DETAILED DESCRIPTION OF THE INVENTION

[0067] In U.S. Pat. No. 6,249,588, its continuation-in-part U.S. Pat. No. 5,995,638, U.S. patent application Ser. No. 09/902,445, Amidror and Hersch, and in U.S. patent application Ser. No. 10/183,550, Amidror disclose methods for the authentication of documents by using the moiré intensity profile. These methods are based on specially designed two-dimensional structures (dot-screens, pinhole-screens, microlens structures), which generate in their superposition two-dimensional moiré intensity profiles of any preferred colors and shapes (such as letters, digits, the country emblem, etc.) whose size, location and orientation gradually vary as the superposed layers are rotated or shifted on top of each other. In reflective mode and with a revealing layer (called master screen in the above mentioned inventions) embodied by an opaque layer with tiny transparent dots or holes (e.g. a film with tiny transparent holes), the amount of reflected light is too low and therefore the moiré shapes are nearly invisible. In addition, in these inventions, the base layer is made of a set (2D array) of similar dots (dot screen) where each dot has a very limited space within which one or a very small number of tiny shapes such as characters, digits or logos must be placed. This space is limited by the 2D frequency of the dot screen, i.e. by its two period vectors. The higher the 2D frequency, the less space there is for placing the tiny shapes which, when superposed with a 2D circular dot screen as revealing layer, produce as 2D moiré an enlargement of these tiny shapes.

[0068] To make the moiré patterns visible under normal light conditions, in reflective mode or in transparent mode without a light table, the present inventors disclose a new category of moiré based methods, in which the base layer is formed by bands incorporating original patterns and the revealing layer is made of a grating of transparent lines. Such a grating is shown in FIG. 1A, where the transparent lines 11 have an aperture τ and the opaque parts 10 have a width T−τ. The moiré patterns, representing the enlarged and transformed original patterns, are very well visible because much more light is able to pass through a grating of transparent lines than through a 2D circular dot screen. For a revealing line grating of period T and aperture τ (FIG. 1A), the relative amount of light able to pass through the transparent part of the grating is τ/T. For a revealing grating made of a dot screen, i.e. horizontally and vertically repeated circular dots with horizontal and vertical repetition period T, and with a dot diameter τ (FIG. 1B), the relative amount of light able to pass through the transparent part of the dot screen is (π/4)*(τ/T)². When comparing the two methods, a line grating allows (4/π)*(T/τ) times more light to pass through its aperture than the corresponding 2D circular dot screen. With an aperture τ/T of 1/4, 5.09 times more light passes through the line grating aperture than through the 2D circular dot screen. With an aperture of τ/T of 1/6, the corresponding ratio is 7.6 and with an aperture of τ/T=1/10, the corresponding ratio is 12.7. Please note that the smaller the aperture, the sharper the revealed moiré patterns.

[0069] It is well known from the prior art that the superposition of two line gratings generates moiré fringes, i.e. moiré lines as shown in FIG. 2 (see for example K. Patorski, The Moiré Fringe Technique, Elsevier 1993, pp. 14-16). In the present invention, we extend the concept of line grating to band grating. A band of width T1 corresponds to one line instance of a line grating (of period T1) and may incorporate as original shapes any kind of patterns, which may vary along the band, such as black white patterns (e.g. typographic characters), variable intensity patterns and color patterns. For example, in FIG. 3, a line grating 31 and its corresponding band grating 32 incorporating in each band the vertically compressed and mirrored letters EPFL are shown. When revealed with a revealing line grating 33, one can observe on the left side the well known moiré fringe 35 and on the right side, band moiré patterns 34 (EPFL), which are an enlargement and transformation of the letters located in the base bands. These band moiré patterns 34 have the same orientation and repetition period as the moiré fringes 35. FIG. 4 gives the base layer of FIG. 3 and FIG. 5 gives its revealing layer. The revealing layer (line grating) may be photocopied on a transparent support and placed on top of the base layer. The reader may verify that when shifting the revealing line grating vertically, the band moiré patterns also undergo a vertical shift. When rotating the revealing line grating, the band moiré patterns are subject to a shearing and their global orientation is accordingly modified.

