Methods and apparatus for authentication of documents by using the intensity profile of moire patterns

New method and apparatus for authenticating security documents such as banknotes, passports, etc. which may be printed on any support, including transparent synthetic materials and traditional opaque materials such as paper. The invention is based on moire patterns occuring between superposed dot-screens. By using a specially designed basic screen and master screen, where at least the basic screen is comprised in the document, a moire intensity profile of a chosen shape becomes visible in their superposition, thereby allowing the authentication of the document. If a microlens array is used as a master screen, the document comprising the basic screen may be printed on an opaque reflective support, thereby enabling the visualization of the moire intensity profile by reflection. Different variants of the invention are disclosed, some of which are specially adapted for use as covert features. Automatic document authentication is supported by an apparatus comprising a master screen, an image acquisition means such as a CCD camera and a comparing processor whose task is to compare the acquired moire intensity profile with a prestored reference image. Depending on the match, the document handling device connected to the comparing processor accepts or rejects the document. An important advantage of the present invention is that it can be incorporated into the standard document printing process, so that it offers high security at the same cost as standard state of the art document production.

BACKGROUND OF THE INVENTION 
The present invention relates generally to the field of anticounterfeiting 
and authentication methods and devices and, more particularly, to a method 
and apparatus for authentication of valuable documents using the intensity 
profile of moire patterns. 
Counterfeiting documents such as banknotes is becoming now more than ever a 
serious problem, due to the availability of high-quality and low-priced 
color photocopiers and desk-top publishing systems (see, for example, 
"Making Money", by Gary Stix, Scientific American, March 1994, pp. 81-83). 
The present invention is concerned with providing a novel security element 
and authentication means offering enhanced security for banknotes, checks, 
credit cards, travel documents and the like, thus making them even more 
difficult to counterfeit than present banknotes and security documents. 
Various sophisticated means have been introduced in prior art for 
counterfeit prevention and for authentication of documents. Some of these 
means are clearly visible to the naked eye and are intended for the 
general public, while other means are hidden and only detectable by the 
competent authorities, or by automatic devices. Some of the already used 
anti-counterfeit and authentication means include the use of special 
paper, special inks, watermarks, micro-letters, security threads, 
holograms, etc. Nevertheless, there is still an urgent need to introduce 
further security elements, which do not considerably increase the cost of 
the produced documents. 
Moire effects have already been used in prior art for the authentication of 
documents. For example, United Kingdom Pat. No. 1,138,011 (Canadian Bank 
Note Company) discloses a method which relates to printing on the original 
document special elements which, when counterfeited by means of halftone 
reproduction, show a moire pattern of high contrast. Similar methods are 
also applied to the prevention of digital photocopying or digital scanning 
of documents (for example, U.S. Pat. No. 5,018,767 (Wicker), or U.K. Pat. 
Application No. 2,224,240 A (Kenrick & Jefferson)). In all these cases, 
the presence of moire patterns indicates that the document in question is 
counterfeit. However, in prior art no advantage is taken of the 
intentional generation of a moire pattern having a particular intensity 
profile, whose existence, and whose precise shape, are used as a means of 
authentifying the document. The only method known until now in which a 
moire effect is used to make visible an image en coded on the document (as 
described, for example, in the section "Background" of U.S. Pat. No. 
5,396,559 (McGrew)) is based on the physical presence of that image on the 
document as a latent image, using the technique known as "phase 
modulation". In this technique, a uniform line grating or a uniform random 
screen of dots is printed on the document, but within the pre-defined 
borders of the latent image on the document the same line grating (or 
respectively, the same random dot-screen) is printed in a different phase, 
or possibly in a different orientation. For a layman, the latent image 
thus printed on the document is hard to distinguish from its background; 
but when a reference transparency consisting of an identical, but 
unmodulated, line grating (respectively, random dot-screen) is superposed 
on the document, thereby generating a moire effect, the latent image 
pre-designed on the document becomes clearly visible, since within its 
pre-defined borders the moire effect appears in a different phase than in 
the background. However, this previously known method has the major flaw 
of being simple to simulate, since the form of the latent image is 
physically present on the document and only filled by a different texture. 
The existence of such a latent image on the document will not escape the 
eye of a skilled person, and moreover, its imitation by filling the form 
by a texture of lines (or dots) in an inversed (or different) phase can 
easily be carried out by anyone skilled in the graphics arts. 
The approach on which the present invention is based completely differs 
from this technique, since no phase modulation techniques are used, and 
furthermore, no latent image is present on the document. On the contrary, 
all the spatial information which is made visible by the moire intensity 
profiles according to the present invention is encoded in the specially 
designed forms of the individual dots which constitute the dot-screens. 
The approach on which the present invention is based further differs from 
that of prior art in that it not only provides full mastering of the 
qualitative geometric properties of the generated moire (such as its 
period and its orientation), but it also enables the intensity levels of 
the generated moire to be quantitatively determined. 
SUMMARY OF THE INVENTION 
The present invention relates to a new method and apparatus for 
authenticating documents such as banknotes, trust papers, securities, 
identification cards, passports, etc. This invention is based on the moire 
phenomena which are generated between two or more specially designed 
dot-screens, at least one of which being printed on the document itself. 
Each dot-screen consists of a lattice of tiny dots, and is characterized 
by three parameters: its repetition frequency, its orientation, and its 
dot shapes. The dot-screens used in the present invention are similar to 
dot-screens which are used in classical halftoning, but they have 
specially designed dot shapes, frequencies and orientations, in accordance 
with the present disclosure. Such dot-screens with simple dot shapes may 
be produced by classical (optical or electronic) means, which are well 
known to people skilled in the art. Dot-screens with more complex dot 
shapes may be produced by means of the method disclosed in co-pending U.S. 
patent application Ser. No. 08/410,767 filed Mar. 27, 1995 (Ostromoukhov, 
Hersch). 
When the second dot-screen (hereinafter: "the master screen") is laid on 
top of the first dot-screen (hereinafter: "the basic screen"), in the case 
where both screens have been designed in accordance with the present 
disclosure, there appears in the superposition a highly visible repetitive 
moire pattern of a predefined intensity profile shape. For example, the 
repetitive moire pattern may consist of any predefined letters, digits or 
any other preferred symbols (such as the country emblem, the currency, 
etc.). 
As disclosed in U.S. Pat. No. 5,275,870 (Halope et al.) it may be 
advantageous in the manufacture of long lasting documents or documents 
which must withstand highly adverse handling to replace paper by synthetic 
material. Transparent sheets of synthetic materials have been successfully 
introduced for printing banknotes (for example, Australian banknotes of 5 
or 10 Australian Dollars). 
The present invention concerns a new method for authenticating documents 
which may be printed on various supports, including (but not limited to) 
such transparent synthetic materials. In one embodiment of the present 
invention, the moire intensity profile shapes can be visualized by 
superposing a basic screen and a master screen which are both printed on 
two different areas of the same document (banknote, etc.). In a second 
embodiment of the present invention, only the basic screen appears on the 
document itself, and the master screen is superposed on it by the human 
operator or the apparatus which visually or optically validates the 
authenticity of the document. In a third embodiment of this invention, the 
basic screen appears on the document itself, and the master screen which 
is used by the human operator or by the apparatus is a sheet of 
microlenses (hereinafter: "microlens array"). An advantage of this third 
embodiment is that it applies equally well to both transparent support, 
where the moire is observed by transmittance, and to opaque support, where 
the moire is observed by reflection. (The term "opaque support" as 
employed in the present disclosure also includes the case of transparent 
materials which have been made opaque by an inking process or by a 
photographic or any other process.) 
The fact that moire effects generated between superposed dot-screens are 
very sensitive to any microscopic variations in the screened layers makes 
any document protected according to the present invention practically 
impossible to counterfeit, and serves as a means to distinguish easily 
between a real document and a falsified one. 
It should be noted that the dot-screens which appear on the document itself 
in accordance with the present invention may be printed on the document 
like any screened (halftoned) image, within the standard printing process, 
and therefore no additional cost is incurred in the document production. 
Furthersore, the dot-screens printed on the document in accordance with the 
present invention need not be of a constant intensity level. On the 
contrary, they may include dots of gradually varying sizes and shapes, and 
they can be incorporated (or dissimulated) within any halftoned image 
printed on the document (such as a portrait, landscape, or any decorative 
motif, which may be different from the motif generated by the moire effect 
in the superposition). To reflect this fact, the terms "basic screen" and 
"master screen" used hereinafter will also include cases where the basic 
screens (respectively: the master screens) are not constant and represent 
halftoned images. (As is well known in the art, the dot sizes in halftoned 
images determine the intensity levels in the image: larger dots give 
darker intensity levels, while smaller dots give brighter intensity 
levels.) 
In the present disclosure different variants of the invention are 
described, some of which are intended to be used by the general public 
(hereinafter: "overt" features), while other variants can only be detected 
by the competent authorities or by automatic devices (hereinafter: 
"covert" features). In the latter case, the information carried by the 
basic screen is masked using any of a variety of techniques, which can be 
classified into three main methods: the masking layer method; the 
composite basic screen method; the perturbation patterns method; and any 
combinations thereof. These different variants of the present invention 
are described in detail later in the present disclosure. Also described in 
the present disclosure is the multichromatic case, in which the 
dot-screens used are multichromatic, thereby generating a multichromatic 
moire effect. 
The terms "print" and "printing" in the pre sent disclosure refer to any 
process for transferring an image on to a support, including by means of a 
lithographic, photographic or any other process. 
The disclosures "A generalized Fourier-based method for the analysis of 2D 
moire envelope-forms in screen superpositions" by I. Amidror, Journal of 
Modem Optics, Vol. 41, 1994, pp. 1837-1862 (hereinafter, "Amidro94") and 
U.S. patent application Ser. No. 08/410,767 (Ostromoukhov, Hersch) have 
certain information an d content which may relate to the present invention 
and aid in understanding thereof.

DETAILED DESCRIPTION 
The present invention is based on the intensity profiles of the moire 
patterns which occur in the superposition of dot-screens. The explanation 
of these moire intensity profiles is based on the duality between 
two-dimensional (hereinafter: "2D") periodic images in the (x,y) plane and 
their 2D spectra in the (u,v) frequency plane through the 2D Fourier 
transform. For the sake of simplicity, the explanation hereinafter is 
given for the monochromatic case, although the present invention is not 
limited only to the monochromatic case, and it relates just as well to the 
moire intensity profiles in the multichromatic case. 
