 5.76 picture even infinite infinite 0 21.65 21.65

wobei n_d, n_F usw. die Brechungsindizes des Materials bei den Wellenlängen der entsprechenden Fraunhoferlinien 587,5618 nm (gelbe He-Linie), 486,1327 nm (blaue H-Linie) und 656,2725 nm (rote H-Linie) sind.The rows show, from top to bottom, those with the 2a and 2 B corresponding surface numbers of the lenses. Area A1 is a so-called dummy area, which is used for better representation in the layout plots ( 2a and 2 B) serves. The columns show, from left to right, the surface number, the surface type (spherical, toroidal, etc.), the apex radius of the surface curvature in the y- and y-direction, the distance to the following surface (the air gap or the lens thickness), the Abbe number v _d , the refractive index n _d and half the diameter of the optically used area in the x and y directions. The Abbe number v _d is defined as: $$v_{\mathrm{i.e}}=\frac{n_{\mathrm{i.e}}-1}{n_f-n_C}$$

where nd, _nF etc. are the refractive indices of the material at the wavelengths of the corresponding Fraunhofer lines _587.5618 nm (yellow He line), 486.1327 nm (blue H line) and 656.2725 nm (red H line) are.

the 3a and 3b show the lens 1 of a second embodiment with a focus setting at infinity. It is also a compact wide-angle lens for a mirrorless camera

Lens 1 with an aperture of f/7.5 and a focal length of 15 mm depicts an object angle of (hFOV, vFOV) = (121°, 78°) and is therefore almost 25% more horizontally than a rectilinear lens of the same focal length (hFOV_rect=98 °) and without perspective distortion. The lens 1 is extremely short with a length of 26.7 mm. The ratio of overall length to sensor diagonal is only 0.62 and thus even smaller than modern smartphone lenses.

The objective 1 has six lenses, namely four toric lenses (without aspheric orders) and 2 spherical lenses as image-forming optical elements 2 . One of the toric lenses is a field lens just in front of the image plane to correct specific distortion, field curvature and astigmatism. The diameters of the other lenses are very small at <11 mm. The screen is flat. The modulation transfer function or contrast transfer function MTF is consistently >50% at 20 lp/mm.

The basic construction data of the lens 1 of the second exemplary embodiment are given in Table 6 below, with the tolerance range of the given values being able to be ±5%. The curved surfaces of the image-forming optical elements of the second embodiment are denoted by B1 to B14. For a more detailed explanation of the basic design data, reference is made to the first exemplary embodiment. Table 6: Basic construction data of the second embodiment Surface surface type Crest radius of surface curvature in y-direction [mm] Crest radius of surface curvature in x-direction [mm] Distance to the following surface (air distance or lens thickness) [mm] refractive index n _d Abbe number v _d Half the diameter of the optically used area in the y-direction [mm] Half the diameter of the optically used area in the x-direction [mm] object spherical infinite infinite infinite B1 spherical infinite infinite 0 5.68 5.68 B2 X toroidal 25.5966 -48.4669 0.22 1.608084 37.20 5.62 5.62 B3 X toroidal 8.5982 80.0715 0.1228 5:13 5:13 B4 X toroidal 10.0181 5.8814 0.4453 1.804742 31.82 4:16 4:16 B5 X toroidal 14.0519 4.9507 2 3.94 3.94 B6 X toroidal 9.9784 12.0444 3 1.804200 46.50 2.37 2.37 B7 Cylinder 23.6124 infinite 0.1526 1.09 1.09 aperture stop spherical infinite infinite 2 1.02 1.02 B9 spherical -15.6308 -15.6308 4 1.804200 46.50 3.20 3.20 B10 spherical -4.9415 -4.9415 0.6229 4.34 4.34 B11 spherical -4.5414 -4.5414 0.22 1.805180 25.36 4.34 4.34 B12 spherical -8.7073 -8.7073 12 5.62 5.62 B13 X toroidal 35.8747 -47.4697 3 1.805180 25.36 19:12 19:12 B14 spherical infinite infinite 0.3 21:10 21:10 picture even infinite infinite 0 21.67 21.67 21.67 21.67

the 4a and 4b show the lens 1 of a third embodiment with a focus setting at infinity. This third exemplary embodiment is a lens 1 designed as a wide-angle lens with a front pupil for forming a rectilinear-stereographic Pannini image, which is designed as a 4/28 mm lens 1 for a 35 mm format. The pupil (system diaphragm) is located directly at the system entrance in front of the first lens.