[0070] FIG. 3 also shows that the base band layer (or more precisely a single set of base bands) has only one spatial frequency component given by period T1. Therefore, while the space between each band is limited by period T1, there is no spatial limitation along the long side of the band. Therefore, a large number of patterns, for example a text sentence, may be place along each band. This is an important advantage over the prior art moiré profile based authentication methods relying on two-dimensional structures (U.S. Pat. No. 6,249,588, its continuation-in-part U.S. Pat. No. 5,995,638, U.S. patent application Ser. No. 09/902,445, Amidror and Hersch, and in U.S. patent application Ser. No. 10/183,550, Amidror).

[0071] In the section “Geometry of straight band grating moirés”, we show that a revealing layer made of a straight line grating (set of transparent lines) generates as band moiré patterns a linear transformation of the original patterns located within the individual bands. This transformation comprises an enlargement, possibly a mirroring, and possibly a shearing of the original patterns.

[0072] FIGS. 6A, 6B and 6C show a further example with a revealing layer having an oblique orientation. FIG. 6A gives the revealing line grating. It can be photocopied on a transparency and used as the revealing layer to be put on top of the base band grating shown in FIG. 6B. FIG. 6C shows the moiré patterns (“1 2 3”) generated when the base band grating and revealing line grating are superposed one on top of the other. A single horizontal base band is shown on top of FIG. 6B.

[0073] By rotating the revealing layer, one can see how the moiré patterns modify their shape. Rotating the revealing layer modifies the angle and therefore the transformation between original shape and moiré shape, yielding a transformation comprising a change of orientation of the moiré band, and a shearing of the moiré pattern.

[0074] We describe first the geometry of moirés obtained by the superposition of a base layer made of straight base bands and of a revealing layer made of a straight line grating. Then we explain how to obtain curvilinear moirés by applying geometric transformations to the base layer and possibly to the revealing layer.

[0075] Please note that all drawings showing base band patterns and revealing line grating layers are strongly enlarged in order to allow to photocopy the drawings and verify the appearance of the moiré patterns. However, in real security documents, the base band periods (T1) the revealing line grating periods (T2) will be much lower, making it very difficult or impossible to make photocopies of the base band patterns with standard photocopiers or desktop systems.

Terminology

[0076] The term security document refers to banknotes, checks, trust papers, securities, identification cards, passports, travel documents, tickets, etc.). It also refers to valuable articles (such as optical disks, CDs, DVDs, software packages, medical products, etc.) which need to be protected by a security device. A security device is a means allowing to verify the authenticity of a valuable item. Generally a security device is incorporated into a document, into the package of a valuable article or into the valuable article itself.

[0077] The term “image” characterizes images used for various purposes, such as illustrations, graphics and ornamental patterns reproduced on various media such as paper, displays, or optical media such as holograms, kinegrams, etc. . . . . Images may have a single channel (e.g. gray or single color) or multiple channels (e.g. RGB color images). Each channel comprises a given number of intensity levels, e.g. 256 levels). Multi-intensity images such as gray-level images are often called bytemaps. Hereinafter, bilevel images (e.g. intensity “0” for black and intensity “1” for white) are called bitmaps.

[0078] Printed images may be printed with standard colors (cyan, magenta, yellow and black, generally embodied by inks or toners) or with non-standard colors (i.e. colors which differ from standard colors), for example fluorescent colors (inks), ultra-violet colors (inks) as well as any other special colors such as metallic or iridescent colors (inks).

[0079] The term moiré pattern image or simply moiré image characterizes the moiré patterns produced by the superposition of a base layer made of base bands (also called base band layer) and of a line grating as the revealing layer. The terms band moiré or band moiré patterns indicate that the considered moiré patterns are produced by the superposition of a base layer made of base bands and of a revealing layer made of a grating of lines.