As is known to people skilled in the art, any monochromatic image can be 
represented in the image domain by a reflectance function, which assigns 
to each point (x,y) of the image a value between 0 and 1 representing its 
light reflectance: 0 for black (i.e. no reflected light), 1 for white 
(i.e. full light reflectance), and intermediate values for in-between 
shades. In the case of transparencies, the reflectance function is 
replaced by a transmittance function defined in a similar way. When m 
monochromatic images are superposed, the reflectance of the resulting 
image is given by the product of the reflectance functions of the 
individual images: 
EQU r(x,y)=r.sub.1 (x,y)r.sub.2 (x,y) . . . r.sub.m (x,y) (1) 
According to a theorem known in the art as "the Convolution theorem", the 
Fourier transform of the product function is the convolution of the 
Fourier transforms of the individual functions (see, for example, "Linear 
Systems, Fourier Transforms, and Optics" by J. D. Gaskill, 1978, p. 314). 
Therefore, denoting the Fourier transform of each function by the 
respective capital letter and the 2D convolution by "**", the spectrum of 
the superposition is given by: 
EQU p(u,v)=r.sub.1 (u,v)r.sub.2 (u,v) . . . r.sub.m (u,v) (2) 
In the present disclosure we are basically interested in periodic images, 
such as line-gratings or dot-screens, and their superpositions. This 
implies that the spectrum of the image on the (u,v)-plane is not a 
continuous one but rather consists of impulses, corresponding to the 
frequencies which appear in the Fourier series decomposition of the image 
(see, for example, "Linear Systems, Fourier Transforms, and Optics" by J. 
D. Gaskill, 1978, p. 113). A strong impulse in the spectrum indicates a 
pronounced periodic component in the original image at the frequency and 
direction represented by that impulse. In the case of a 1-fold periodic 
image, such as a line-grating, the spectrum consists of a ID "comb" of 
impulses through the origin; in the case of a 2-fold periodic image the 
spectrum is a 2D "nailbed" of impulses through the origin. 
Each impulse in the 2D spectrum is characterized by three main properties: 
its label (which is its index in the Fourier series development); its 
geometric location in the spectrum plane (which is called: "the impulse 
location"), and its amplitude. To the geometric location of any impulse is 
attached a frequency vector f in the spectrum plane, which connects the 
spectrum origin with the geometric location of the impulse. In terms of 
the original image, the geometric location of an impulse in the spectrum 
determines the frequency and the direction of the corresponding periodic 
component in the image, and the amplitude of the impulse represents the 
intensity of that periodic component in the image. 
The question of whether or not an impulse in the spectrum represents a 
visible periodic component in the image strongly depends on properties of 
the human visual system. The fact that the eye cannot distinguish fine 
details above a certain frequency (i.e. below a certain period) suggests 
that the human visual system model includes a low-pass filtering stage. 
When the frequencies of the original image elements are beyond the limit 
of frequency visibility, the eye can no longer see them; but if a strong 
enough impulse in the spectrum of the image superposition falls closer to 
the spectrum origin, then a moire effect becomes visible in the superposed 
image. 
According to the Convolution theorem (Eqs. (1), (2)), when m line-gratings 
are superposed in the image domain, the resulting spectrum is the 
convolution of their individual spectra. This convolution of combs (or 
nailbeds) can be seen as an operation in which frequency vectors from the 
individual spectra are added vectorially, while the corresponding impulse 
amplitudes are multiplied. More precisely, each impulse in the 
spectrum-convolution is generated during the convolution process by the 
contribution of one impulse from each individual spectrum: its location is 
given by the sum of their frequency vectors, and its amplitude is given by 
the product of their amplitudes. This enables us to introduce an indexing 
method for denoting each of the impulses of the spectrum-convolution in a 
unique, unambiguous way. The general impulse in the spectrum-convolution 
will be denoted the "(k.sub.1,k.sub.2, . . . ,k.sub.m)-impulse," where m 
is the number of superposed gratings, and each integer k.sub.i is the 
index (harmonic), within the comb (the Fourier series) of the i-th 
spectrum, of the impulse that this i-th spectrum contributed to the 
impulse in question in the convolution. Using this formal notation the 
geometric location of the general (k.sub.1,k.sub.2, . . . 
,k.sub.m)-impulse in the spectrum-convolution is given by the vectorial 
sum (or linear combination): 
EQU f.sub.k.sbsb.1.sub.,k.sbsb.2.sub., . . . ,k.sbsb.m =k.sub.1 f.sub.1 
+k.sub.2 f.sub.2 +. . . +k.sub.m f.sub.m (3) 
and the impulse amplitude is given by: 
EQU a.sub.k.sbsb.1.sub.,k.sbsb.2.sub., . . . ,k.sbsb.m =a.sup.(1).sub.k.sbsb.1 
a.sup.(2).sub.k.sbsb.2. . . a.sup.(m).sub.k.sbsb.m (4) 
where f.sub.i denotes the frequency vector of the fundamental impulse in 
the spectrum of the i-th grating, and k.sub.i f.sub.i and 
a.sup.(i).sub.k.sbsb.i are respectively the frequency vector and the 
amplitude of the k.sub.i -th harmonic impulse in the spectrum of the i-th 
grating. 
A (k.sub.1,k.sub.2, . . . ,k.sub.m)-impulse of the spectrum-convolution 
which falls close to the spectrum origin, within the range of visible 
frequencies, represents a moire effect in the superposed image. See for 
example the moire effect in the two-grating superposition of FIG. 1C, 
which is represented in the spectrum convolution by the (1,-1)-impulse 
shown by 11 in FIG. 1F (obviously, this impulse is also accompanied by its 
respective symmetrical twin 12 to the opposite side of the spectrum 
origin, namely, the (-1,1)-impulse. The range of visible frequencies is 
schematically represented in FIG. 1F by circle 10). We call the m-grating 
moire whose fundamental impulse is the (k.sub.1,k.sub.2, . . . 
,k.sub.m)-impulse in the spectrum-convolution a "(k.sub.1,k.sub.2, . . . 
,k.sub.m)-moire"; the highest absolute value in the index-list is called 
the "order" of the moire. For example, the 2-grating moire effect of FIGS. 
1C and 1F is a (1,-1)-moire, which is a moire of order 1. It should be 
noted that in the case of doubly periodic images, such as in dot-screens, 
each superposed image contributes two perpendicular frequency vectors to 
the spectrum, so that in Eqs. (3) and (4) m represents twice the number of 
superposed images. 
The vectorial sum of Eq. (3) can also be written in terms of its Cartesian 
components. If f.sub.i are the frequencies of the m original gratings and 
.theta..sub.i are the angles that they form with the positive horizontal 
axis, then the coordinates (f.sub.u,f.sub.v) of the (k.sub.1,k.sub.2, . . 
. ,k.sub.m)-impulse in the spectrum-convolution are given by: 
EQU f.sub.u.spsb.k.sub.1.spsb.,k.sub.2.spsb., . . . ,k.sub.m =k.sub.1 f.sub.1 
cos.theta..sub.1 +k.sub.2 f.sub.2 cos.theta..sub.2 +. . . +k.sub.m f.sub.m 
cos.theta..sub.m 
EQU f.sub.v.spsb.k.sub.1.spsb.,k.sub.2.spsb., . . . ,k.sub.m =k.sub.1 f.sub.1 
sin.theta..sub.1 +k.sub.2 f.sub.2 sin.theta..sub.2 +. . . +k.sub.m f.sub.m 
sin.theta..sub.m (5) 
Therefore, the frequency, the period and the angle of the (k.sub.1,k.sub.2, 
. . . ,k.sub.m)-impulse (and of the (k.sub.1,k.sub.2, . . . 
,k.sub.m)-moire it represents) are given by the length and the direction 
of the vector f.sub.k.sbsb.1.sub.,k.sbsb.2.sub., . . . ,k.sbsb.m, as 
follows: 
##EQU1## 
Note that in the special case of the (1,-1)-moire between m=2 gratings, 
where a moire effect occurs due to the vectorial sum of the frequency 
vectors f.sub.1 and -f.sub.2, these formulas are reduced to the well-known 
formulas of the period and angle of the moire effect between two gratings: 
##EQU2## 
(where T.sub.1 and T.sub.2 are the periods of the two original gratings 
and .alpha. is the angle difference between them, .theta..sub.2 
-.theta..sub.1). When T.sub.1 =T.sub.2 this is further simplified into the 
well-known formulas: 
##EQU3## 
The moire patterns obtained in the superposition of periodic structures can 
be described at two different levels. The first, basic level only deals 
with geometric properties within the (x,y)-plane, such as the periods and 
angles of the original images and of their moire patterns. The second 
level also takes into account the amplitude properties, which can be added 
on top of the planar 2D descriptions of the original structures or their 
moire patterns as a third dimension, z=g(x,y), showing their intensities 
or gray-level values. (In terms of the spectral domain, the first level 
only considers the impulse locations (or frequency vectors) within the 
(u,v)-plane, while the second level also considers the amplitudes of the 
impulses.) This 3D representation of the shape and the intensity 
variations of the moire pattern is called "the moire intensity profile". 
The present disclosure is based on the analysis, using the Fourier 
approach, of the intensity profiles of moire patterns which are obtained 
in the superposition of periodic layers such as line-gratings, 
dot-screens, etc. This analysis is described in the following section for 
the simple case of line-grating superpositions, and then, in the next 
section, for the more complex case of dot-screen superpositions. 
Moires between superposed line-gratings 
Assume that we are given two line-gratings (like in FIG. 1A and FIG. 1B). 
The spectrum of each of the line-gratings (see FIG. 1D and FIG. 1E, 
respectively) consists of an infinite impulse-comb, in which the amplitude 
of the n-th impulse is given by the coefficient of the n-harmonic term in 
the Fourier series development of that line-grating. When we superpose 
(i.e. multiply) two line-gratings the spectrum of the superposition is, 
according to the Convolution theorem, the convolution of the two original 
combs, which gives an oblique nailbed of impulses (see FIG. 1F). Each 
moire which appears in the grating superposition is represented in the 
spectrum of the superposition by a comb of impulses through the origin 
which is included in the nailbed. If a moire is visible in the 
superposition, it means that in the spectral domain the fundamental 
impulse-pair of the moire-comb (11 and 12 in FIG. 1F) is located close to 
the spectrum origin, inside the range of visible frequencies (10); this 
impulse-pair determines the period and the direction of the moire. Now, by 
extracting from the spectrum-convolution only this infinite moire-comb 
(FIG. 1H) and taking its inverse Fourier transform, we can reconstruct, 
back in the image domain, the isolated contribution of the moire in 
question to the image superposition; this is the intensity profile of the 
moire (see FIG. 1G). 