The basic construction data of the lens 1 of the third exemplary embodiment are given in Table 7 below, with the tolerance range of the given values being able to be ±5%. The curved surfaces of the image-forming optical elements of the third embodiment are denoted by C1 to C11. For a more detailed explanation of the basic design data, reference is made to the first exemplary embodiment. Table 7: Basic construction data of the third embodiment Surface surface type Crest radius of surface curvature in y-direction [mm] Crest radius of surface curvature in x-direction [mm] Distance to the following surface (air distance or lens thickness) [mm] refractive index Abbe number v _d Half the diameter of the optically used area in the y-direction [mm] Half the diameter of the optically used area in the x-direction [mm] object spherical infinite infinite infinite C1 spherical infinite infinite 5 7.45 7.45 aperture stop spherical infinite infinite 2.88 1, 607024 61.03 3.50 3.50 C3 spherical -9.346435 -9.346435 4.24 4.71 4.71 C4 spherical -7.108486 -7.108486 0.10 1.845594 23.81 6:13 6:13 C5 spherical -7.009878 -7.009878 0.75 6:13 6:13 C6 spherical -9.796425 -9.796425 0.10 1.669499 55.07 7:16 7:16 C7 X toroidal 44.649850 60.154512 3:19 9.85 9.85 C8 X toroidal -38.897582 -31.622852 30:14 1.846660 23.78 10.04 10.04 C9 X toroidal -17.177055 -22.760910 1 11.00 15.60 C10 X toroidal -27.063758 -49.241884 1 12.00 16.00 picture even infinite infinite 0 21.63 21.63

the 5a and 5b show the objective 1 of a fourth exemplary embodiment with a focusing setting at infinity. This fourth exemplary embodiment is a lens 1 designed as a wide-angle lens with a high opening for mirrorless cameras for larger image formats, such as a 35 mm format or similar. Lens 1 is a compact 2.8/21mm stereographic/rectilinear (d=1) Pannini lens for a 36mm × 24mm full-frame mirrorless camera close to the image plane with an object angle range (hFOV, vFOV) = (92.7 °, 59.6°). The horizontal maximum field angle corresponds to f = 17 mm with rectilinear imaging. The objective 1 has two toric lenses directly in front of the flat image surface 3 .

The basic construction data of the lens 1 of the fourth exemplary embodiment are given in Table 8 below, with the tolerance range of the given values being able to be ±5%. The curved surfaces of the image-forming optical elements of the fourth embodiment are denoted by D1 to D19. For a more detailed explanation of the basic design data, reference is made to the first exemplary embodiment. Table 8: Basic construction data of the fourth embodiment Surface surface type Crest radius of surface curvature in y-direction [mm] Crest radius of surface curvature in x-direction [mm] Distance to the following surface (air distance or lens thickness) [mm] refractive index n _d Abbe number v _d Half the diameter of the optically used area in the y-direction [mm] Half the diameter of the optically used area in the x-direction [mm] object spherical infinite infinite infinite D1 spherical infinite infinite 0.00 23.26 23.26 D2 spherical 15.714051 15.714051 1.00 1.487491 70.41 13.52 13.52 D3 spherical 10.300123 10.300123 12:19 10:23 10:23 D4 spherical -24.982293 -24.982293 10.66 1.654373 54.45 9:21 9:21 D5 spherical -12.248854 -12.248854 1.00 1.729215 29.62 7.60 7.60 D6 spherical -26.734382 -26.734382 0.10 7.29 7.29 D7 spherical 12.105863 12.105863 11:18 1.621801 60.04 6.00 6.00 D8 spherical -138.736214 -138.73621 0.10 4.24 4.24 aperture stop spherical infinite infinite 2.44 4:19 4:19 D11 spherical -7.021519 -7.021519 1.00 1.755201 27.58 4.55 4.55 D12 spherical -11.818525 -11.818525 0.10 5.50 5.50 D13 spherical 45.009436 45.009436 1.00 1.755201 27.58 8.56 8.56 D14 spherical 24.773075 24.773075 4.35 1.620411 60.32 9.28 9.28 D15 spherical -39.706614 -39.706614 5.59 9.87 9.87 D16 X toroidal -10.738960 -12.086868 2.00 1.487491 70.41 13.00 9.00 D17 X toroidal -11.501059 -20.888060 1.87 12:35 11.00 D18 X toroidal -12.122687 -64.746762 1.40 1.620411 60.32 15:16 15:16 D19 cylindrical -26.197053 infinite 1.00 17.67 17.67 picture even infinite infinite 0.00 21.65 21.65

the 6a and 6b show the lens 1 of a fifth embodiment with a focus setting at infinity. This fifth exemplary embodiment is also a lens 1 designed as a wide-angle lens with a high aperture for mirrorless cameras for larger image formats, such as a 35 mm format or similar. Lens 1 is a compact 2.8/21mm stereographic/rectilinear (d=1) Pannini lens for a 36mm × 24mm full-frame mirrorless camera close to the image plane with an object angle range (hFOV, vFOV) = (92.7 °, 59.6°). The horizontal maximum field angle corresponds to f = 17 mm with rectilinear imaging. The objective 1 has two toric lenses directly in front of the cylindrically curved image surface 3 .