[0080] The base layer may comprise several different sets of base bands. Different sets of base bands are characterized by having different geometric layouts, e.g. their orientations, period or the geometric transform characterizing the layout of a set of curvilinear base bands may vary. The terms “set of base bands” or “base band grating” are equivalent.

[0081] In the present invention, we use the term line gratings in a generic way: a line grating may be embodied by a set of transparent lines (e.g. FIG. 1A, 11) on an opaque or partially opaque support (e.g. FIG. 1A, 10), by cylindric microlenses or by diffractive devices acting as cylindric microlenses. Sometimes, we use instead of the term “line grating” the term “grating of lines”. In the present invention, these two terms should be considered as equivalent.

[0082] In the literature, line gratings are generally set of parallel lines, where the transparent (or white) part (FIG. 2) is half the full width, i.e. with a ratio of τT=1/2. In the present invention, regarding the line gratings used as revealing layers, the relative width of the transparent part (aperture) will be generally lower than 1/2, for example 1/3, 1/5, 1/8, or 1/10. In the case that the line grating is embodied by an optical device such as cylindric microlenses or diffractive devices acting as cylindric microlense, an even smaller relative sampling width may chosen.

[0083] In the present invention, we assume that base bands and line gratings may be rectilinear, i.e. formed by respectively straight bands and straight lines, or curvilinear, i.e. formed respectively by curved bands and curved lines. In addition, gratings of lines need not be made of continous lines. A revealing line grating may be made of interrupted lines and still be able to produce band moiré patterns.

[0084] The term “printing” is not limited to a traditional printing process, such as the deposition of ink on a substrate. Hereinafter, it has a broader signification and encompasses any process allowing to create a pattern or to transfer a latent image on a substrate, for example engraving, photolithography, light exposition of photo-sensitive media, etching, perforating, embossing, thermoplastic recording, foil transfer, inkjet, dye-sublimation, etc. . . . .

The Geometry of Straight Band Grating Moirés

[0085] The example given in FIG. 7 shows in detail that the superposition of a base band layer 71 with base band period T1 and a revealing layer line grating 72 with line period T2 produces band moiré patterns 73 which are a transformed instance of the patterns (triangles) located in the base bands, where the transformation comprises an enlargement. Since the revealing line grating has a larger period T2 than the base band period T1, it samples different instances of base band triangles at successively different relative positions within the base bands 74.

[0086] FIG. 8 shows that the moiré patterns are a transformation of the original base band patterns 81 that are located in the present embodiment within each repetition of the base bands 82, 83, . . . of the base band layer. Patterns laid out within individual bands need not be repetitive. Single base band example 81 incorporates non repetitive patterns. In the general case, the patterns incorporated in successive base bands should be similar in order to produce moiré patterns which are a transformation (including an enlargement) of the base band patterns.

[0087] By purely geometric considerations, one can derive the transformations between the individual bands B₀, B₁, B₂, . . . incorporating the original patterns (original base band space) and the x-y space where the moiré appears (moiré space). For this purpose, consider the geometry described in FIG. 9.

[0088] Each individual band B_i of the band grating B₀, B₁, B₂, . . . is given by one band of period T1. Without loss of generality, we assume for the sake of the explanation that base bands are horizontal, i.e. their boundaries are parallel to the x-axis.

[0089] For the present geometric explanation, we assume that successive horizontal bands B₀, B₁, B₂ . . . are simply translated replications of the base band B₀. In the present case (FIG. 9), the translation is perpendicular to the band orientation and the corresponding translation vector is (0, T1).

[0090] The revealing layer is made of a grating of single lines (called impulses when their width becomes infinitely small, see R. N. Bracewell, Two Dimensional Imaging, Prentice Hall, 1995, pp 120-122, 125-127). Single lines L₀, L₁, L₂ . . . are defined by their line equation

y=(tan θ)x+k*(< italic>T2/cos θ (eq. 1)

[0091] where k is an integer giving the index of the line L_k. Line impulses have a slope of tan θ, where θ is the angle between line impulses and the base line grating. Without loss of generality, we assume that the origin of the x-y coordinate system is at the intersection between the lower boundary of band B₀ and line impulse L₀ (FIG. 9).