We denote by c.sub.n the amplitude of the n-th impulse of the moire-comb. 
If the moire is a (k.sub.1,k.sub.2)-moire, the fundamental impulse of its 
comb is the (k.sub.1,k.sub.2)-impulse in the spectrum-convolution, and the 
n-th impulse of its comb is the (nk.sub.1,nk.sub.2)-impulse in the 
spectrum-convolution. Its amplitude is given by: 
EQU c.sub.n =a.sub.nk.sbsb.1.sub.,nk.sbsb.2 
and according to Eq. (4): 
EQU c.sub.n =a.sup.(1).sub.nk.sbsb.1 a.sup.(2).sub.nk.sbsb.2 
where a.sup.(i).sub.i and a.sup.(2).sub.i are the respective impulse 
amplitudes from the combs of the first and of the second line-gratings. In 
other words: 
Result 1: The impulse amplitudes of the moire-comb in the 
spectrum-convolution are determined by a simple term-by-term 
multiplication of the combs of the original superposed gratings (or 
subcombs thereof, in case of higher order moires). 
For example, in the case of a (1,-1)-moire (as in FIG. 1F) the amplitudes 
of the moire-comb impulses are given by: c.sub.n =a.sub.n,-n 
=a.sup.(1).sub.n a.sup.(2).sub.-n. 
However, this term-by-term multiplication of the original combs (i.e. the 
term-by-term product of the Fourier series of the two original gratings) 
can be interpreted according to a theorem, which is the equivalent of the 
Convolution theorem in the case of periodic functions, and which is known 
in the art as the T-convolution theorem (se e "Fourier theorems" by 
Champeney, 1987, p. 166; "Trigonometric Series Vol. 1" by Zygmund, 1968, 
p. 36): 
T-convolution theorem: Let f(x) and g(x) be functions of period T 
integrable on a one-period interval 0,T) and let {F.sub.n } and {G.sub.n } 
(for n=0,.+-.1,.+-.2, . . . ) be their Fourier series coefficients. Then 
the function: 
##EQU4## 
(where .intg..sub.T means integration over a one-period interval), which 
is called "the T-convolution of f and g", and denoted by "f*g," is also 
periodic with the same period T and has Fourier series coefficients 
{H.sub.n } given by: H.sub.n =F.sub.n G.sub.n for all integers n. 
The T-convolution theorem can be rephrased in a more illustrative way as 
follows: If the spectrum of f(x) is a comb with fundamental frequency of 
1/T and impulse amplitudes {F.sub.n }, and the spectrum of g(x) is a comb 
with the same fundamental frequency and impulse amplitudes {G.sub.n }, 
then the spectrum of the T-convolution f*g is a comb with the same 
fundamental frequency and with impulse amplitudes of {F.sub.n G.sub.n }. 
In other words, the spectrum of the T-convolution of the two periodic 
images is the product of the combs in their respective spectra. 
Using this theorem, the fact that the comb of the (1,-1)-moire in the 
spectral domain is the term-by-term product of the combs of the two 
original gratings (Result 1) can be interpreted back in the image domain 
as follows: 
The intensity profile of the (1,-1)-moire generated in the superposition of 
two line-gratings with identical periods T is the T-convolution of the two 
original line-gratings. If the periods are not identical, they must be 
first normalized by stretching and rotation transformations, as disclosed 
in Appendix A of "Amidror94." This result can be further generalized to 
also cover higher-order moires: 
Result 2: The intensity profile of the general (k.sub.1,k.sub.2)-moire 
generated in the superposition of two line-gratings with periods T.sub.1 
and T.sub.2 and an angle difference a can be seen from the image-domain 
point of view as a normalized T-convolution of the images belonging to the 
k.sub.1 -subcomb of the first grating and to the k.sub.2 -subcomb of the 
second grating. In more detail, this can be seen as a 3-stage process: 
(1) Extracting the k.sub.1 -subcomb (i.e. the partial comb which contains 
only every k.sub.1 -th impulse) from the comb of the first original 
line-grating, and similarly, extracting the k.sub.2 -subcomb from the comb 
of the second original grating. 
(2) Normalization of the two subcombs by linear stretching- and 
rotation-transformations in order to bring each of them to the period and 
the direction of the moire, as they are determined by Eq. (3). 
(3) T-convolution of the images belonging to the two normalized subcombs. 
(This can be done by multiplying the normalized subcombs in the spectrum 
and taking the inverse Fourier transform of the product). 
In conclusion, the T-convolution theorem enables us to present the 
extraction of the moire intensity profile between two gratings either in 
the image or in the spectral domains. From the spectral point of view, the 
intensity profile of any (k.sub.1,k.sub.2)-moire between two superposed 
(=multiplied) gratings is obtained by extracting from their 
spectrum-convolution only those impulses which belong to the 
(k.sub.1,k.sub.2)-moire comb, thus reconstructing back in the image domain 
only the isolated contribution of this moire to the image of the 
superposition. On the other hand, from the point of view of the image 
domain, the intensity profile of any (k.sub.1,k.sub.2)-moire between two 
superposed gratings is a normalized T-convolution of the images belonging 
to the k.sub.1 -subcomb of the first grating and to the k.sub.2 -subcomb 
of the second grating. 
Moires between superposed dot-screens 
The moire extraction process described above for the superposition of 
line-gratings can be generalized to the superposition of doubly periodic 
dot-screens, where the moire effect obtained in the superposition is 
really of a 2D nature: 
Let f(x,y) be a doubly periodic image (for example, f(x,y) may be a 
dot-screen which is periodic in two orthogonal directions, .theta..sub.1 
and .theta..sub.2 +90.degree., with an identical period T.sub.1 in both 
directions). Its spectrum F(u,v) is a nailbed whose impulses are located 
on a lattice L.sub.1 (u,v), rotated by the same angle .theta..sub.1 and 
with period of 1/T.sub.1 ; the amplitude of a general 
(k.sub.1,k.sub.2)-impulse in this nailbed is given by the coefficient of 
the (k.sub.1,k.sub.2)-harmonic term in the 2D Fourier series development 
of the periodic function f(x,y). 
The lattice L.sub.1 (u,v) can be seen as the 2D support of the nailbed 
F(u,v) on the plane of the spectrum, i.e. the set of all the nailbed 
impulse-locations. Its unit points (0,1) and (1,0) are situated in the 
spectrum at the geometric locations of the two perpendicular fundamental 
impulses of the nailbed F(u,v), whose frequency vectors are f.sub.1 and 
f.sub.2. Therefore, the location w.sub.1 in the spectrum of a general 
point (k.sub.1,k.sub.2) of this lattice is given by a linear combination 
of f.sub.1 and f.sub.2 with the integer coefficients k.sub.1 and k.sub.2 ; 
and the location w.sub.2 of the perpendicular point (-k.sub.2,k.sub.1) on 
the lattice can also be expressed in a similar way: 
##EQU5## 
Let g(x,y) be a second doubly periodic image, for example a dot-screen 
whose periods in the two orthogonal directions .theta..sub.2 and 
.theta..sub.2 +90.degree. are T.sub.2. Again, its spectrum G(u,v) is a 
nailbed whose support is a lattice L.sub.2 (u,v), rotated by .theta..sub.2 
and with a period of 1/T.sub.2. The unit points (0,1) and (1,0) of the 
lattice L.sub.2 (u,v) are situated in the spectrum at the geometric 
locations of the frequency vectors f.sub.3 and f.sub.4 of the two 
perpendicular fundamental impulses of the nailbed G(u,v). Therefore the 
location w.sub.3 of a general point (k.sub.3,k.sub.4) of this lattice and 
the location w.sub.4 of its perpendicular twin (-k.sub.4,k.sub.3) are 
given by: 
##EQU6## 
Assume now that we superpose (i.e. multiply) f(x,y) and g(x,y). According 
to the Convolution theorem (Eqs. (1) and (2)) the spectrum of the 
superposition is the convolution of the nailbeds F(u,v) and G(u,v); this 
means that a centered copy of one of the nailbeds is placed on top of each 
impulse of the other nailbed (the amplitude of each copied nailbed being 
scaled down by the amplitude of the impulse on top of which it has been 
copied). 
FIG. 2A shows the locations of the impulses in such a spectrum-convolution 
in a typical case where no moire effect is visible in the superposition 
(note that only impulses up to the third harmonic are shown). FIGS. 2B and 
2C, however, show the impulse locations received in the 
spectrum-convolution in typical cases in which the superposition does 
generate a visible moire effect, say a 
(k.sub.1,k.sub.2,k.sub.3,k.sub.4)-moire. As we can see, in these cases the 
DC impulse at the spectrum origin is closely surrounded by a whole cluster 
of impulses. The cluster impulses closest to the spectrum origin, within 
the range of visible frequencies, are the 
(k.sub.1,k.sub.2,k.sub.3,k.sub.4)-impulse of the convolution, which is the 
fundamental impulse of the moire in question, and its perpendicular 
counterpart, the (-k.sub.2,k.sub.1,-k.sub.4,k.sub.3)-impulse, which is the 
fundamental impulse of the moire in the perpendicular direction. 
(Obviously, each of these two impulses is also accompanied by its 
respective symmetrical twin to the opposite side of the origin). The 
locations (frequency vectors) of these four impulses are marked in FIGS. 
2B and 2C by: a, b, -a and -b. Note that in FIG. 2B the impulse-cluster 
belongs to the second order (1,2,-2,-1)-moire, while in FIG. 2C the 
impulse-cluster belongs to the first order (1,0,-1,0)-moire, and consists 
of another subset of impulses from the spectrum-convolution. 
The impulse-cluster surrounding the spectrum origin is in fact a nailbed 
whose support is the lattice which is spanned by a and b, the locations of 
the fundamental moire impulses (k.sub.1,k.sub.2,k.sub.3,k.sub.4) and 
(-k.sub.2,k.sub.1,-k.sub.4,k.sub.3). This infinite impulse-cluster 
represents in the spectrum the 2D (k.sub.1,k.sub.2,k.sub.3,k.sub.4)-moire, 
and its basis vectors a and b (the locations of the fundamental impulses) 
determine the period and the two perpendicular directions of the moire. 
