Manufacturing security documents by intaglio-printing a three-dimensional object

Abstract

A method is provided for manufacturing a security document by intaglio-printing a rendering of a three-dimensional object on the security document. The method comprises at least the following: S1) providing a model (1) of the object with a plurality of surfaces (A, B, C, D) in three-dimensional space; S2) attributing, to each point on the surfaces (A, B, C, D), a grayscale or color value; S3) using mapping functions (M) for mapping the surfaces (A, B, C, D) into two-dimensional parameter spaces, thereby defining grayscale or color images (2) in the parameter spaces, wherein the grayscale or color value of each point in said images is given by the grayscale or color of the point on the surface corresponding to the point of the image; S4) rendering, in the parameter spaces, the grayscale or color images (2) as halftone images (3) using halftone patterns, wherein at least some of the halftone patterns are different from each other; S5) using inverses of the mapping functions for mapping the halftone images (3) as textures onto the surfaces (A, B, C, D); S6) projecting the model with the surfaces (A, B, C, D) carrying the textures into a two-dimensional printing plane using a three-dimensional projection (P), thereby projecting said halftone images (3) and textures into the printing plane; S7) rendering, within the printing plane, the surfaces (A, B, C, D) with the textures into a two-dimensional array of printing pixels and forming an intaglio printing plate (5) with recesses corresponding to the array of the printing pixels; and S8) printing, using the intaglio printing plate (5), the rendering of the three-dimensional object onto said security document.

Description

[0001] P194096PC00 2024-11-22.DOCX

[0002] 1

[0003] Manufacturing security documents by intaglio-printing a three-dimensional object

[0004] Technical Field

[0005] The invention relates to a method for manufacturing a security document by intaglio-printing a rendering of a three-dimensional object thereon. The invention also relates to a security document manufactured by this method.

[0006] Background Art

[0007] Intaglio-printing of grayscale or color images requires great skill. Typically, an image, such as a portrait, is engraved into a printing plate by forming recesses in the printing plate. The size and density of recesses is varied in order to generate the desired gray level or color.

[0008] Such security documents may, e.g., include banknotes, passports, or other documents that should be hard to counterfeit and easy to check for authenticity.

[0009] Aspects

[0010] The problem to be solved is to provide an efficient method for manufacturing a security document of this type.

[0011] This problem is solved by the method of claim 1.

[0012] Accordingly, the method for manufacturing a security document by intaglio-printing a rendering of a three-dimensional object thereon comprises at least the following:

[0013] SI : Providing a model of the object to be rendered. The model comprises a plurality of surfaces in three-dimensional space.

[0014] S2: Attributing, to each point on the surfaces, a grayscale value or color value: This allows to give the object better contrast and definition.

[0015] S3: Using mapping functions for mapping said surfaces into two- dimensional parameter spaces, thereby defining color or grayscale images in said parameter spaces. The color or grayscale value of each point in said images is given by P194096PC00 2024-11-22.DOCX

[0016] 2 the color or grayscale value of the point on the surface that corresponds to the point of the image.

[0017] S4: Rendering, in said parameter spaces, said color or grayscale images as halftone images using halftone patterns. At least some of the halftone patterns are different from each other. This results in making the rendered images more distinct.

[0018] S5: Using inverses of said mapping functions for mapping said halftone images as textures onto said surfaces. The surfaces are now textured with the halftone patterns.

[0019] S6: Projecting said model with said surfaces carrying the textured, i.e., mapped halftone images, into a two-dimensional printing plane using a three-dimensional projection, thereby projecting said textures into said printing plane.

[0020] S7: Rendering, within said printing plane, the surfaces with the textures into a two-dimensional array of printing pixels and forming an intaglio printing plate with recesses corresponding to the array of the printing pixels. This renders the textured surfaces into printing pixels in a 2D-plane, and the printing pixels are then used to form recesses in an intaglio printing plate.

[0021] S8: Printing, using the intaglio printing plate, the rendering of the three-dimensional object onto the security document.