[0092] FIG. 10 shows that successive lines L₀, L₁, L₂, . . . of the revealing line grating sample within the parallelogram P₀′ of the base layer different bands B₀, B₁, B₂ . . . . Since vertical bands are replicates of band B0, the revealing line grating samples different (replicated) instances of the same base band patterns.

[0093] Let us consider the parallelogram P₀ defined by the intersection of line impulses L₀ and L₁ (FIG. 10) with the base grating band B₀.

[0094] Line segment l₀₁ of line L₁ intersecting band B₁ samples the same space as its translated version l₀₁′ in band B₀. Line segment l₀₂ of line L₂ intersecting band B₂ samples the same space as its translated version l₀₂′ in band B₀, etc. . . . .

[0095] Therefore, successive line segments l_0j of line impulses L_j intersecting band B_j sample the same space as their translated versions l_0j′. This establishes a linear mapping between parallelogram P₀′ and parallelogram P₀ located within band B₀.

[0096] Similarly, as shown in FIG. 11, a linear mapping exists between parallelogram P_-1 and parallelogram P_-1′, parallelogram P₀ and parallelogram P₀′, parallelogram P₁ and parallelogram P₁′, etc. . . . . The parallelograms making up band B₀ are mapped to parallelograms making up band B₀′. In a similar manner, the parallelograms Q_i composing band B₁ are mapped to parallelograms Q_i′ making up band B₁′ and so on for all the bands.

[0097] This establishes a linear mapping (here an affine mapping) from the x-y plane comprising the base line grating to the x_m-y_m plane comprising the moiré. Parameters a,b,c,d of the transformation 1

[0098] are obtained by enforcing the mapping of the fixed point (λ,T1)−>(λ,T1) and of the point (x_i,0)−>(x _i,, T1) (see FIG. 10).

[0099] These parameters are

a=1, b=0, c=T1/x_i and d=(x_i� ��λ)/xi, (eq. 3)

[0100] where λ=T1/tan θ.

[0101] x_i is the x-coordinate of the intersection of L₁ and the upper boundary of band B₀, i.e. x_i is given by the set of equations

y=(tan θ)x+(T2/cos θ)

y=T1 (eq. 4)

[0102] Solving for x gives

x_i=(T1/tan θ)−(T2/sin θ), when θ<>0 (eq. 5)

[0103] Recall that bands B₁, B₂, . . . are translated replicates of band B₀. Therefore, moiré bands B₁′, B₂′ . . . (FIG. 11) are also replicates of moiré band B₀′. According to FIG. 9, parallelogram P₀ is mapped to parallelogram P₀′ in moiré band B₀′ and at the same time to parallelogram P₀″ in moiré band B_-1′. Therefore, moiré band B₀′ is translated by (0,h) in respect to moiré band B_-1′, where according to FIG. 10, 2

[0104] Thanks to the linear mapping property, tiny visually significant patterns located within the replicated individual bands, on top of which the revealing layer is applied yield as band moiré patterns their original patterns, sheared, enlarged, and possibly mirrored.

[0105] Theoretically, when the revealing layer is made of lines being line impulses, the band moiré image is a sampled and transformed version of the patterns located within the individual bands. However, in practical applications, the grating of lines is a rect function with an aperture τ/T1 ([Amidror00], p. 21). Such a grating of lines used as the revealing layer generate moiré patterns which are a transformed low pass version of the original patterns located within the individual base bands.

[0106] One may also slightly translate the content of one band B_i in respect to its previous band B_i-1 by a value s₁. This has the effect of translating horizontally by s₁ the location of l₀₁′, by 2* s₁ the location of l₀₁′, etc. . . . . This yields a different linear mapping whose parameters can be calculated following a similar approach as the one described above.