This impulse-cluster is the 2D generalization of the 1D moire-comb that we 
had in the case of line-grating superpositions. We will call the infinite 
impulse-cluster of the (k.sub.1,k.sub.2,k.sub.3,k.sub.4)-moire the 
"(k.sub.1,k.sub.2,k.sub.3,k.sub.4)-cluster," and we will denote it by: 
"M.sub.k.sbsb.1.sub.,k.sbsb.2.sub.,k.sbsb.3.sub.,k.sbsb.4 (u,v)." If we 
extract from the spectrum of the superposition only the impulses of this 
infinite cluster, we get the 2D Fourier series development of the 
intensity profile of the (k.sub.1,k.sub.2,k.sub.3,k.sub.4)-moire; in other 
words, the amplitude of the (i,j)-th impulse of the cluster is the 
coefficient of the (i,j)-harmonic term in the Fourier series development 
of the moire intensity profile. By taking the inverse 2D Fourier transform 
of this extracted cluster we can analytically reconstruct in the image 
domain the intensity profile of this moire. If we denote the intensity 
profile of the (k.sub.1,k.sub.2,k.sub.3,k.sub.4)-moire between the 
superposed images f(x,y) and g(x,y) by 
"m.sub.k.sbsb.1.sub.,k.sbsb.2.sub.,k.sbsb.3.sub.,k.sbsb.4 (x,y)," we 
therefore have: 
EQU m.sub.k.sbsb.1.sub.,k.sbsb.2.sub.,k.sbsb.3.sub.,k.sbsb.4 (x,y)=F.sup.-1 
{M.sub.k.sbsb.1.sub.,k.sbsb.2.sub.,k.sbsb.3.sub.,k.sbsb.4 (u,v)}(12) 
The intensity profile of the (k.sub.1,k.sub.2,k.sub.3,k.sub.4)-moire 
between the superposed images f(x,y) and g(x,y) is therefore a function 
m.sub.k.sbsb.1.sub.,k.sbsb.2.sub.,k.sbsb.3.sub.,k.sbsb.4 (x,y) in the 
image domain whose value at each point (x,y) indicates quantitatively the 
intensity level of the moire in question, i.e. the particular intensity 
contribution of this moire to the image superposition. Note that although 
this moire is visible both in the image superposition 
f(x,y).multidot.g(x,y) and in the extracted moire intensity profile 
m.sub.k.sbsb.1.sub.,k.sbsb.2.sub.,k.sbsb.3.sub.,k.sbsb.4 (x,y), the latter 
does not contain the fine structure of the original images f(x,y) and 
g(x,y) but only the isolated form of the extracted 
(k.sub.1,k.sub.2,k.sub.3,k.sub.4)-moire. Moreover, in a single image 
superposition f(x,y).multidot.g(x,y) several different moires may be 
visible simultaneously; but each of them will have a different moire 
intensity profile m.sub.k.sbsb.1.sub.,k.sbsb.2.sub.,k.sbsb.3.sub.,k.sbsb.4 
(x,y) of its own. 
Let us now find the expressions for the location, the index and the 
amplitude of each of the impulses of the 
(k.sub.1,k.sub.2,k.sub.3,k.sub.4)-moire cluster. If a is the frequency 
vector of the (k.sub.1,k.sub.2,k.sub.3,k.sub.4)-impulse in the convolution 
and b is the orthogonal frequency vector of the 
(-k.sub.2,k.sub.1,-k.sub.4,k.sub.3)-impulse, then we have: 
##EQU7## 
The index-vector of the (i,j)-th impulse in the 
(k.sub.1,k.sub.2,k.sub.3,k.sub.4)-moire cluster is, therefore: 
EQU i(k.sub.1,k.sub.2,k.sub.3,k.sub.4)+j(-k.sub.2,k.sub.1,-k.sub.4,k.sub.3)=(ik 
.sub.1 -jk.sub.2, ik.sub.2 +jk.sub.1, ik.sub.3 -jk.sub.4, ik.sub.4 
+jk.sub.3). (14) 
And furthermore, since the geometric locations of the 
(k.sub.1,k.sub.2,k.sub.3,k.sub.4)- and 
(-k.sub.2,k.sub.1,-k.sub.4,k.sub.3)-impulses are a and b (they are the 
basis vectors which span the lattice L.sub.M (u,v), the support of the 
moire-cluster), the location of the (i,j)-th impulse within this 
moire-cluster is given by the linear combination ia+jb: 
EQU ia+jb=(ik.sub.1 -jk.sub.2)f.sub.1 +(ik.sub.2 +jk.sub.1)f.sub.2 +(ik.sub.3 
-jk.sub.4)f.sub.3 +(ik.sub.4 +jk.sub.3)f.sub.4 (15) 
As we can see, the (k.sub.1,k.sub.2,k.sub.3,k.sub.4)-moire cluster is the 
infinite subset of the full spectrum-convolution which only contains those 
impulses whose indices are given by Eq. (14), for all integer i,j. 
Finally, the amplitude c.sub.i,j of the (i,j)-th impulse in the 
(k.sub.1,k.sub.2,k.sub.3,k.sub.4)-moire cluster is given by: 
EQU c.sub.i,j 
=a.sub.ik.sbsb.1.sub.-jk.sbsb.2.sub.,ik.sbsb.2.sub.+jk.sbsb.1.sub.,ik.sbsb 
.3.sub.-jk.sbsb.4.sub.,ik.sbsb.4.sub.+jk.sbsb.3 (16) 
and according to Eq. (4) we obtain: 
EQU c.sub.i,j =a.sup.(1).sub.ik.sbsb.1.sub.-jk.sbsb.2 
a.sup.(2).sub.ik.sbsb.2.sub.+jk.sbsb.1 
a.sup.(3).sub.ik.sbsb.3.sub.-jk.sbsb.4 
a.sup.(4).sub.ik.sbsb.4.sub.+jk.sbsb.3 (17) 
But since we are dealing here with the superposition of two orthogonal 
layers (dot-screens) rather than with a superposition of four independent 
layers (gratings), each of the two 2D layers may be inseparable. 
Consequently, we should rather group the four amplitudes in Eq. (17) into 
pairs, so that each element in the expression corresponds to an impulse 
amplitude in the nailbed F(u,v) or in the nailbed G(u,v): 
EQU c.sub.ij 
=a.sup.(f).sub.ik.sbsb.1.sub.-jk.sbsb.2.sub.,ik.sbsb.2.sub.+jk.sbsb.1 
a.sup.(g).sub.ik.sbsb.3.sub.-jk.sbsb.4.sub.,ik.sbsb.4.sub.+jk.sbsb.3(18) 
This means that the amplitude c.sub.i,j of the (i,j)-th impulse in the 
(k.sub.1,k.sub.2,k.sub.3,k.sub.4)-moire cluster is the product of the 
amplitudes of its two generating impulses: the (ik.sub.1 -jk.sub.2, 
ik.sub.2 +jk.sub.1)-impulse of the nailbed F(u,v) and the (ik.sub.3 
-jk.sub.4, ik.sub.4 +jk.sub.3)-impulse of the nailbed G(u,v). This can be 
interpreted more illustratively in the following way: 
Let us call "the (k.sub.1,k.sub.2)-subnailbed of the nailbed F(u,v)" the 
partial nailbed of F(u,v) whose fundamental impulses are the 
(k.sub.1,k.sub.2)- and the (-k.sub.2,k.sub.1)-impulses of F(u,v); its 
general (i,j-impulse is the 
i(k.sub.1,k.sub.2)+j(-k.sub.2,k.sub.1)=(ik.sub.1 -jk.sub.2, ik.sub.2 
+jk.sub.1)-impulse of F(u,v). Similarly, let the 
(k.sub.3,k.sub.4)-subnailbed of the nailbed G(u,v) be the partial nailbed 
of G(u,v) whose fundamental impulses are the (k.sub.3,k.sub.4)- and the 
(-k.sub.4,k.sub.3)-impulses of G(u,v); its general (i,j)-impulse is the 
(ik.sub.3 -jk.sub.4, ik.sub.4 +jk.sub.3)-impulse of G(u,v). It therefore 
follows from Eq. (18) that the amplitude of the (i,j)-impulse of the 
nailbed of the (k.sub.1,k.sub.2,k.sub.3,k.sub.4)-moire in the 
spectrum-convolution is the product of the (i,j)-impulse of the 
(k.sub.1,k.sub.2)-subnailbed of F(u,v) and the (i,j)-impulse of the 
(k.sub.3,k.sub.4)-subnailbed of G(u,v). This means that: 
Result 3: (2D generalization of Result 1): The impulse amplitudes of the 
(k.sub.1,k.sub.2,k.sub.3,k.sub.4)-moire cluster in the 
spectrum-convolution are the term-by-term product of the 
(k.sub.1,k.sub.2)-subnailbed of F(u,v) and the 
(k.sub.3,k.sub.4)-subnailbed of G(u,v). 
For example, in the case of the simplest first-order moire between the 
dot-screen f(x,y) and g(x,y), the (1,0,-1 ,0)-moire (see FIG. 2C), the 
amplitudes of the moire-cluster impulses in the spectrum-convolution are 
given by: c.sub.i,j =a.sup.(f).sub.i,j a.sup.(g).sub.-i,-j. This means 
that in this case the moire-cluster is simply a term-by-term product of 
the nailbeds F(u,v) and G(-u,-v) of the original images f(x,y) and 
g(-x,-y). For the second-order (1,2,-2,-1)-moire (see FIG. 2B) the 
amplitudes of the moire-cluster impulses are: c.sub.i,j 
=a.sup.(f).sub.i-2j,2i+j a.sup.(g).sub.-2i+j,-i-2j. 
Now, since we also know the exact locations of the impulses of the 
moire-cluster (according to Eq. (14)), the spectrum of the isolated moire 
in question is fully determined, and given analytically by: 
##EQU8## 
where .delta..sub.f (u,v) denotes an impulse located at the 
frequency-vector f in the spectrum. Therefore, we can reconstruct the 
intensity profile of the moire, back in the image domain, by formally 
taking the inverse Fourier transform of the isolated moire cluster. 