[0022] To make the process harder to imitate, at least one of the halftone patterns may be non-repetitive over the halftone image, with "non-repetitive" as defined below.

[0023] The quality of the rendering may be improved by taking into account that the maximum spatial frequency, in the printing plane, supported in S7 and S8 is limited by the resolution of the manufacturing of the printing plate and the resolution of the printing process. In the following, we assume this maximum spatial frequency to be flmax.

[0024] In order not to exceed fl max in the printing plane, the halftone patterns used in step S4 should have a spatial frequency f2 selected to result, in the printing plane, in a spatial frequency fl smaller than the maximum spatial frequency fl max.

[0025] In some embodiments, the method may comprise checking in S7, if the maximum spatial frequency flmax is exceeded. If this is the case, at least one of the halftone patterns used in step S4 is modified and the steps S5, S6, S7 are repeated at least for the surface where the modified halftone pattern is used. Modifying the halftone pattern may include rotating the halftone pattern. P194096PC00 2024-11-22.DOCX

[0026] 3

[0027] In some embodiments, the method may comprise calculating, using the maximum spatial frequency fl max, the three-dimensional projection, and the mapping functions, at least one maximum spatial frequency f2max in the parameter spaces in S4. In that case, halftone patterns having spatial frequencies f2 smaller than the maximum spatial frequency f2max are used in the respective parameter spaces.

[0028] To generate a tilt effect on the security document, at least one of the halftone patterns comprises a first set of parallel line elements and a second set of parallel line elements, with the line elements of the first set extending transversally to the line elements of the second set. Even though this pattern is then transformed by means of the inverse of the mapping function and the three-dimensional projection, which will, in general, distort the halftone pattern, the preferential directions of the two sets of line elements will still be transversal, which allows to generate a tilt effect similar to the classic intaglio tilt effects, but the sets of lines will "hug" the contour of the surface, which makes the security feature more remarkable and therefore easier to detect and verify.

[0029] Brief Description of the Drawings

[0030] The technology will be better understood and objects other than those set forth above will become apparent when consideration is given to the following detailed description thereof. Such description makes reference to the annexed drawings, wherein:

[0031] Fig. 1 shows an example of a photograph that was used to create a 3D model,

[0032] Fig. 2 shows the 3D model made from the photograph of Fig. 1 rendered in 2D space with shadowing added,

[0033] Fig. 3 shows the 3D model of Fig. 2 but with surface grayscale values added,

[0034] Fig. 4 shows a possible choice of surfaces A - E on the 3D model,

[0035] Figs. 5 - 7 show a representation of surfaces A - C in two-dimensional parameter spaces,

[0036] Figs. 8 - 10 show the surfaces (or parts thereof) of Figs. 5 - 7 but rendered as halftone images using halftone patterns,

[0037] Fig. 11 shows a rendering of the 3D model with the textured surfaces, P194096PC00 2024-11-22.DOCX

[0038] 4

[0039] Fig. 12 shows part of surface A (as in Fig. 4) but with a different halftone pattern having first and second sets of parallel line elements to give rise to a tilting effect,

[0040] Fig. 13 shows part of a rendering of the 3D model using a surface textured as in Fig. 12,

[0041] Fig. 14 shows part of a rendering of the 3D model (in the region X denoted in Fig. 13) with a halftone pattern generating a large spatial frequency,

[0042] Fig. 15 shows the part of Fig. 14 with a halftone pattern generating a lower spatial frequency,

[0043] Fig. 16 shows steps of some embodiments of the method, and

[0044] Fig. 17 an example of a halftone pattern comprising alphanumeric elements.

[0045] Notes:

[0046] - Figs. 1 - 7 represent grayscale images that have been Jarvis-dithered for printing in a patent drawing, i.e., the dithering is added for the purpose of making the patent drawings printable. In the method, the images are true grayscale or color images, where each pixel represents a grayscale or color value. In contrast, the other figures 9ff show halftone patterns and textures illustrating embodiments of the present technique.