[0107] When rotating the revealing layer, we modify angle θ and the linear transformation changes accordingly. When translating the revealing layer, we just modify the origin of the coordinate system. Up to a translation, the moiré patterns remain identical.

[0108] In the special case where the band grating (base layer) and the revealing layer have the same orientation, θ=0, (and assuming no translation between successive horizontal bands, i.e. s₁=0), the moiré patterns are simply a vertically scaled version of the patterns embedded in the replicated base bands, where the vertical scaling factor is T2/(T2 mod T1). One can easily verify by simple algebraic and trigonometric manipulations that for θ=0, and T1<T2<2*T1, the parameters in eq. 3 are c=0 and d=T2/(T2−T1).

[0109] FIG. 13 illustrates a vertical scaling example. FIG. 13, 130 shows a succession of base bands with a period T1 and incorporating a vertically reduced letter “P”. In the present examples, the the period T2 of the revealing layer is modified. Three cases may be considered. When the ratio T2/T1 is inferior to 1, the moiré patterns are the mirrored and scaled base band patterns. In FIG. 13, 131, the ratio T2a/T1 is 0.95. Thus the scaling factor d=1/(1−T1/T2) is equal to 1/(1−1/0.95)=−19. The moiré patterns (132) are the mirrored image of the base band patterns (d<0). When T1=T2 (133), the revealing layer reveals exactly the same part of each base band and the scaling factor is infinite. When the ratio T2/T1 is superior to 1, the moiré patterns are the scaled base band patterns. In FIG. 13, 134 the ratio T2c/T1 is 1.05. Thus the scaling factor d is equal to 20. The moiré patterns (135) are the base band patterns scaled by a factor 20.

[0110] With a ratio T2/T1 inferior to 1, i.e. T2<T1 (FIG. 13, 136), the base band patterns are sampled by more revealing lines of the revealing layer and their corresponding revealed moiré patterns are therefore more accurate. In this case, we may create mirrored base band patterns. Mirrored base band patterns are more difficult to perceive and may therefore be more easily hidden (see section “Combined multiple orientation band moirés”).

Generation of Band Patterns

[0111] FIG. 9 incorporates the basis layer with the band grating B₀, B₁, B₂, . . . and the revealing layer with the revealing line grating L₀, L₁, L₂. Parallelogram P₀, replicated over base bands B₁, . . . ,B₆ yields the moiré parallelogram P₀′. Replicating parallelogram P₀ over base bands B_-1, . . . , B_-6 yields moiré parallelogram P₀″. Similarly replicating parallelogram P₁ over base bands B₁, . . . ,B₆ yields the moiré parallelogram P₁′ and over base bands B_-1, . . . ,B_-6 yields moiré parallelogram P₀″. Successive parallelograms of base band B₀ cover successive moiré parallelograms.

[0112] Since the forward transformation from band patterns to moiré patterns is known, the inverse of the matrix of eq. 2 specifies the reverse transformation from moiré patterns to band patterns. For the reverse transformation, we obtain 3$\begin{array}{c}[x\\ y\end{array}]=[\begin{array}{cc}p& q\end{array}r& s& ]·[x_m\\ y_m& ]\left(\mathrm{eq}.7\right)$

[0113] The parameters are p=1, q=0, r=T1/(λ−x_i) and s=x_i/(x_i−λ).

[0114] The reverse transformation may be useful for conceiving the patterns to be generated in the base bands which, when overlaid with the revealing layer, will produce the desired moiré patterns at a given angle between base layer and revealing layer.

[0115] In order to define the base and the revealing layers, one needs to define the moiré patterns that are to be visualized within the moiré bands, knowing that base band parallelograms P_i are mapped to moiré band parallelograms P_i′ and P_i″. The layout of the band moiré patterns and their corresponding base band patterns influence the selection of the base band period T1, the revealing line grating period T2 and the preferred angle θ. Good results are obtained with periods T1 and T2 which vary only by a small percentage (e.g. 5% to 10%). Angle θ should be small, generally below 30 degrees.