Practically, this can be done either by interpreting the moire cluster as 
a 2D Fourier series, and summing up the corresponding sinusoidal functions 
(up to the desired precision); or, more efficiently, by approximating the 
continuous inverse Fourier transform of the isolated moire-cluster by 
means of the inverse 2D discrete Fourier transform (using FFT). 
As in the case of grating superposition, the spectral domain term-by-term 
multiplication of the moire-clusters can be interpreted directly in the 
image domain by means of the 2D version of the T-convolution theorem: 
2D T-convolution theorem: Let f(x,y) and g(x,y) be doubly periodic 
functions of period T.sub.x, T.sub.y integrable on a one-period interval 
(0.ltoreq.x.ltoreq.T.sub.x,0.ltoreq.y.ltoreq.T.sub.y), and let {F.sub.m,n 
} and {G.sub.m,n } (for m,n=0,.+-.1,.+-.2, . . . ) be their 2D Fourier 
series coefficients. Then the function: 
##EQU9## 
(where .intg..intg..sub.T.sbsb.x.sub.T.sbsb.y means integration over a 
one-period interval), which is called "the T-convolution off and g" and 
denoted by "f**g," is also doubly periodic with the same periods T.sub.x, 
T.sub.y and has Fourier series coefficients {H.sub.m,n } given by: 
H.sub.m,n =F.sub.m,n G.sub.m,n for all integers m,n. 
According to this theorem we have the following result, which is the 
generalization of Result 2 to the general 2D case: 
Result 4: The intensity profile of the 
(k.sub.1,k.sub.2,k.sub.3,k.sub.4)-moire in the superposition of f(x,y) and 
g(x,y) is a T-convolution of the (normalized) images belonging to the 
(k.sub.1,k.sub.2)-subnailbed of F(u,v) and the 
(k.sub.3,k.sub.4)-subnailbed of G(u,v). Note that, before applying the 
T-convolution theorem, the images must be normalized by stretching and 
rotation transformations, to fit the actual period and angle of the moire, 
as determined by Eq. (3) (or by the lattice L.sub.3 (u,v) of the 
(k.sub.1,k.sub.2,k.sub.3,k.sub.4)-moire, which is spanned by the 
fundamental vectors a and b). As shown in Appendix A in "Amidror94," 
normalizing the periodic images by stretching and rotation does not affect 
their impulse amplitudes in the spectrum, but only the impulse locations. 
These results can be easily generalized to any (k.sub.1, . . . 
,k.sub.m)-moire between any number of superposed images by a simple, 
straightforward extension of this procedure. 
A preferred case: the (1,0,-1,0)-moire 
A preferred moire for the present invention relates to the special case of 
the (1,0,-1,0)-moire. A (1,0,-1,0)-moire becomes visible in the 
superposition of two dot-screens when both dot-screens have identical or 
almost identical frequencies and an angle difference .alpha. which is 
close to 0 degrees (this is illustrated, in the spectral domain, by FIG. 
2C). As shown in the example following Result 3, in the special case of 
the (1,0,-1,0)-moire the impulse amplitudes of the moire-cluster are 
simply a term-by-term product of the nailbeds F(u,v) and G(-u,-v) 
themselves: C.sub.i,j =a.sup.(f).sub.i,j a.sup.(g).sub.-i,-j. Since the 
impulse locations of this moire-cluster are also known, according to Eq. 
(3), we can obtain the intensity profile of the (1,0,-1,0)-moire by 
extracting this moire-cluster from the full spectrum-convolution, and 
taking its inverse Fourier transform. 
However, according to Result 4, the intensity profile of the 
(1,0,-1,0)-moire can also be interpreted directly in the image domain: in 
this special case the moire intensity profile is simply a T-convolution of 
the original images f(x,y) and g(-x,-y) (after undergoing the necessary 
stretching and rotations to make their periods, or their supporting 
lattices in the spectrum, coincide). 
Let us see now how T-convolution fully explains the moire intensity profile 
forms and the striking visual effects observed in superpositions of 
dot-screens with any chosen dot shapes, such as in FIG. 3 or FIG. 4. In 
these figures the moire is obtained by superposing two dot-screens having 
identical frequencies, with just a small angle difference a; this implies 
that in this case we are dealing, indeed, with a (1,0,-1,0)-moire. In the 
example of FIG. 4, dot-screen 41 consists of black "1"-shaped dots, and 
dot-screens 40 and 41 consist of black circular dot shapes. Each of the 
dot-screens 40, 41 and 42 consists of gradually increasing dots, with 
identical frequencies, and the superposition angle between the dot-screens 
is 4 degrees. 
Case 1: As can be seen in FIG. 4, the form of the moire intensity profiles 
in the superposition is most clear-cut and striking where one of the two 
dot-screens is relatively dark (see 43 and 44 in FIG. 4). This happens 
because the dark screen includes only tiny white dots, which play in the 
T-convolution the role of very narrow pulses with amplitude 1. As shown in 
FIG. SA, the T-convolution of such narrow pulses 50 (from one dot-screen) 
and dots 51 of any chosen shape (from a second dot-screen) gives dots 52 
of the same chosen shape, in which the zero values remain at zero, the 1 
values are scaled down to the value A (the volume or the area of the 
narrow white pulse divided by the total cell area, T.sub.x 
.multidot.T.sub.y), and the sharp step transitions are replaced by 
slightly softer ramps. This means that the dot shape received in the 
normalized moire-period is practically identical to the dot shape of the 
second screen, except that its white areas turn darker. However, this 
normalized moire-period is stretched back into the real size of the 
moire-period T.sub.M, as it is determined by Eqs. (5) and (6) (or in our 
case, according to Eq. (8), by the angle difference a alone, since the 
screen frequencies are fixed; note that the moire period becomes larger as 
the angle .alpha. tends to 0 degrees). This means that the moire intensity 
profile form in this case is essentially a magnified version of the second 
screen, where the magnification rate is controlled only by the angle 
.alpha.. This magnification property of the moire effect is used in the 
present invention as a "virtual microscope" for visualizing the detailed 
structure of the dot-screen printed on the document. 
Case 2: A related effect occurs in the superposition where one of the two 
dot-screens contains tiny black dots (see 45 and 46 in FIG. 4). Tiny black 
dots on a white background can be interpreted as "inversed" pulses of 
0-amplitude on a constant background of amplitude 1. As shown in FIG. 5B, 
the T-convolution of such inversed pulses 53 (from one dot-screen) and 
dots 54 of any chosen shape (from a second dot-screen) gives dots 55 of 
the same chosen shape, where the zero values are replaced by the value B 
(the volume under a one-period cell of the second screen divided by 
T.sub.x .multidot.T.sub.y)) and the 1 values are replaced by the value B-A 
(where A is the volume of the "hole" of the narrow black pulse divided by 
T.sub.x .multidot.T.sub.y). This means that the dot shape of the 
normalized moire-period is similar to the dot shape of the second screen, 
except that it appears in inverse video and with slightly softer ramps. 
And indeed, as it can be seen in FIG. 4, wherever one of the screens in 
the superposition contains tiny black dots, the moire intensity profile 
appears to be a magnified version of the second screen, but this time in 
inverse video. 
Case 3: When none of the two superposed screens contains tiny dots (either 
white or black), the intensity profile form of the resulting moire is 
still a magnified version of the T-convolution of the two original 
screens. This T-convolution gives, as before, some kind of blending 
between the two original dot shapes, but this time the resulting shape has 
a rather blurred or smoothed appearance. 
The orientation of the (1,0,-1,0)-moire intensity profiles 
Although the (1,0,-1,0)-moire intensity profiles inherit the shapes of the 
original screen dots, they do not inherit their orientation. Rather than 
having the same direction as the dots of the original screens (or an 
intermediate orientation), the moire intensity profiles appear in a 
perpendicular direction. This fact is explained as follows: 
As we have seen, the orientation of the moire is determined by the location 
of the fundamental impulses of the moire-cluster in the spectrum, i.e. by 
the location of the basis vectors a and b (Eq. (13)). In the case of the 
(1,0,-1,0)-moire these vectors are reduced to: 
##EQU10## 
And in fact, as it can be seen in FIG. 2C, when the two original 
dot-screens have the same frequency, these basis vectors are rotated by 
about 90 degrees from the directions of the frequency vectors f.sub.i of 
the two original dot-screens. This means that the (1,0,-1,0)-moire cluster 
(and the moire intensity profile it generates in the image domain) are 
rotated by about 90 degrees relative to the original dot-screens f(x,y) 
and g(x,y). Note that the precise period and angle of this moire can be 
found by formulas (8) which were derived for the (1,-1)-moire between two 
line-gratings with identical periods T and angle difference of .alpha.. 
Obviously, the fact that the direction of the moire intensity profile is 
almost perpendicular to the direction of the original dot-screens is a 
property of the (1,0,-1,0)-moire between two dot-screens having identical 
frequencies; in other cases the angle of the moire may be different. In 
all cases the moire angle can be found by Eqs. (5) and (6). 
Further details about more complex moires and moires of higher order are 
disclosed in detail in "Amidror94". In general, in order to obtain a 
(k.sub.1,k.sub.2,k.sub.3,k.sub.4)-moire in the superposition of two 
dot-screens, the frequencies f.sub.i and the angles .theta..sub.i of the 
two dot-screens have to be chosen in accordance with Eqs. (5) and (6), so 
that the frequency of the (k.sub.1,k.sub.2,k.sub.3,k.sub.4)-impulse is 
located close to the origin of the frequency spectrum, within the range of 
visible frequencies. 
Authentication of documents using the intensity profile of moire patterns 
The present invention concerns a new method for authenticating documents, 
which is based on the intensity profile of moire patterns. In one 
embodiment of the present invention, the moire intensity profiles can be 
visualized by superposing the basic screen and the master screen which 
both appear on two different areas of the same document (banknote, etc.). 
In a second embodiment of the present invention, only the basic screen 
appears on the document itself, and the master screen is superposed on it 
by the human operator or the apparatus which visually or optically 
validates the authenticity of the document. In a third embodiment of this 
invention, the basic screen appears on the document itself, and the master 
screen which is used by the human operator or by the apparatus is a 
microlens array. An advantage of this third embodiment is that it applies 
equally well to both transparent support (where the moire is observed by 
transmittance) and to opaque support (where the moire is observed by 
reflection). Since the document may be printed on traditional opaque 
support (such as white paper), this embodiment offers high security 
without requiring additional costs in the document production. 