[0047] - Some of the figures have been cut off at the edges (e.g., Figs. 8 - 10) to allow for larger zooming into the patterns.

[0048] Modes for Carrying Out the Invention

[0049] Definitions

[0050] "Grayscale" indicates that the brightness of each point of an object, such as of a surface or an image, is defined by more than two brightness values (i.e., not only by black-and-white values but, e.g., by at least four, in particular by at least 256, different gray levels).

[0051] A "halftone pattern" is a rule for rendering a grayscale or color image as a binary halftone image consisting of pixels that are either printed or nonprinted. The printed pixels generate, on an average, regions having the grayscale or color value of the image. The printed pixels form a pattern of geometric elements, such as lines and / or dots, with the elements being thicker / larger or thinner / smaller as a function of the desired grayscale or color value. P194096PC00 2024-11-22.DOCX

[0052] 5

[0053] A halftone pattern is "non-repetitive" over a halftone image if there is no direction, within the halftone pattern, along which the same elements repeat over the whole halftone image. There may be local areas where the elements repeat, but not over the whole halftone image. Advantageously, any such area with repeating elements extends over less than half of the halftone image.

[0054] A "halftone image" is an image of black or white pixels simulating a grayscale or color image by suitable selection of the black and white pixels.

[0055] A "three-dimensional projection" maps the three-dimensional space of the object into the two-dimensional printing plane. In some embodiments, the projection may be a parallel projection or a point projection.

[0056] 3D Model Generation

[0057] The present technique may use a three-dimensional model (in the following, simple called "the model"). The model has a plurality of surfaces in three- dimensional space, which may, e.g., be represented by a set of 3D coordinates and / or surface parameters.

[0058] The model may, e.g., be generated by recording a real-life object using a 3D camera. It may also be calculated ab-initio using mathematical representations of, e.g., spheres, cylinders, polynomial surfaces, 3D-spline surfaces, etc. It may also be generated interactively, by a draftsperson, using 3D modelling software.

[0059] Figs. 1 - 3 illustrate an example of a model created by a draftsperson. The draftsperson started from a grayscale or color photograph of a person's head as shown in Fig. 1. Then, the draftsperson manually created the model using 3D modelling software. Fig. 2 illustrates a 2D-rendinging of this model with uniformly colored surfaces and shading added by the 2D modelling software. The draftsperson generated the surfaces, e.g., by composing the model from basic 3D shapes, such as spheres and polynomial surfaces.

[0060] A model as shown in Fig. 2, where the grayscale values of the surfaces are calculated based on shading (simulated lights and shadows) alone yields poor contrast in context with the rendering technique below. Therefore, for better printing results, grayscale or color values are applied to the surfaces of the model. These values can, e.g., be derived not only from the shading (light and shadows) but also from the reflectivity of the surfaces of the original object. For example, in the photograph of Fig. 1, the person wore a dark sweater, a bright shirt, a half-dark coat, and he had a white beard. Therefore, darker and brighter grayscale values are attributed to the respective parts of the model as illustrated in Fig. 3. P194096PC00 2024-11-22.DOCX

[0061] 6

[0062] Model Surfaces

[0063] Before printing, surfaces may be defined on the model. Fig. 4 shows an example of four surfaces A, B, C, and D in the model of Fig. 3, with A representing a region of bare skin of the face, B the beard, C the clothes, and D the hair. Each surface may, e.g., be described by a set of triangles or a polynomial surface, with a grayscale (optionally including color) value attributed to each point thereof.

[0064] For each surface, a mapping function M is provided that maps the 3D coordinates of the surface into a 2D parameter space, i.e.

[0065] (u, v) = M(x, y, z), (1) with x, y, z being the 3D coordinates of points on the surface in the 3D space of the model and u, v being the coordinates of the 2D parameter space.