[0116] Bi-level base band patterns may be easily generated by standard software, such as Adobe Illustrator or Adobe Photoshop. Base band patterns may also incorporate scanned and possibly edited bitmaps incorporating the desired repetitive or non-repetitive patterns.

[0117] Variable intensity base band patterns may be created by inserting within each base band a dithered image, either black-white or color. The resulting moiré patterns will also be a variable intensity image, either black-white or color.

[0118] FIGS. 12A, 12B and 12C illustrate the layout of the base band patterns once a desired non-trivial moiré pattern image has been defined and the preferred orientation of the revealing line grating has been chosen. According to FIG. 9, moiré parallelograms P_i′ (in FIG. 12A, 121) are mapped to base band parallelograms P_i (in FIG. 12B, 122). The forward transformation given in eq. 2 specifies the mapping of the base band parallelograms (FIG. 12B) to the moiré band parallelograms in the moiré image space (FIG. 12A). FIG. 12C shows a part of the base layer made of a repetition of the base band shown in FIG. 12B.

[0119] In order to build a base band capable of yielding a desired band moiré pattern image (FIG. 12A), the base band image (bytemap or bitmap) is traversed pixel by pixel and scanline by scanline. At each pixel, the current base band parallelogram P_i (e.g. 122) and moiré band parallelogram P_i′ (e.g. 121) may be identified. According to the forward transformation, the corresponding pixel in the corresponding moiré parallelogram P_i′ is located and its intensity is obtained, possibly by interpolation between neighbouring pixels. That intensity is assigned to the current base band pixel intensity. This algorithm generates one single base band (FIG. 12B). By replicating the base band vertically, one generates the base band grating FIG. 12C).

[0120] One may optimize that algorithm by associating to a unit horizontal pixel displacement in the base band a displacement vector in the moiré band image computed according to (eq. 2). Scanning the base band horizontally corresponds in the moiré band image (FIG. 12A) to an oblique scan according to the computed displacement vector. After reaching one of the vertical boundaries of the moiré band image given by its height h, the next position is the current position modulo the height h of the band moiré parallelograms (for the calculation of h, see eq. 6).

[0121] FIG. 12A shows only one instance of the produced moiré patterns. With many vertically replicated base bands, one obtains vertically several instances of the moiré pattern shown in FIG. 12A. To obtain lateral replications of the moiré pattern, the base band pattern shown in FIG. 12B needs to be replicated horizontally along the base bands. However, one may also choose to have different moiré patterns on the left and right side of the moiré pattern shown in FIG. 12A. This would mean that the corresponding different base band patterns would need to be inserted on the left and on the right side of the pattern shown in FIG. 12B.

[0122] In order to offer a strong security against counterfeiting attempts and provide at the same time beautiful security documents, one may halftone a global image (grayscale or color) laid out over the document with a particular microstructure pattern fitted within each band of the base layers. For this purpose, one may use the method described in U.S. patent application Ser. No. 09/902,227, Images and security documents protected by microstructures, inventors R. D. Hersch, E. Forler, B. Wittwer, P. Emmel. This invention teaches how to synthesize microstructure patterns from which a global image is synthesized. Given a bitmap representation of the desired microstructure patterns, that method generates a complex dither matrix incorporating the microstructure patterns. The dither matrix is then used to dither the global image and produce the base layer. In the resulting dithered image, such a dither matrix has the effect of modifying the thicknesses of individual microstructure patterns according to the corresponding local intensities within the global image.

[0123] However, dither matrices incorporating microstructure patterns may be synthesized by other means. Oleg Veryovka and John Buchanan in their article “Texture-based Dither Matrices” Computer Graphics Forum Vol. 19, No. 1, pp 51-64, show how to build a dither matrix from an arbitrary grayscale texture or grayscale image. They apply histogram equilibration to ensure a uniform distribution of dither threshold levels. One may obtain the grayscale image from bitmap patterns by simply applying a low-pass filter on the bitmap patterns. The result is of lower quality than the method proposed in U.S. patent application Ser. No. 09/902,227, but may work for simple patterns.