The basic construction data of the lens 1 of the fifth exemplary embodiment are given in Table 9 below, with the tolerance range of the given values being able to be ±5%. The curved surfaces of the image-forming optical elements of the fifth embodiment are denoted by E1 to E19. Reference is made to the first exemplary embodiment for a more detailed explanation of the basic design data. Table 9: Basic construction data of the fifth embodiment Surface surface type Crest radius of surface curvature in y-direction [mm] Crest radius of surface curvature in x-direction [mm] Distance to the following surface (air distance or lens thickness) [mm] refractive index n _d Abbe number v _d Half the diameter of the optically used area in the y-direction [mm] half the diameter of the optically used area in x-direction [mm] object spherical infinite infinite infinite E1 spherical infinite infinite 0.00 25.75 25.75 E2 spherical 16.490360 16.490360 1:13 1.487491 70.41 14.61 14.61 E3 spherical 10.527654 10.527654 15.85 10.53 10.53 E4 spherical -20.903016 -20.903016 4.69 1.604917 61:15 8.54 8.54 E5 spherical -10.401604 -10.401604 0.95 1.665828 34.66 8.34 8.34 E6 spherical -23.033559 -23.033559 0.61 8:27 8:27 E7 spherical 17.969902 17.969902 5.12 1.579737 57.03 6.00 6.00 E8 spherical -28.456772 -28.456772 2:16 6:14 6:14 aperture stop spherical infinite infinite 0.48 4.71 4.71 E11 spherical -11.822192 -11.822192 0.95 1.755191 27.58 4.75 4.75 E12 spherical -15.152096 -15.152096 0.10 4.40 4.40 E13 spherical 27.938190 27.938190 6.66 1.755200 27.58 6.25 6.25 E14 spherical 12.905729 12.905729 5.34 1.487492 70.41 8.07 8.07 E15 spherical -24.458006 -24.458006 3.49 8.60 8.60 E16 X toroidal -9.217725 -10.603640 2.00 1.743964 44.85 8.68 8.68 E17 X toroidal -14.114991 -26.633295 3.22 11.91 11.91 E18 X toroidal -19.452989 235.745697 2.50 1.620411 60.28 18.75 18.75 E19 X toroidal -61.798784 168.774134 1.74 21.68 21.68 picture cylindrical infinite -84.493392 0.00 21.65 21.65

6c shows the distortion grating of the lens 1 of the fifth embodiment. The ideal grating is shown thin, the grating obtained by means of the lens 1 of the fifth exemplary embodiment is shown highlighted (heavy lines) and has the characteristics of Pannini imaging: vertical lines remain almost vertical, while the spacing of vertical lines decreases monotonically in the horizontal direction ( horizontal barrel distortion).

6d shows the course of the modulation transfer function of the lens 1 of the fifth exemplary embodiment.

The modulation transfer function (MTF) gives the contrast of structures of a certain period, characterized by the spatial frequency of periodic structures in units of line pairs per millimeter (English: cycles per millimeter). A Nyquist frequency of 100 line pairs/mm results for typical pixel periods of modern image sensors of modern DSLR or system cameras of about 5 microns. Very fine structures that are still reproducibly imaged with such an image sensor are at around the Nyquist frequency/3, i.e. slightly more than 30 Lp/mm. The MTF graph shows the contrast starting from coarse periodic structures to these fine structures in the range 0 to 30 lp/mm for various pixels in the image field (see legend next to the diagram). The image performance is 10 lp/mm for almost that entire image field over 80% (shown in the diagram as a relative value of 0.8) and at 30 lp/mm over 30%, with the contrast in the center of the image being higher than at the edges. This means that the picture performance of this Pannini lens is as high as that of a very good, conventional, rotationally symmetrical lens for a professional full-frame system camera.

the 7a and 7b show the lens 1 of a sixth embodiment with a focus setting at infinity. This sixth exemplary embodiment is also a lens 1 designed as a wide-angle lens with a high aperture for mirrorless cameras for larger image formats, such as a 35 mm format or similar. Lens 1 is a compact stereographic/rectilinear (d = 1) Pannini lens 2.8/18mm for a 36mm x 24mm mirrorless full frame camera with a moderate free distance to the image plane with an object angle range (hFOV, vFOV) = (104° , 66°). The horizontal image area is significantly larger than with a purely rectilinear imaging lens (104° compared to 88°) and reduces 3D deformations at the image edges. In addition to rotationally symmetrical lenses, the objective 1 contains four toric lenses (lenses 1, 3, 4 and 5) in the front area of the objective 1, especially in the area nearer the pupil (one of which is aspherical). These are all designed with a spherical basic shape (no higher polynomial coefficients).