[0066] Figs. 5 - 7 show the surfaces A - C mapped into parameter space, with the grayscale values (or colors) of the mapped points. Surface D is not shown in the following figures, but it is treated the same way. In other words, the surfaces A - D are represented by grayscale or color images in their respective parameter space.

[0067] (Note: Fig. 5 includes more than just surface A (e.g., it also includes area beneath the beard, but these parts will be covered, during final rendering, by the other surfaces and be invisible. Therefore, they need not necessarily be used in steps S4ff as described below).

[0068] Texture Generation

[0069] In the next step, textures need to be generated for the surfaces A - E. This happens in parameter space and is illustrated in Figs. 8 - 10, each of which shows a part of one of the surfaces A - C (D is processed in the same manner).

[0070] For generating the textures, the grayscale or color bitmap images in parameter space (as shown in Figs. 5 - 7) are rendered using halftone patterns, with at least some of the halftone patterns differing from each other, i.e., not all of the surfaces are rendered by means of the same halftone pattern.

[0071] For example, and as shown in Fig. 8, the skin (surface A) is rendered using a halftone pattern that contains repetitive geometric elements. The beard (surface B, Fig. 9), on the other hand, is rendered using a non-repetitive halftone pattern that contains curved elements that are bent to simulate the structure of the hair. The clothing (surface C, Fig. 10) is rendered using a halftone pattern that contains repetitive geometric elements, but with a halftone pattern that may be different from the P194096PC00 2024-11-22.DOCX

[0072] 7 one of the skin. The hair (surface D, not shown) may be rendered using a halftone pattern of repetitive line elements that illustrate a general hair direction.

[0073] Once the bitmap images in parameter space have been generated, they are mapped back, using the inverse M1of the mapping function M, onto the surfaces of the 3D model, where they are used as textures for the surfaces.

[0074] Fig. 11 shows the model with the textured surfaces A, B, C, D as rendered into 2D space.

[0075] Note that Fig. 11 does not show the eyes. The eyes may, e.g., be also treated as two separate surfaces, which are mapped into their parameter spaces, where the color or grayscale images are rendered as halftone images using suitable halftone patterns, and the resulting halftone (bitmap) images are mapped as textures back onto the eye surfaces of the 3D model.

[0076] Printing Plate and Intaglio Printing

[0077] In a next step, the textured 3D model, e.g., as shown in Fig. 11 (with textured surfaces) is rendered into a two-dimensional printing plane, such as the drawing plane of Fig. 11 using a three-dimensional projection P. As mentioned above, this projection P maps the three-dimensional space of the object into the two- dimensional printing plane, such as by using point projection or parallel projection.

[0078] The result of this step may be a bitmap "printing-plane image", such as shown in Fig. 11, which consists of a 2D-array of printing pixels. The values of the printing pixels of the printing-plane image are calculated from the projected textures of the surfaces. For example, a given printing pixel value p(i, j) of printing pixel i, j may be calculated from the weighted sum of the texture pixels overlapping, in printing space, with the printing pixel i, j, with the weights depending on the amount of overlap of the texture pixels with the given printing pixel i, j. Other techniques are known to the skilled person, see, e.g., https: / / en.wikipedia.org / wiki / Texture_mapping.

[0079] The values of the printing pixels are then used to form recesses in an intaglio printing plate. For example, a recess may be formed in the printing place if the gray level of the corresponding printing pixel is below a given threshold or, if the printing pixels are black-and-white pixels, if the corresponding printing pixel is black.

[0080] In another embodiment, if the printing pixels are grayscale pixels, the depth of the recesses formed in the printing plate are a function of the grayscale values of their pixels, with deeper recesses being formed for darker pixels.

[0081] In more general terms, the printing pixels may be calculated from the projected textures of the surface elements. P194096PC00 2024-11-22.DOCX

[0082] 8

[0083] If the textures already take account of light and shadow, as in the examples above, it is not necessary nor very useful to calculate shading (i.e., to simulate a non-uniform illumination and shade the surfaces according to their attitude in respect to the illumination) when rendering the model into the printing plane.