[0124] A further method for creating a dither matrix incorporating the desired base band patterns consists in creating a dither matrix which modifies the intensities of respectively the pattern (foreground) or of the pattern background according to the image local intensity to be reproduced. To create such a dither matrix, let us consider the base band patterns as a mask, and let us modify the values of a standard dither matrix, for example a dither matrix producing small clustered dots (see. H. R. Kang, Digital Color Halftoning, SPIE Press, 1999, pp. 214-225). One may chose to scale and possibly shift the initial dither values within the base band pattern mask so as to fit within the first part of a partition (e.g. half) of the full range of dither values and the dither values outside the mask so as to fit within the second part of the partition (e.g. half) of the full range of dither values. Such a modified dither matrix incorporating base band patterns is shown in FIG. 14, 144. A corresponding dithered base band part of the global image is shown in FIG. 14, 146. At dark tones, the pattern is black and the pattern background is dark. At intermediate tones, the pattern is close to black and the pattern background is close to white.

[0125] The partition of the full range of dither values may be proportional to the relative surfaces of the pattern (foreground) and of its corresponding pattern background.

[0126] As an illustration of the result, FIG. 14, 141 shows a global image, 142 represents the bitmap incorporating the microstructure patterns. 144 shows an enlargement of the modified dither matrix fitted within a single base band and incorporating the base band patterns (microstructure). 145 shows the resulting dithered base band layer. The base layer is the dithered global image and its base bands incorporate the microstructure patterns. The dithering process creates the microstructure patterns within each individual base band. In the present case, base bands differ one from another by the intensity of the patterns or by the intensity of their background. One may also create a dither matrix combining thickness modification (according to U.S. patent application Ser. No. 09/902,227, see above) and modification of the patterns foreground, respectively background intensity values.

[0127] One may also generate color patterns in the basic bands within a global image by the color difference method disclosed in European Patent application 99 114 740.6 (inventors R. D. Hersch, N. Rudaz, filed Jul. 28, 1999, assignees: Orell-Füssli and EPFL) and in the publication by N. Rudaz, R. D. Hersch, Protecting identity documents with a just noticeable microstructure, Conf. Optical Security and Counterfeit Deterrence Techniques IV, 2002, SPIE Vol. 4677, pp. 101-109.

Curvilinear Band Moirés

[0128] In addition to periodic band moiré patterns, one may also create interesting curvilinear band moiré patterns. It is known from the Fourier analysis of geometrically transformed periodic structures [Amidror98] that the moiré in the superposition of two geometrically transformed periodic layers is a geometric transformation of the moiré formed between the original periodic layers.

[0129] For specifying curvilinear band moiré patterns, le us consider according to [Amidror98] a geometric transformation g₁(x,y) between a curvilinear line grating r₁(x,y) and its corresponding original periodic line grating p₁(x′), i.e. r₁(x,y)=p(g(x,y)). If we keep the same coefficients c_m as in the Fourier serie decomposition of p(x′), then 4

[0130] We also consider the geometric transformation g₂(x,y) between a revealing curvilinear line grating r₁(x,y) and its corresponding original periodic revealing line grating p₂(x′) 5

[0131] Coefficients c_m and c_n are respectively the coefficients of the Fourier series development of the original periodic straight line grating p₁(x′) and of the revealing periodic straight line grating p₂(x′).

[0132] Then, the superposition between the curvilinear line grating r₁(x,y) and the possibly curvilinear revealing layer r₂(x,y) is given by 6

[0133] Appearing moirés m(x,y) are given by partial sums within eq 8, i.e. by combinations of integer multiples of specific (m,n) terms. Such combinations form z*(k₁,k₂) terms (with z integer). 7

[0134] Each combination of (k₁,k₂) specifies a different moiré. The most visible moirés are those with low values for (k₁,k₂), for example (1,−1).

[0135] Eq. 11 defines the geometry of curvilinear line moiré (k₁,k₂). In order to to generate curvilinear moiré bands incorporating