The method for authenticating documents comprises the steps of: 
a) creating on a document a basic screen with at least one basic screen dot 
shape; 
b) creating a master screen with a master screen dot shape (where the 
master screen may be either a dot-screen or a microlens array); 
c) superposing the master screen and the basic screen, thereby producing a 
moire intensity profile; 
d) comparing said moire intensity profile with a prestored moire intensity 
profile, and depending on the result of the comparison, accepting or 
rejecting the document. 
In accordance with the third embodiment of this invention, the master 
screen may also be made of a microlens array. Microlens arrays are 
composed of microlenses arranged for example on a square or a rectangular 
grid with a chosen frequency (see, for example, "Microlens arrays" by 
Hutley et al., Physics World, July 1991, pp. 27-32). They have the 
particularity of enlarging on each grid element only a very small region 
of the underlying source image, and therefore they behave in a similar 
manner as screens comprising small white dots, having the same frequency. 
However, since the substrate between neighboring microlenses in the 
microlens array is transparent and not black, microlens arrays have the 
advange of letting the incident light pass through the array. They can 
therefore be used for producing moire intensity profiles either by 
reflection or by transmission, and the document including the basic screen 
may be printed on any support, opaque or transparent. 
The comparison in step d) above can be done either by human biosystems (a 
human eye and brain), or by means of an apparatus described later in the 
present disclosure. In the latter case, comparing the moire intensity 
profile with a prestored moire intensity profile can be made by matching 
techniques, to which a reference is made in the section "Computer-based 
authentication of documents by matching prestored and acquired moire 
intensity profiles" below. 
The prestored moire intensity profile (also called: "reference moire 
intensity profile") can be obtained either by image acquisition, for 
example by a CCD camera, of the superposition of a sample basic screen and 
a master screen, or it can be obtained by precalculation. The 
precalculation can be done, as explained earlier in the present 
disclosure, either in the image domain (by means of a normalized 
T-convolution of the basic screen and the master screen), or in the 
spectral domain (by extracting from the convolution of the frequency 
spectrum of the basic screen and the frequency spectrum of the master 
screen those impulses describing the 
(k.sub.1,k.sub.2,k.sub.3,k.sub.4)-moire, and by applying to said impulses 
an inverse Fourier transform). In the case where a microlens array is used 
as a master screen, the frequency spectrum of the microlens array is 
considered to be the frequency spectrum of the equivalent dot-screen, 
having the same frequency and orientation as the microlens array. 
In the case where the basic screen is formed as a part of a halftoned image 
printed on the document, the basic screen will not be distinguishable by 
the naked eye from other areas on the document. However, when 
authenticating the document according to the present invention, the moire 
intensity profile will become immediatly apparent 
Any attempt to falsify a document produced in accordance with the present 
invention by photocopying, by means of a desk-top publishing system, by a 
photographic process, or by any other counterfeiting method, be it digital 
or analog, will inevitably influence (even if slightly) the size or the 
shape of the tiny screen dots of the basic (or master) screens comprised 
in the document (for example, due to dot-gain or ink-propagation, as is 
well known in the art). But since moire effects between superposed 
dot-screens are very sensitive to any microscopic variations in the 
screened layers, this makes any document protected according to the 
present invention practically impossible to counterfeit, and serves as a 
means to distinguish between a real document and a falsified one. 
Furthermore, unlike previously known moire-based anticounterfeiting 
methods, which are only effective against counterfeiting by digital 
equipment (digital scanners or photocopiers), the present invention is 
equally effective in the cases of analog or digital equipment 
The invention is elucidated by means of the Examples below which are 
provided in illustrative and non-limiting manner. 
EXAMPLE 1 
Basic Screen and Master Screen on Same Document 
Consider as a first example a banknote comprising a basic screen with a 
basic screen dot shape of the digit "1" (like 51 in FIG. 5A). Such a 
dot-screen can either be generated according to state of the art 
halftoning methods such as the ordered dither methods described in 
"Digital Halftoning" by R. Ulichney, 1988 (Chapter 5), or by contour based 
screening methods as disclosed in co-pending U.S. patent application Ser. 
No. 08/410,767 filed Mar. 27, 1995 (Ostromoukhov, Hersch). It should be 
noted that the term "dither matrix" used in the present disclosure is 
equivalent to the term "threshold array" used in "Digital Halftoning" by 
Ulichney. 
In a different area of the banknote a master screen is printed, for 
example, with a master screen dot shape of small white dots (like 50 in 
FIG. 5A), giving a dark intensity level. The banknote is printed on a 
transparent support. 
In this example both the basic screen and the master screen are produced 
with the same dot frequency, and the generated moire is a (1,0,-1 
,0)-moire. In order that the produced moire intensity profile shapes be 
upright (90 degrees orientation), the screen dot shapes of the basic and 
the master screens are required to have an orientation close to 180 
degrees (or 0 degrees), according to the explanation given in the section 
"The orientation of the (1,0,-1,0)-moire intensity profiles" above. 
FIG. 6 shows an example of a basic screen with a basic screen dot shape of 
the digit "1", which is generated with varying intensity levels using the 
dither matrix shown in FIG. 7A. FIG. 7B shows a magnified view of a small 
portion of this basic screen, and how it is built by the dither matrix of 
FIG. 7A. FIG. 8 shows an example of a master screen which is generated 
with the dither matrix shown in FIG. 9A (with darker intensity levels than 
the basic screen, in order to obtain small white dots). FIG. 9B shows a 
magnified view of a small portion of this master screen, and how it is 
built by the dither matrix of FIG. 9A. Note that FIG. 6 and FIG. 8 are 
reproduced here on a 300 dot-per-inch printer in order to show the screen 
details; on the real banknote the screens will normally be reproduced by a 
system whose resolution is at least 1270 or 2540 dots-per-inch. The moire 
intensity profile which is obtained when the basic screen and the master 
screen are superposed has the form of the digit "1", as shown by 43 in 
FIG. 4. 
EXAMPLE 2 
Basic Screen on Document and Master Screen on Separate Support 
As an alternative to Example 1, a banknote may contain a basic screen, 
which is produced by screen dots of a chosen size and shape (or possibly, 
by screen dots of varying size and shape, being incorporated in a 
halftoned image). The banknote is printed on a transparent support. The 
master screen may be identical to the master screen described in Example 
1, but it is not printed on the banknote itself but rather on a separate 
transparent support, and the banknote can be authenticated by superposing 
the basic screen of the banknote with the separate master screen. For 
example, the superposition moire may be visualized by laying the banknote 
on the master screen, which may be fixed on a transparent sheet of plastic 
and attached on the top of a box containing a diffuse light source. 
EXAMPLE 3 
Basic Screen on Document and Master Screen Made of a Microlens Array 
In the present example, the master screen has the same frequency as in 
Example 2, but it is made of a microlens array. The basic screen is as in 
Example 2, but the document is printed on a reflective (opaque) support. 
In the case where the basic screen is formed as a part of a halftoned 
image printed on the document, the basic screen will not be 
distinguishable by the naked eye from other areas on the document. 
However, when authenticated under the microlens array, the moire intensity 
profile will become immediatly apparent. Since the printing of the basic 
screen on the document is incorporated in the standard printing process, 
and since the document may be printed on traditional opaque support (such 
as white paper), this embodiment offers high security without requiring 
additional costs in the document production. 
The multichromatic case 
As previously mentioned, the present invention is not limited only to the 
monochromatic case; on the contrary, it may largely benefit from the use 
of different colors in any of the dot-screens being used. 
One way of using colored dot-screens in the present invention is similar to 
the standard multichromatic printing technique, where several (usually 
three or four) dot-screens of different colors (usually: cyan, magenta, 
yellow and black) are superposed in order to generate a full-color image 
by halftoning. By way of example, if one of these colored dot-screens is 
used as a basic screen according to the present invention, the moire 
intensity profile that will be generated with a black-and-white master 
screen will closely approximate the color of the color basic screen. If 
several of the different colored dot-screens are used as basic screens 
according to the present invention, each of them will generate with a 
black-and-white master screen a moire intensity profile approximating the 
color of the basic screen in question. 
Another possible way of using colored dot-screens in the present invention 
is by using a basic screen whose individual screen elements are composed 
of sub-elements of different colors. (Note that the term "screen element" 
is used hereinafter to indicate a full 2D period of the dot-screen; it 
refers both to the screen dot which appears within this 2D period and to 
the background area which fills the rest of the period). An example of 
such a basic screen is illustrated in FIG. 14A, in which each of the 
screen dots of the basic screen has a triangular shape, and is sub-divided 
into sub-elements of different colors, as indicated by the different 
hachures in FIG. 14A, where each type of hachure represents a different 
color (for example: cyan, magenta, yellow and black). When a 
black-and-white master screen is superposed on such a multichromatic basic 
screen, a similar multichromatic moire effect is obtained, where not only 
the shape of the moire profiles is determined by the screen elements of 
the basic screen but also their colors. For example, in the case of the 
basic screen shown in FIG. 14A, the moire profiles obtained will be 
triangular, and each of them will be sub-divided into colored zones like 
in FIG. 14A. An important advantage of this method as an 
anticounterfeiting means is gained from the extreme difficulty in printing 
perfectly juxtaposed sub-elements of the screen dots, due to the high 
precision it requires between the different colors in multi-pass color 
printing. Only the best high-performance security printing equipment which 
is used for printing security documents such as banknotes is capable of 
giving the required precision in the alignment (hereinafter: 
"registration") of the different colors. Registration errors which are 
unavoidable when counterfeiting the document on lower-performance 
equipment will cause small shifts between the different colored 
sub-elements of the basic screen elements; such registration errors will 
be largely magnified by the moire effect, and they will significantly 
corrupt the form and the color of the moire profiles obtained by the 
master screen. 
In practice, a multichromatic basic screen like the one shown in FIG. 14A 
can be generated by the same method as that described in "Example 1" 
above, with one dither matrix for each of the colors of the multichromatic 
basic screen. In the example of FIG. 14A, each screen element is generated 
by four dither matrices: one for the cyan pixels, one for the magenta 
pixels, one for the yellow pixels, and finally, one for the black pixels. 