[0084] Hence, in more general terms, the method may comprise: including, in the textures, a shading of the object in view of a non-uniform illumination (i.e., the halftone images of step S4 include shading information and the textures generated in step S5 also include the shading information).

[0085] In that case, in step S7, the rendering of the surfaces with the textures into the two-dimensional array of printing pixels may assume a uniform illumination of the surfaces, i.e., no further shading information is added in step S7.

[0086] Suitable techniques for manufacturing the printing plate, such as based on etching, are, e.g., described in https: / / en.wikipedia.org / wiki / Intaglio_(print- making).

[0087] Finally, the printing plate manufactured in this way may be used for intaglio printing the object onto the security document.

[0088] Tilt Effect

[0089] As mentioned, the method may include steps to generate a tilt effect of the security document. This is illustrated in examples of Figs. 12 and 13. Fig. 12 shows the halftone image in parameter space corresponding to Fig. 8 of the first embodiment but with a different halftone pattern, and Fig. 13 shows the result as rendered into the printing plane.

[0090] The halftone patterns used in step S4 may comprise a first set of parallel line elements and a second set of parallel line elements, with the line elements of the first set extending transversally to the line elements of the second set. An example of such lines is shown in a region R of Fig. 12, where vertical lines V are provided in an area corresponding to the number "10" and horizontal lines H are formed in a background area around this number.

[0091] In the examples, the lines H and V are perpendicular to each other, but they may, in general, be transversal to each other, e.g., under a smallest mutual angle of at least 30°.

[0092] In the array of printing pixels as well as in the intaglio print, as shown in Fig. 13, the sets H, V of line elements will still be transversal to each other and, at least locally, the line elements within each set will still be substantially parallel, even though the pattern might be slightly distorted according to the curvature of P194096PC00 2024-11-22.DOCX

[0093] 9 the respective surface and according to the tilt of the surface in respect to the projection direction. Fig. 13 shows the sets H and V as rendered and intaglio-printed.

[0094] The tilt effect of the security document can be checked by tilting the security document about tilt axes T1 and T2, respectively. For example, when initially viewing the surface of the security document perpendicularly and then tilting the document about Tl, the line set H will become darker while the brightness of the line set V will remain substantially constant.

[0095] In contrast to conventional tilting elements of this type, however, the present tiling element is mapped to a 3D surface, apparently extending along it, thereby generating a very characteristic effect that is easier to check for tampering.

[0096] Print Resolution

[0097] Even in modern intaglio printing, the spatial resolution of what can be printed is limited, e.g., due to the maximum resolution of the process used for creating the recesses in the printing plate as well as because of, e.g., ink flow processes during the actual intaglio printing step.

[0098] Hence, in general, the process of printing the rendering in step S8 typically supports a maximum spatial frequency flmax only. Patterns having larger spatial frequency fl will blur. For example, this maximum spatial frequency fl max may be evaluated by printing sets of parallel lines with widths D and mutual distances D, i.e., with a period 2 D. In that case, if the contrast between the darkest and brightest points of the periodic line pattern falls below a given threshold of, e.g., 0.5, for D = DI, the maximum spatial frequency flmax in the printing plane is given by flmax = 1 / (2 D1). (2)

[0099] A halftone pattern as used in step S4 has one or more spatial frequencies f2 in parameter space. The spatial frequency f2 may vary over the halftone pattern and be a function of the direction within parameter space. For example, a set of parallel lines with a period 2 D2 in parameter space would have spatial frequency of 1 / (2 D2) in a direction perpendicular to the lines and a spatial frequency of 0 in a direction parallel to the lines.