The basic construction data of the lens 1 of the sixth exemplary embodiment are given in Table 10 below, with the tolerance range of the given values being able to be ±5%. The curved surfaces of the image-forming optical elements of the sixth embodiment are denoted by F1 to F30. Reference is made to the first exemplary embodiment for a more detailed explanation of the basic design data. The surfaces F28 and F29 are flat plates in front of the image sensor. For example, this can be protective glass for the image sensor, optionally with an integrated low-pass and/or infrared filter. Table 10: Basic construction data of the sixth embodiment Surface surface type Crest radius of surface curvature in y-direction [mm] Crest radius of surface curvature in x-direction [mm] Distance to the following surface (air distance or lens thickness) [mm] refractive index n _d Abbe number v _d Half the diameter of the optically used area in the y-direction [mm] Half the diameter of the optically used area in the x-direction [mm] object spherical infinite infinite infinite F1 spherical infinite infinite 0 39.90 39.90 F2 X toroidal 37.722836 27.689931 5 1.641421 56.48 23.24 23.24 F3 X toroidal 21.47906 35.009513 4 21:42 21:42 F4 spherical 36.20163 36.201637 1.2 1.567632 63.42 18.81 18.81 F5 spherical 20.78268 20.782687 8.1738 15.47 15.47 F6 X toroidal 22.854153 -298.31126 1.2 1.643045 56.21 13:22 13:22 F7 X toroidal 15.421035 13.691903 4.4212 10:26 10:26 F8 X toroidal -140.89212 -1032.9409 6.8354 1.716442 30.76 10:31 10:31 F9 X toroidal -27.213747 60.431732 2.2772 9.00 9.00 F10 X toroidal -31.847556 136.41774 4.5455 1.749051 34.89 8.77 8.77 F11 X toroidal -58.323331 -32.773264 0.1 8.46 8.46 F12 spherical 37.120258 37.120258 1.2 1.719649 46.89 7.62 7.62 F13 spherical 8.9354474 8.9354474 3.8400 1.650688 42.51 6.71 6.71 F14 spherical 87.634394 87.634394 2.2648 6.50 6.50 F15 spherical -21.047456 -21.047456 4.4791 1.62041 60.32 6:18 6:18 F16 spherical -17.528962 -17.528962 0.1 6:21 6:21 aperture stop spherical 1.00E+18 1.00E+18 0.1 5.99 5.99 F18 spherical 14.860628 14.860628 3.9661 1.62041 60.32 6.75 6.75 F19 spherical -24.982767 -24.982767 0.1 6.79 6.79 F20 spherical -28.191947 -28.191947 1.2 1.755201 27.58 6.76 6.76 Q21 spherical 161.77107 161.77107 0.1 6.83 6.83 Q22 spherical 15.006438 15.006438 1.2 1.755201 27.58 6.97 6.97 Q23 spherical 10.561844 10.561844 4.4053 6.65 6.65 Q24 spherical -13.637531 -13.637531 1.2 1.632956 35.09 6.75 6.75 Q25 spherical 133.48804 133.48804 0.1 8:19 8:19 Q26 aspheric 118.10318 118.10318 4.4539 1.743972 44.85 8.65 8.65 F27 aspheric -14.527636 -14.527636 22 9.10 9.10 Q28 spherical 1.00E+18 1.00E+18 2.5 1.516800 64:17 20.49 20.49 Q29 spherical 1.00E+18 1.00E+18 1.0375921 21:24 21:24 picture spherical 1.00E+18 1.00E+18 0 21.74 21.74

Tables 11 and 12 show the coefficients of the aspheric surfaces F26 and F27 according to the definition equation of the vertex shape.

wobei z die Pfeilhöhe, R den Scheitel-Krümmungsradius der Linsen, r den radialen Abstand mit $r=\sqrt{x^2+y^2},$

k die Kegelschnittkonstante und A, B, C, D, E, F, G, H, J den Verformungskoeffizienten der jeweiligen Ordnung bezeichnen. Bei sphärischen Linsen sind A = B = C = D = E = F = G = H = J = 0 sowie k = 0. Die Koeffizienten F, G, H und J (14. bis 20. Ordnung) sowie die Kegelschnittkonstante bzw. konische Konstante sind für alle angeführten asphärischen Flächen gleich Null. Der jeweilige Scheitelradius kann Tabelle 10 entnommen werden. Tabelle 11: Koeffizienten der asphärischen Fläche F26 Parameter Wert Y Radius 118,103182 Konische Konstante (K) 0 4.Ordnung - Koeffizient A -7,2715E-06 6.Ordnung - Koeffizient B 6,2427E-07 8.Ordnung - Koeffizient C 2,1914E-09 10.Ordnung - Koeffizient D -2,0630E-11 12.Ordnung - Koeffizient E 2,1543E-13 Tabelle 12: Koeffizienten der asphärischen Fläche F27 Parameter Wert Y Radius -14,5276361 Konische Konstante (K) 0 4.Ordnung - Koeffizient A 4,78E-05 6.Ordnung - Koeffizient B 1,73E-07 8.Ordnung - Koeffizient C 8,69E-09 10.Ordnung - Koeffizient D -2,81E-11 12.Ordnung - Koeffizient E 5,74E-13 The vertex shape is described by the equation: $$\begin{array}{l}\mathrm{e.g}:=\frac{{\mathrm{right}}^2\text{/}\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="DE102021103323A1\_0024.png,DE102021103323A1\_0026.png"\; state="\{\{state\}\}">R</figure-callout>}}{1+\sqrt{1-\left(1+k\right)*\frac{{\mathrm{right}}_2}{{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="DE102021103323A1\_0033.png,DE102021103323A1\_0038.png"\; state="\{\{state\}\}">R</figure-callout>}}^2}}}+A{\mathrm{right}}^4+B{\mathrm{right}}^6+C{\mathrm{right}}^{\mathrm{8th}}+D{\mathrm{right}}^{10}+E{\mathrm{right}}^{12}+f{\mathrm{right}}^{14}+G{\mathrm{right}}^{16}\\ +H{\mathrm{right}}^{18}+J{\mathrm{right}}^{20}\end{array}$$

where z is the vertex height, R is the vertex radius of curvature of the lenses, r is the radial distance $\mathrm{right}=\sqrt{x^2+y^2},$