Each of these single-color dither matrices is built in the same way as 
described in "Example 1", where only the dither matrix elements of the 
single color in question are numbered, while all the other dither matrix 
elements of the other colors are masked out (set to zero). For example, 
FIG. 14B shows a possible dither matrix for generating the magenta part of 
the screen elements shown in FIG. 14A, and FIG. 14C shows a possible 
dither matrix for generating the black part of the screen elements of FIG. 
14A. 
Covert anticounterfeit and authentication means 
While some anticounterfeit and authentication means are intended to be used 
by the general public ("overt" features), other means are meant to remain 
hidden, only detectable by the competent authorities or by automatic 
authentication devices ("covert" features). The present invention also 
lends itself particularly well to the latter case. In fact, a first step 
in this direction can be taken by incorporating the dot-screens which are 
printed on the document in accordance with the present disclosure within 
any halftoned image printed on the document (such as a portrait, 
landscape, or any decorative motif, which may be different from the motif 
generated by the moire effect in the superposition). 
However, in cases where the present invention is to be used as a covert 
feature, it may be desirable that the document, even when inspected under 
a strong magnifying glass, should not reveal the information carried by 
the basic screen (i.e. the nature and the shapes of the moire intensity 
profiles which appear when the master screen is superposed). 
This can be achieved by masking the information carried by the basic 
screen, in order to obtain a masked basic screen. A masked basic screen 
can be obtained in a variety of ways, which can be classified into several 
methods as follows: 
(a) The masking layer method. In this method a masked basic screen is 
obtained by superposing a new layer with any geometric or decorative forms 
(such as a multitude of circles, triangles, letters, etc.) on top of the 
basic screen. For example, the masking of the basic screen can be carried 
out by superposition of circles placed at random positions, with radiuses 
varying randomly between a minimal predefined value and a maximal 
predefined value. 
(b) The composite basic screen method. In this method the masked basic 
screen is a composite basic screen which is composed of two or more 
different dot-screens, each carrying its own information, that are 
superposed on each other. 
(c) The perturbation patterns method. In this method a masked basic screen 
is obtained by altering the basic screen itself. This can be done by 
introduction of perturbation patterns into the basic screen by means of 
mathematical, statistical or logical Boolean operations. An example of 
this method is the introduction of any sort of statistical noise into the 
basic screen. The perturbation patterns can alter the original dither 
matrix used to generate the basic screen. 
(d) Any combination of methods (a) (b) and (c). 
As will become clear in the explanation below, if the new superposed 
masking layer (or respectively, the inserted perturbation) is 
non-periodic, or if it is periodic but it has a different period and/or 
orientation than the basic screen, this masking effect will not hamper the 
appearance of the moire intensity profiles when the master screen is 
superposed, but it will prevent the visualization of the information 
carried by the basic screen without using the master screen (for example, 
by a mere inspection of the document under a microscope). 
Furthermore, since masked basic screens are generated by a computer 
program, they can be made so complex that even professionals in the 
graphic arts cannot re-engineer them without having the original computing 
programs specially developed for creating them. Masking of information 
carried by a basic screen will now be exemplified by means of three 
techniques described below, which are provided in illustrative and 
non-limiting manner. Techniques 1 and 2 are provided as examples of a 
composite basic screen method, and technique 3 is given as an example of a 
perturbation patterns method. 
Technique 1: The composite basic screen method with a single master screen 
This technique, which illustrates the composite basic screen method, will 
be most clearly understood by means of the following case. Assume we are 
given two regular dot-screens with identical frequencies (a dot-screen is 
called "regular" when its two main directions are perpendicular and have 
the same frequency). These dot-screens are superposed, preferably at such 
an angle difference that no moire is visible in their superposition (in 
the case of two regular dot-screens, the angle difference may be, for 
example, about 45 degrees). Assume, now, that each of the two superposed 
dot-screens is made of a non-trivial screen dot shape (preferably, a 
different dot shape for each of the dot-screens). This is illustrated in 
FIGS. 11A and 11B, in which one of the dot-screens has a screen dot shape 
of "EPFL/LSP" (110), while the other dot-screen has a screen dot shape of 
"USA/$50" (111). When these two dot-screens are superposed with said angle 
difference, their superposed screen dots intersect each other, generating 
a complex and intricate microstructure. When looking under a magnifying 
glass or a microscope the microstructure of this superposition looks 
scrambled and unintelligible, as illustrated in FIG. 11C. Such a 
superposed screen will be called hereinafter "a composite screen", and a 
basic screen which consists of a composite screen will be called "a 
composite basic screen". 
Now, since both of the dot-screens which make the composite basic screen 
have identical frequencies and they only differ in their orientations (and 
in their screen dot shapes), the same master screen can be used for both 
screens. When this master screen is superposed on top of the composite 
basic screen in an angle close to the orientation of the first screen, a 
moire intensity profile is generated between the first screen of the 
composite basic screen and the master screen. This moire intensity profile 
has the shape of the screen dot of the first screen; however, due to the 
angle difference of 45 degrees (in the present example), the second screen 
does not generate a visible moire intensity profile with the master 
screen, so that only the moire intensity profile due to the first screen 
is visible. However, when the master screen is rotated by about 45 degrees 
(in the present example), the first moire intensity profile becomes 
invisible, and it is the second screen of the composite basic screen which 
generates with the master screen a visible moire intensity profile, whose 
shape corresponds this time to the screen dot shape of the second screen. 
It should be understood that the description given here also holds for 
cases in which the master screen is a microlens array. 
Thus, although the composite basic screen appearing on the document is 
scrambled and unintelligible, two different moire intensity profiles (in 
the example of FIG. 11C: the texts "EPFL/LSP" and "USA/$50") can become 
clearly visible when the appropriate master screen is superposed on the 
composite basic screen, each of the two moire intensity profiles being 
visible in a different orientation of the master screen. 
Since the microstructure of the composite basic screen is unintelligible, 
and the individual screen dot shapes can only be made visible by 
superposing the appropriate master screen on top of the composite basic 
screen, it is therefore clear that if the master screen is not rendered 
public, the present technique becomes a covert anticounterfeit means, 
which can only be detected by the competent authorities or by automatic 
devices which possess the master screen. 
This method is not limited to composite basic screens which are composed of 
two superposed dot-screens. On the contrary, further advantages can be 
obtained by using a composite basic screen which consists of more than two 
superposed dot-screens, possibly of different colors. For example, a 
composite basic screen may consist of three dot-screens with different dot 
shapes which are superposed with angle differences of 30 degrees (in which 
case no superposition moire is generated, as already known in the art of 
color printing). In this case, three different moire intensity profiles 
will be obtained by the master screen at angle differences of 30 degrees. 
However, some benefits can also be gained by using a composite basic 
screen in which some of the superposed dot-screens do generate a weak, 
visible moire effect; this weak visible moire effect may have a nice 
geometric form and serve as a decorative pattern on the document, while 
more dominant and completely different moire intensity profiles (for 
example: "EPFL/LSP" or "USA/$50") are revealed on top of this decorative 
pattern by using the master screen. (A weak moire effect can be generated, 
for example, by using for the basic screens in question lower gray levels, 
i.e. smaller screen dots.) 
It should be noted that a composite basic screen printed on the document in 
accordance with the present invention need not necessarily be of a 
constant intensity level. On the contrary, it may include dots of 
gradually varying sizes and shapes, and it can be incorporated (or 
dissimulated) within any halftoned image printed on the document (such as 
a portrait, landscape, or any decorative motif, which may be different 
from the motif generated by the moire effects in the superposition). In 
the case of a composite basic screen, intensity level variations can be 
obtained, for instance, by varying the dot size and shape of each of the 
superposed screens independently (for example, using the dither matrix 
method, as illustrated in FIGS. 6, 7A and 7B for the simple case of a 
"1"-shaped screen dot). 
It should be also noted that although the present disclosure has been 
illustrated, for the sake of simplicity, by examples with regular screens, 
this invention is by no means limited only to the case of regular screens, 
and similar results can be also obtained in the case of non-regular 
dot-screens (a dot-screen is called "non-regular" when its two main 
directions are not perpendicular and/or have different frequencies). 
However, in the case of technique 1, in a composite basic screen which is 
composed of non-regular dot-screens, each of the non-regular dot-screens 
which form the composite basic screen should have approximately the same 
internal angle between its two main directions and approximately the same 
frequencies in the respective directions (so that the same master screen 
will be appropriate for all the individual screens which together form the 
composite basic screen). 
Technique 2: The composite basic screen method with multiple master screens 
In this variant of the composite basic screen method, the composite basic 
screen may be composed of two (or more) superposed basic dot-screens, each 
having not only a different screen dot shape, but also different 
frequencies, and in the case of non-regular dot-screens, even different 
internal angles and/or different frequencies in the two main directions of 
each dot-screen. Therefore in this variant each dot-screen in the 
composite basic screen requires a different master screen for generating 
its moire intensity profile. 
This multiple master screen variant offers a higher degree of security, 
since each of the moire intensity profiles hidden in the composite basic 
screen can only be revealed by its own, special master screen. 
Furthermore, this variant even enables the introduction of a hierarchy of 
security levels, each security level being protected by a different master 
screen (or a different combination of master screens). For example, one of 
the master screens can be intended for the general public, while the other 
master screens remain available only to the competent authorities or to 
automatic authentication devices. In this case, one of the moire intensity 
profiles can serve as a public authentication means of the document, while 
the other moire intensity profiles hidden in the same composite basic 
screen are not accessible to the general public. 
It should be noted that as is the case in technique 1, the composite basic 
screen printed on the document may include dots of gradually varying sizes 
and shapes, and it can be incorporated (or dissimulated) within any 
halftoned image printed on the document, as already explained in the case 
of technique 1. 
Note that any of the master screens in the multiple master screen variant 
can also be implemented by a microlens array with the appropriate angles 
and frequencies. 
Technique 3: The Irregular sub-element alterations technique 
This technique is an example of the perturbation patterns method, in which 
a basic screen (or a composite basic screen) on the document is rendered 
unintelligible by means of the introduction of perturbation patterns. 
Perturbation patterns can be introduced into the basic screen to render it 
unintelligible in several different ways. For the sake of example, in the 
present technique this is done by means of irregular sub-element 
alterations. This can be most clearly illustrated by means of the 
following example. 