[0100] In order not to exceed the spatial frequency fl in the printing plane, the halftone patterns used in step S4 should have a spatial frequency f2 below a largest spatial frequency f2max in parameter space. f2max depends, in general, on the mapping function M (or its inverse M1) as well as on the projection P. P194096PC00 2024-11-22.DOCX i o

[0101] This is illustrated in the examples of Fig. 14 and 15, which illustrates two different halftone patterns being used as textures for the skin surface A, e.g., in the region X of Fig. 11. In contrast to Fig 11, however, the halftone patterns are simple sets of parallel lines 10a (Fig. 14) and 10b (Fig. 15), respectively.

[0102] In parameter space (not shown), the patterns of lines 10a, 10b may have the same value of the maximum frequency f2, but said maximum is reached for along directions in parameter space. In other words, the halftone patterns of lines 10a, 10b are, in parameter space, identical except for being rotated in respect to each other.

[0103] However, as shown in Figs. 14 and 15, the largest spatial frequency fl in the printing plane differs for the patterns in that the pattern of lines 10a shown in Fig. 14 has a largest frequency fla larger than the largest frequency fib of the lines 10b shown in Fig. 15.

[0104] In particular, fla may be larger than the allowable maximum spatial frequency flmax while fib isn't.

[0105] This illustrates that the printing quality can be improved by using, in step S4, halftone patterns that have, in the parameter spaces, a spatial frequency f2 selected to result, in the printing plane, in a spatial frequency fl smaller than the maximum spatial frequency fl max.

[0106] For example, in step S7, a check can be made if the maximum spatial frequency flmax is exceeded and, if yes, at least one of the halftone patterns used in step S4 may be modified.

[0107] As illustrated with the example of Figs. 14 and 15, this modification can involve a simple rotation of the halftone pattern in parameter space, for example if the halftone pattern has different spatial frequencies along different directions in parameter space.

[0108] In other embodiments, the maximum allowable spatial frequencies in parameter space may be calculated if the three-dimensional project P is known in advance, together with the mapping functions M (or their inverses M1). This allows to select, in parameter space, only such halftone patterns and orientations that will result in sufficiently low spatial frequencies in the printing plane.

[0109] Example Steps

[0110] Fig. 16, illustrates, by way of example, steps that may be carried out in the method.

[0111] The model of the object comprising a plurality of surfaces in three- dimensional space is provided (Step SI), and grayscale or color values are added to P194096PC00 2024-11-22.DOCX

[0112] 11 the points of the surfaces (Step S2), thereby resulting in a model 1 of surfaces with grayscale or color values (as it is, e.g., shown in Figs. 3 and 4).

[0113] The surfaces are mapped into their two-dimensional parameter spaces (Step S3), thereby defining color or grayscale images 2 in the parameter spaces (as they are, e.g., shown Figs. 5 - 7).

[0114] The color or grayscale images 2 are rendered, in parameter space, as halftone images 3 using the halftone patterns (Step S4, with the halftone images 3, e.g., as shown in Figs. 8 - 10).

[0115] The halftone images 3 are mapped back, using the inverses M1of the mapping functions, for mapping the halftone images as textures onto the surfaces, thereby generating a textured 3D model 4 (Step S5, with the textured 3D model 4, e.g., as shown in Fig. 11).

[0116] The textured 3D model 4 is projected into the printing plane using the three-dimensional projection P, thereby projecting the textures into the printing plane and forming the array of printing pixels, which is used to form the intaglio printing plate 5 (Steps S6, S7).

[0117] The printing plate 5 is then used for intaglio-printing the security documents 6 using an intaglio printer 7 (Step S8)

[0118] Notes

[0119] As mentioned, the halftone pattern may comprise non-repetitive elements. Fig. 17 shows an example where the halftone pattern comprises alphanumeric elements, such as letters and digits, with varying "boldness" (font weight) to represent different gray levels. This type of pattern is easily recognized and therefore easier to verify.

[0120] Hence, in some embodiments, at least some of the halftone patterns comprises alphanumeric elements.

[0121] The intaglio-printed image may be machine readable. To make such machine detection easier, at least one ink used for printing may be an IR absorbing ink, an IR transparent ink, a magnetic ink, or an ink comprising a taggant, such as a fluorescent taggant.