k denote the conic section constant and A, B, C, D, E, F, G, H, J denote the strain coefficient of the respective order. For spherical lenses, A = B = C = D = E = F = G = H = J = 0 and k = 0. The coefficients F, G, H and J (14th to 20th order) and the conic constant or conic constants are zero for all aspheric surfaces listed. The respective crest radius can be found in Table 10. Table 11: Coefficients of the aspheric surface F26 parameter value Y radius 118.103182 Conic constant (K) 0 4th order - coefficient A -7.2715E-06 6th order - coefficient B 6.2427E-07 8th order - coefficient C 2.1914E-09 10th order - coefficient D -2.0630E-11 12th order - coefficient E 2.1543E-13 Table 12: Coefficients of the aspheric surface F27 parameter value Y radius -14.5276361 Conic constant (K) 0 4th order - coefficient A 4.78E-05 6th order - coefficient B 1.73E-07 8th order - coefficient C 8.69E-09 10th order - coefficient D -2.81E-11 12th order - coefficient E 5.74E-13

7c shows the distortion grating of the lens 1 of the sixth embodiment. The ideal grating is shown thin, the grating obtained by means of the lens 1 of the sixth exemplary embodiment is shown highlighted (heavy lines) and has the characteristics of Pannini imaging: vertical lines remain almost vertical, while the spacing of vertical lines decreases monotonically in the horizontal direction ( horizontal barrel distortion).

the 8a and 8b show the lens 1 of a seventh embodiment with a focus setting at infinity. This seventh exemplary embodiment is a lens 1 designed as a wide-angle lens for a single-lens reflex camera for larger image formats, such as a 35 mm format or similar. Lens 1 is a stereographic/rectilinear (d=1) Pannini lens 2.8/15 mm for a single lens reflex camera (free back focus <38.5 mm suitable for bayonet sizes from all major manufacturers). The imaged object angle is (hFOV, vFOV) = (121°, 78°), i.e. horizontally almost 25% more than a rectilinear lens of this focal length (hFOV_rect = 98°) and without perspective distortion. The image performance, contrast and correction of chromatic aberrations is on the level of a premium ultra wide-angle lens such as the rectilinear Distagon ZE/ZF 2.8/15 mm from ZEISS. The lenses 1 to 6 are toric, one lens is aspheric (rotationally symmetrical; directly in front of aperture 4), the other lenses are spherical.

The basic construction data of the lens 1 of the seventh exemplary embodiment are given in Table 13 below, with the tolerance range of the given values being able to be ±5%. The curved surfaces of the image-forming optical elements of the seventh embodiment are denoted by G1 to G34. Reference is made to the first exemplary embodiment for a more detailed explanation of the basic design data. The surfaces G33 and G34 are flat plates in front of the image sensor. For example, this can be protective glass for the image sensor, optionally with an integrated low-pass and/or infrared filter. Table 13: Basic construction data of the seventh embodiment Surface surface type Crest radius of surface curvature in y-direction [mm] Crest radius of surface curvature in x-direction [mm] Distance to the following surface (air distance or lens thickness) [mm] refractive index n _d Abbe number v _d Half the diameter of the optically used area in the y-direction [mm] Half the diameter of the optically used area in the x-direction [mm] object spherical infinite infinite infinite G1 spherical infinite infinite 0.0000 95.35 95.35 G2 X toroidal 64.2614 391.0810 2.0000 1.740151 51.24 44.00 60.96 G3 X toroidal 43.3883 47.5417 11.1476 42.64 42.64 G4 X toroidal 90.3465 42.9685 13.7524 1.772500 49.62 38.00 42.09 G5 X toroidal -561.6142 59.7751 8.9093 40.21 40.21 G6 X toroidal 224.2608 35.9615 2.0000 1.774476 44.81 26.00 26.71 G7 X toroidal 20.8860 21.4185 12.3333 16.70 19.76 G8 X toroidal -124.5899 74.4002 4.3490 1.812640 30.77 15.00 19.52 G9 X toroidal -108.5011 -2449.9520 0.1000 18.75 18.75 G10 X toroidal 57.3814 63.0516 2.0000 1.772500 49.62 16.52 16.52 G11 X toroidal 27.3627 19.0346 5.8868 13.82 13.82 G12 X toroidal 532.0721 -387.7512 2.0000 1.646896 57.73 13.67 13.67 G13 X toroidal 23.8293 27.2252 5.4020 12.72 12.72 G14 spherical 36.6704 36.6704 6.6577 1.518678 61.59 13.00 13.00 G15 spherical -54.4414 -54.4414 1.3986 13.00 13.00 G16 spherical -32.0092 -32.0092 3.4481 1.621037 60.26 12.97 12.97 G17 spherical -23.6910 -23.6910 0.1000 13:18 13:18 G18 spherical 20.3795 20.3795 3.5068 1.814681 24.72 10.95 10.95 G19 spherical 29.5298 29.5298 2.4248 1.768647 26.38 10.03 10.03 G20 spherical 31.5299 31.5299 5.9425 9:13 9:13 G21 aspheric -22.8042 -22.8042 2.0000 1.772500 49.62 7.09 7.09 G22 aspheric -27.6482 -27.6482 0.1000 6.65 6.65 aperture stop spherical infinite infinite 0.1000 6.46 6.46 G24 spherical 299.7027 299.7027 2.0000 1.818803 24.59 6.46 6.46 G25 spherical 25.8718 25.8718 2.5837 1.627143 36.02 6.45 6.45 G26 spherical 88.9411 88.9411 1.4347 6.59 6.59 G27 X toroidal -88.9946 -93.0029 2.0000 1.833238 24.85 6.75 6.75 G28 X toroidal 85.7658 80.1435 0.4791 7.06 7.06 G29 spherical 65.7730 65.7730 5.3434 1.718137 52.50 7.24 7.24 G30 spherical -11.5281 -11.5281 2.0000 1.835037 25.72 7.51 7.51 G31 spherical -18.0142 -18.0142 35.1000 8.00 8.00 G33 spherical infinite infinite 2.5000 1.516800 64:17 20.73 20.73 G34 spherical infinite infinite 1.0000 21:28 21:28 picture even infinite infinite 0.0000 21.63 21.63