[0122] Hence, in some embodiments, the step of printing may comprise using an ink that is at least one of the group consisting of IR absorbing inks, IR transparent inks, magnetic inks, and inks comprising a taggant, such as a fluorescent taggant. P194096PC00 2024-11-22.DOCX

[0123] 12

[0124] The security document manufactured by the present technique may be a banknote, a passport, or another document that should be hard to counterfeit and easy to check for authenticity

[0125] While there are shown and described presently preferred embodi- ments, it is to be distinctly understood that the invention is not limited thereto but may be otherwise variously embodied and practiced within the scope of the following claims.

Claims

P194096PC00 2024-11-22.DOCX13Claims1. A method for manufacturing a security document by intaglioprinting a rendering of a three-dimensional object thereon, which method comprises51) providing a model (1) of the object comprising a plurality of surfaces (A, B, C, D) in three-dimensional space,52) attributing, to each point on the surfaces (A, B, C, D), a grayscale or color value,53) using mapping functions (M) for mapping said surfaces (A, B, C, D) into two-dimensional parameter spaces, thereby defining grayscale or color images (2) in said parameter spaces, wherein the grayscale or color value of each point in said images is given by the grayscale or color value of the point on the surface corresponding to the point of the image,54) rendering, in said parameter spaces, said grayscale or color images (2) as halftone images (3) using halftone patterns, wherein at least some of the halftone patterns are different from each other,55) using inverses of said mapping functions (M) for mapping said halftone images (3) as textures onto said surfaces (A, B, C, D),56) projecting said model with said surfaces (A, B, C, D) carrying the textures into a two-dimensional printing plane using a three-dimensional projection (P), thereby projecting said textures into said printing plane,57) rendering, within said printing plane, the surfaces (A, B, C, D) with the textures into a two-dimensional array of printing pixels and forming an intaglio printing plate (5) with recesses corresponding to the array of the printing pixels, and58) printing, using the intaglio printing plate (5), the rendering of the three-dimensional object onto said security document (6).

2. The method of claim 1 wherein at least one of the halftone patterns is non-repetitive over the halftone image (3).

3. The method of any of the preceding claims wherein, in S7 and S8, the printing of the rendering supports a maximum spatial frequency fl max in the printing plane, and wherein the method comprisesP194096PC00 2024-11-22.DOCX14 using, in S4, halftone patterns that have, in the parameter spaces, a spatial frequency f2 selected to result, in the printing plane, in a spatial frequency fl smaller than the maximum spatial frequency fl max.

4. The method of claim 3 comprising, checking in S7, if the maximum spatial frequency flmax is exceeded and, if yes, modifying at least one of the halftone patterns in step S4.

5. The method of any of the claims 3 or 4 comprising, modifying the at least one halftone pattern by rotation.

6. The method of any of the claims 3 to 5 comprising, calculating, using the maximum spatial frequency fl max, the three- dimensional projection (P), and the mapping functions (M), at least one maximum spatial frequency f2max in the parameter spaces in S4, and using halftone patterns having spatial frequencies f2 smaller than the maximum spatial frequency f2max.

7. The method of any of the preceding claims wherein at least one of the halftone patterns comprises a first set (H) of parallel line elements and a second set (V) of parallel line elements, with the line elements of the first set (H) extending transversally to the line elements of the second set (V).

8. The method of any of the preceding claims wherein, in S7, the printing pixels are calculated from the projected textures.

9. The method of any of the preceding claims comprising including a shading, in said textures, of the object in view of a non- uniform illumination.

10. The method of any of the preceding claims wherein, in S7, the rendering of the surfaces (A, B, C, D) with the textures into the two-dimensional array of printing pixels assumes a uniform illumination of the surfaces (A, B, C, D).

11. The method of any of the preceding claims wherein at least some of the halftone patterns comprises alphanumeric elements.P194096PC00 2024-11-22.DOCX1512. A security document manufactured using the method of any of the preceding claims.5

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