Tables 14 and 15 show the coefficients of the aspheric surfaces G21 and G22 according to the definition equation of the vertex shape. Table 14: Coefficients of the aspheric surface G21 parameter value Y radius -22.804200 Conic constant (K) 0 4th order - coefficient A 0.000139131 6th order - coefficient B 9.321000E-07 8th order - coefficient C -1.585800E-08 10th order - coefficient D 6.957600E-11 Table 15: Coefficients of the aspheric surface G22 parameter value Y radius -27.648200 Conic constant (K) 0 4th order - coefficient A 0.000171981 6th order - coefficient B 9.384600E-07 8th order - coefficient C -1.126200E-08 10th order - coefficient D 5.169000E-11 12th order - coefficient E -1.850900E-13

the 8c until 8e show transverse aberration diagrams for the lens 1 of the seventh embodiment.

Transverse aberration diagrams are often used to measure the image performance of lenses such as B. photographic lenses: The ordinate characterizes the deviation of a ray in the image plane Δy' from the point of passage of the main ray of the same ray bundle of an object point in the image plane depending on the pupil coordinate y _EP (abscissa). The scale of the ordinate is 0.025 mm. The transverse aberration is shown for different field points, characterized by the "relative field" -1 ≤ x_normalized ≤ 1 and -1 ≤ y_normalized ≤ 1, where x_normalized = x/18 mm and y_normalized = y/12 mm (i.e. image field of a full-frame camera 36 mm × 24 mm) or also (in brackets) by the object-side field angle, expressed by the x, y component of the principal ray in the object space relative to the optical axis z, for three different wavelengths of light, namely 486 nm, 587 nm and 656 nm. The beam deviations are well within the +/-25 micron scale for all field points, all wavelengths and across the entire aperture: This means that both the image sharpness and the color errors for a camera with a modern full-frame image sensor are over the entire field of view are very well corrected.

8f shows the course of the modulation transfer function of the lens 1 of the seventh embodiment.

As with lens 1 of the fifth exemplary embodiment, the image performance at 10 lp/mm for almost the entire image field is over 80% (indicated in the diagram as a relative value of 0.8) and at 30 lp/mm over 30%, with the contrast in the center of the image is higher than at the edge. This means that the picture performance of this Pannini lens is as high as that of a very good, conventional, rotationally symmetrical lens for a professional full-frame system camera.

the 9a and 9b show the lens 1 of an eighth embodiment with a focusing setting at infinity, which has the same characteristics as the lens of the seventh embodiment (stereographic-rectilinear (d=1) Pannini lens 2.8/15mm). The objective 1 does not contain any toric surfaces, but instead 4 cylindrical lenses (lenses 1 to 4) and, as in the seventh exemplary embodiment, an aspherical and otherwise spherical lens. The image performance is comparable to that of the seventh embodiment. However, the front lens diameter is larger and the system is heavier. Use in cinematography would be possible here.

The basic design data of the lens 1 of the eighth exemplary embodiment are given in Table 16 below, with the tolerance range of the given values being able to be ±5%. The curved surfaces of the image-forming optical elements of the eighth embodiment are denoted by H1 to H34. Reference is made to the first exemplary embodiment for a more detailed explanation of the basic design data. The surfaces H33 and H34 are flat plates in front of the image sensor. For example, this can be protective glass for the image sensor, optionally with an integrated low-pass and/or infrared filter. Table 16: Basic construction data of the eighth embodiment Surface surface type Crest radius of surface curvature in y-direction [mm] Crest radius of surface curvature in x-direction [mm] Distance to the following surface (air distance or lens thickness) [mm] refractive index n _d Abbe number v _d Half the diameter of the optically used area in the y-direction [mm] Half the diameter of the optically used area in the x-direction [mm] object spherical infinite infinite infinite H1 spherical infinite infinite 0.0000 138:14 138:14 H2 cylindrical 52.7797333 infinite 6.1212 1.649091 33.75 50.00 109.00 H3 X toroidal 43.7019233 1234.6786 16.0226 43.70 87.67 H4 spherical 81.1934487 81.1934487 12.6841 1.758552 50.29 55.26 55.26 H5 cylindrical infinite 74.5305096 5.0599 50.36 50.36 H6 cylindrical infinite 71.1497869 2.0000 1.678241 31:33 37.00 42.84 H7 spherical 32.0775894 32.0775894 15.0614 29.28 29.28 H8 cylindrical infinite 172.124643 4.2314 1.846057 23.87 21.00 30.90 H9 cylindrical -184.107889 infinite 0.1000 30:19 30:19 H10 spherical 26.177257 26.177257 2.0000 1.772500 49.62 18:23 18:23 H11 spherical 12.3346843 12.3346843 7.8762 12:17 12:17 H12 spherical 131.967638 131.967638 2.0000 1.772500 49.62 12.75 12.75 H13 spherical 18.8408495 18.8408495 7.2076 11.60 11.60 H14 spherical 34.2094316 34.2094316 5.1036 1.636829 34.96 13.00 13.00 H15 spherical -39.5143747 -39.5143747 1.4420 13.00 13.00 H16 spherical -23.3116845 -23.3116845 2.6618 1.486562 84.47 13:10 13:10 H17 spherical -19.1550453 -19.1550453 0.1000 13.20 13.20 H18 spherical 32.8029406 32.8029406 3.1860 1.846660 23.78 9.82 9.82 H19 spherical 417.224372 417.224372 2.0000 1.772500 49.62 9:11 9:11 H20 spherical 34.514103 34.514103 2.2663 7.94 7.94 H21 aspheric -15.0630912 -15.0630912 2.0000 1.772500 49.62 8.03 8.03 H22 aspheric -16.6392759 -16.6392759 1.0000 7.59 7.59 aperture stop spherical infinite infinite 1.0000 6.65 6.65 H24 spherical -99.1925698 -99.1925698 2.0000 1.809425 31.69 6.84 6.84 H25 spherical -560.395904 -560.395904 3.2878 1.486562 84.47 7:19 7:19 H26 spherical -20.9915023 -20.9915023 0.1000 7.57 7.57 H27 spherical -124.014065 -124.014065 2.0000 1.846660 23.78 7.70 7.70 H28 spherical 30.3251839 30.3251839 0.1000 7.99 7.99 H29 spherical 30.1437681 30.1437681 5.7879 1.506145 78.81 8.04 8.04 H30 spherical -12.6685526 -12.6685526 2.0000 1.833476 26.01 8:31 8:31 H31 spherical -15.9602156 -15.9602156 35.1000 9.00 9.00 H33 spherical infinite infinite 2.5000 1.516800 64:17 20.69 20.69 H34 spherical infinite infinite 1.0000 21:29 21:29 picture even infinite infinite 0.0000 21.84 21.84

Tables 17 and 18 show the coefficients of the aspheric surfaces H21 and H22 according to the definition equation of the vertex shape. Table 17: Coefficients of the aspherical surface H21 parameter value Y radius -15.063091 Conic constant (K) 0 4th order - coefficient A 0.000145577 6th order - coefficient B 1.502400E-06 8th order - coefficient C -1.403700E-08 10th order - coefficient D 4.551600E-11 Table 18: Coefficients of the aspherical surface H22 parameter value Y radius -16.639276 Conic constant (K) 0 4th order - coefficient A 0.000168764 6th order - coefficient B 8.622900E-07 8th order - coefficient C 5.969400E-09 10th order - coefficient D -1.936900E-10 12th order - coefficient E 1.286300E-12

10 shows an exemplary imaging optical device 10, which is designed as a photo camera. The imaging optical device 10 has an objective 1 with a plurality of image-forming optical elements 2 which are arranged along the optical axis z. The image-forming optical elements 2 are in 10 stylized in the form of four lenses. The actual structure of the lens 1 can correspond to the first to eighth exemplary embodiments, for example. In addition to the lens 1, the imaging optical device 10 has an image surface 12 arranged in a housing 11, which in 10 is shown flat, but alternatively also curved, z. B. cylindrically curved, can be. The image area corresponds to the image sensor, which can be designed, for example, as a full-frame image sensor 12 (image circle diameter 43.2 mm).

11 shows a mobile terminal 100 in a schematic representation. The mobile terminal 100 can be a smartphone, for example. The mobile terminal 100 has a camera as imaging optical device 10, which in turn has a lens 1, which is designed as a Pannini lens.

12 shows a set 200, a lens 1, for example formed as with reference to one of 2 until 9 explained, and includes a user guide 201 with instructions 202. The instructions 202 are directed to the use of the lens 1 and specify that the lens 1 is to be used with a flat image surface 3 or, alternatively, with a curved image surface 3 .

As an alternative to the lens 1, the set 200 can also include an imaging optical device 10 with such a lens 1.

13 shows footprints across the field of view in different lens planes. Reference is made to the relevant explanations in the above description.

It is understood that other embodiments may be utilized and structural or logical changes may be made without departing from the scope of the present invention. The features of the various exemplary embodiments described herein can be combined with one another unless specifically stated otherwise. The description of the embodiments is, therefore, not to be taken in a limiting sense, and the scope of the present invention is defined by the appended claims. In the figures, identical or similar elements are provided with identical reference symbols, insofar as this is appropriate.

As used herein, the term "and/or" when used in a series of two or more items means that each of the listed items can be used alone, or any combination of two or more of the listed items can be used. For example, if a lens is described as having elements A, B, and/or C, the lens can be A alone, B alone, C alone, A and B in combination, A and C in combination, B and C in combination, or A , B and C in combination.

Reference List

1

lens

2

image-forming optical element

3

screen

4

aperture stop

10

imaging optical device

11

Housing

12

screen

100

mobile device

200

set

201

Instructions for use

202

Instruction

A1-A12

curved surfaces of the image-forming optical elements of the first embodiment

B1 - B14

curved surfaces of the image-forming optical elements of the second embodiment

C1 - C11

curved surfaces of the image-forming optical elements of the third embodiment

D1 - D19

curved surfaces of the image-forming optical elements of the fourth embodiment

E1 - E19

curved surfaces of the image-forming optical elements of the fifth embodiment

F1 - F30

curved surfaces of the image-forming optical elements of the sixth embodiment

G1 - G34

curved surfaces of the image-forming optical elements of the seventh embodiment

H1 - H34

curved surfaces of the image-forming optical elements of the eighth embodiment

i.e

Distance from the perspective center to the coordinate origin

dHE

distance of the main planes

DCA

optically clear diameter

DSA

subaperture diameter

FOV

field of view angle field of view

hFOV

horizontal field of view angle horizontal field of view

vFOV

vertical field of view angle vertical field of view

f

focal length

f'

image-side focal length

H

Object coordinate horizontal direction

H'

horizontal image coordinate

i

Radiation incidence angle at the interface

in the

pixel

lp

line pairs

MTF

modulation transfer function

n, n'

Refractive indices of the media at the interface

no'

Refractive index between last optical surface and image surface

O

coordinate origin

If

object point

P

Meeting point of the object beam with the panosphere

personal computer

perspective center

s

object beam

SAR

subaperture ratio

t

main beam

and

Ray angle to the optical axis

uk'

Beam angle to the optical axis between the last optical surface and the image surface

v

Object coordinate vertical direction

v'

vertical image coordinate

w

Ray height at the interface to the optical axis

x

horizontal direction

y

vertical direction

e.g

optical axis

(x,y,z)

Coordinates of an object point in object space

(x',y')

Coordinates of a pixel in the image plane

φ

Angle between y tangent plane and optical axis

φx,1, φx,2

refractive powers in x-direction

φy,1, φy,2

refractive powers in the y-direction

φ

total breaking power

ϑ

object angle

QUOTES INCLUDED IN DESCRIPTION

This list of documents cited by the applicant was generated automatically and is included solely for the better information of the reader. The list is not part of the German patent or utility model application. The DPMA assumes no liability for any errors or omissions.

Patent Literature Cited

• US 2013/0022345 A1 [0025]

Claims (15)

Objective (1), having image-forming optical elements (2) with non-rotationally symmetrically acting curved surfaces, which are arranged along an optical axis (z), characterized in that the image-forming optical elements (2) are designed and arranged in such a way that a image area (3) is a Pannini map with the following characteristics: - equal reproduction scale at the image center along each direction in the image area (3), - straight vertical mapping of straight vertical lines throughout the image field, and - barrel distortion of object space in horizontal direction. Lens (1) after claim 1 , characterized in that the image-forming optical elements (2) have optically effective surfaces which are symmetrical to the x-axis and y-axis of the objective (1) and/or that the objective (1) has field diaphragms which are symmetrical to the x- Axis and y-axis of the lens (1) are. Objective (1) according to one of the preceding claims, characterized in that the image-forming optical elements (2) have cylindrically curved surfaces, toric surfaces and/or free-form surfaces. Objective (1) according to one of the preceding claims, characterized in that the objective has a field stop with a shape deviating from a circular and rectangular shape. Lens (1) after claim 4 , characterized in that the field stop is shaped three-dimensionally. Lens (1) according to one of the preceding claims, characterized in that the image-forming optical elements (2) are designed and arranged in such a way that the image to be formed can be formed on a flat image surface (3). Lens (1) according to one of Claims 1 until 5 , characterized in that the image-forming optical elements (2) are designed and arranged in such a way that the image to be formed can be formed on a cylindrically curved image surface (3). Lens (1) according to one of the preceding claims, characterized in that the lens (1) is a camera lens or a projection lens. Lens (1) according to one of the preceding claims, characterized in that the lens (1) is an attachment lens or a back adapter lens. Imaging optical device (10) with an objective (1) according to one of the preceding claims. Imaging optical device (10) after claim 10 , trained as a camera. Imaging optical device (10) after claim 11 , wherein the camera has a flat or cylindrically curved image surface (3). Imaging optical device (10) after claim 10 , designed as a projection device. Mobile terminal (100) with an imaging optical device (10) according to one of Claims 10 until 13 . Set (200) comprising: - a lens (1) according to any one of Claims 1 until 9 or an imaging optical device (10) according to one of Claims 10 until 13 , and - a user manual (201) with instructions (202) that specify the use of the lens (1) or the imaging optical device (10) with a flat or curved image surface (3).

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