Assume we are given a dot-screen whose screen dot has the shape of 
"EPFL/LSP" as in FIG. 11A. Each part of the screen dot (in the present 
example, each individual letter) can be further divided into a certain 
number of sub-elements. For example, FIG. 12A shows a possible way to 
divide the letter "E" into sub-elements. This division into sub-elements 
should be done in such a way that missing sub-elements (such as 120 in 
FIG. 12B) render the letter unrecognizable, as shown for example in FIGS. 
12B-12D. Moreover, additional segments or shifting of sub-elements (such 
as 121 in FIG. 12F) can also be used to render the letter unintelligible, 
as shown for example in FIGS. 12E and 12F. 
Since the moire intensity profiles in the screen superposition are obtained 
by T-convolution, a small rate of perturbations (in the present example: 
sub-element alterations) in a screen element will hardly influence the 
resulting moire intensity profile, due to the averaging effect of the 
T-convolution. Therefore, if any of the "EPFL/LSP"-shaped screen dots of 
the dot-screen is slightly altered in order to make each individual letter 
unintelligible, but each occurrence of the screen dot "EPFL/LSP" is 
altered in a different way, such that on average each sub-element of each 
letter appears in most occurrences, and each of the extra sub-elements 
only appears in a small rate of occurrence, then the influence on the 
T-convolution will only be negligible. Therefore, the resulting moire 
intensity profile when the master screen is superposed remains almost as 
clear as in the unaltered case, although the basic screen itself is 
unintelligible even under a strong magnifying glass. 
In practice, an irregular alteration of sub-elements can be obtained by 
dividing the basic screen into large super-tiles, each super-tile 
consisting of m.times.n screen dots ("EPFL/LSP", in the present example) 
where m,n are integer numbers, preferably larger than 10. Each occurrence 
of the screen dot within the super-tile is slightly altered in the way 
explained above, each occurrence in a different way, but the large 
super-tile itself is repeated periodically throughout the basic screen. 
FIG. 13 shows a magnified example of such a basic screen which is based on 
the "EPFL/LSP"-shaped screen dot of FIG. 11A. Note that the same 
super-tile can also be used for performing intensity level variations and 
halftoning with the basic screen (using the dither matrix method, as 
illustrated in FIGS. 6, 7A and 7B for the simple case of a "1"-shaped 
screen dot). 
The irregular sub-element alterations technique can be practically 
implemented in 5 steps as described below: 
1. A computer program divides each part of the screen element (in the case 
of the example above: each of the letters E,P,F,L,L,S,P) into a predefined 
number of sub-elements. 
2. Then, the computer program generates for each of the screen element 
parts (letters, in the present example) a series of variants, by omitting, 
shifting, exchanging or adding sub-elements, as illustrated in FIGS. 
12B-12F. 
3. The designer or the graphist then selects a certain number N of variants 
(for example, N=10) for each of the different screen element parts 
(letters, in the present example), choosing from the variants generated in 
step 2 those in which the original form is the least recognizable. 
4. Then, the designer or a computer program generates the large super-tile 
(which consists of m.times.n screen elements) by choosing for each 
occurrence of any screen element part within each of the m.times.n screen 
elements a different variant (from the set of N variants selected for this 
screen element part in step 3): This is done in a statistically uniform 
way, where each sub-element is missing in only up to 10%-20% of the 
occurrences of the screen element part in the super-tile, and each 
additional sub-element appears in no more than 10%-20% of the occurrences 
of the screen element part in the super-tile. 
5. This super-tile is then used, as already known in the art, for 
generating the masked basic screen for the case of the irregular 
sub-element alterations technique. 
The irregular sub-element alterations technique can also be used for 
performing intensity level variations and halftoning with the masked basic 
screen. This can be done using the dither matrix method, as illustrated in 
FIGS. 6, 7A and 7B for the simple case of a "1"-shaped screen dot, but 
this time using an altered super-dither matrix whose size equals that of 
the super-tile. This altered super-dither matrix can be obtained, for 
example, by first preparing an elementary dither matrix which corresponds 
to the original, unaltered screen element. Then, variants of this 
elementary dither matrix are obtained by performing the sub-element 
alterations (the omitting, shifting, exchanging or adding of sub-elements) 
inside copies of the original elementary dither matrix, and these variants 
are then incorporated into the altered super-dither matrix, in accordance 
with steps 1-5 above. After incorporating the sub-element alterations 
within the super-dither matrix, dither threshold levels in the 
super-dither matrix can be renumbered so as to generate a continuous 
sequence of threshold levels. 
In the case of a multicolor basic screen, a similar effect can also be 
obtained by irregular alterations in the color of the sub-elements. 
Furthermore, as shown in FIGS. 16A and 16B, in the multichromatic case the 
screen dots of the basic screen can be divided into sub-elements of 
different colors, while the background (the area between the screen dots) 
can be divided into sub-elements of other colors (for example, brighter 
colors). By way of example, the colors of the sub-elements of the screen 
dots can be arbitrarily chosen from one set of colors (161) and the colors 
of the background sub-elements can be arbitrarily chosen from a second set 
of colors (162) (for example, brighter colors). The multichromatic basic 
screen thus obtained can be generated as already explained in the section 
"The multichromatic case" above. This method turns the basic screen into a 
multichromatic mosaic of sub-elements, making it even more unintelligible; 
and moreover, it renders counterfeiting the document even more difficult 
due to the high registration accuracy required, as already explained in 
the section "The multichromatic case" above. Since registration errors are 
almost unavoidable in a falsified document having such a multichromatic 
basic screen, the moire profiles obtained will be fuzzy and corrupted in 
their shape as well as in their color, thereby making the falsification 
obvious. 
It should be noted that the perturbation patterns method, and in particular 
the irregular sub-element alterations technique, can be used as a covert 
anticounterfeit and authentication means even with a single basic screen. 
However, this method can also be used in any combination with the masking 
layer method and/or the composite basic screen method, thereby further 
enhancing the security offered by the individual methods. 
Computer-based authentication of documents by matching prestored and 
acquired moire intensity profiles 
Since for a basic screen of frequency f.sub.1 and f.sub.2 and for a master 
screen of frequency f.sub.3 and f.sub.4 the resulting 
(k.sub.1,k.sub.2,k.sub.3,k.sub.4)-moire has the frequencies: 
EQU a=(a.sub.u,a.sub.v) 
EQU b=(b.sub.u,b.sub.v) 
which are given by Eq. (13), the orientations .phi..sub.1, .phi..sub.2 and 
the periods T.sub.1, T.sub.2 of the moire's main axes are, according to 
Eq. (6): 
##EQU11## 
As explained earlier in the present disclosure, the prestored moire 
intensity profile can be obtained either by acquisition or by 
precalculation. However, in order to take into account the influence of 
the image acquisition device, for example a CCD camera, it is advantageous 
to obtain the prestored moire intensity profile by the acquisition of the 
moire intensity profile produced by the superposition of the master screen 
and an original document incorporating the basic screen. Since the 
acquisition of the prestored moire intensity profile only occurs once, a 
careful adjustment of the superposition ensures that the orientations of 
the main axes of the acquired prestored moire intensity profile correspond 
exactly to the precalculated orientations .phi..sub.1,.phi..sub.2. Hence, 
the periods P.sub.1,P.sub.2 of the acquired presored moire intensity 
profile (in terms of the acquisition device units, for example, pixels), 
correspond to the precalculated periods T.sub.1,T.sub.2 (in terms of 
document space units). The periods P.sub.1,P.sub.2 in terms of the 
acquisition device units can be found by intersecting the prestored moire 
intensity profile with a straight line parallel to one of the two main 
axes, say the first axis, of the prestored moire intensity profile. A 
discrete straight line segment representing the intensity profile along 
this straight line is obtained by resampling the straight line at the 
acquired moire intensity profile resolution. The period P.sub.1 of the 
resulting discrete straight line segment is measured, and period P.sub.2 
of the prestored moire intensity profile along the other main axis may be 
obtained for example by calculating P.sub.2 =P.sub.1 (T.sub.2 /T.sub.1). 
Consider, as an exemple, FIG. 15A, showing a prestored moire intensity 
profile which is schematically represented in the drawing by triangular 
elements 150. In this example, the main axes of the prestored moire 
intensity profile are axis 151 at orientation .phi..sub.1 and axis 152 at 
orientation .phi..sub.2. Along the first main axis 151 the period of the 
prestored moire intensity is P.sub.1, and along the second main axis 152 
the period of the prestored moire intensity is P.sub.2. 
Note that hereinafter the prestored moire intensity profile will also be 
called "prestored moire image", since the prestored moire intensity 
profile is stored in the same way as a digital grayscale or color image. 
For the same reason, an acquired moire intensity profile will also 
hereinafter be called "acquired moire image". 
The acquired moire intensity profiles obtained by acquiring the 
superposition of the master screen and a non-counterfeited document will 
always have the same geometry as the prestored moire intensity profile, up 
to a rotation angle error, a scaling error a, and a translation error 
(.tau..sub.x,.tau..sub.y) which is also called "phase differences". These 
errors in the acquired moire image may occur due to the limited accuracy 
of the feeding mechanism positioning the basic screen beneath the master 
screen and the image acquisition means (e.g. the CCD camera). FIG. 15B 
shows an example of an acquired moire intensity profile originating from 
the superposition of the master screen and of a non-counterfeited 
document. When the errors .delta., .sigma. and (.tau..sub.x,.tau..sub.y) 
are corrected, as explained below, the geometrically corrected acquired 
moire image will perfectly match the prestored moire image. However, in 
the case of a counterfeited document, even after these geometric 
corrections have been carried out the acquired moire intensity profile 
will not match the prestored moire intensity profile (due to differences 
in intensity profile, in moire shape or even due to the lack of 
periodicity in the acquired moire image). 
In order to find out and correct the rotation angle error .delta. and the 
scaling error .sigma., different methods can be used. As an example, which 
is provided in an illustrative and non-limiting manner, the method 
described below relies on the intersection of lines with the aquired moire 
intensity profile. The goal is to obtain a line (such as line 159 in FIG. 
15B) which intersects the acquired moire intensity profile along its main 
direction. For this purpose, a line is first drawn along the main 
direction of the prestored moire intensity profile (such as line 155 in 
FIG. 15B). Since this line possibly does not intersect any moire shapes 
(represented in the drawing by triangular elements), further parallel 
lines are generated, such as line 157, until moire shapes are intersected. 
Then the resulting line is rotated, until it shows a periodic intensity 
signal (for example line 159 shows the periodic intensity signal 1510 in 
FIG. 15C). The angle .delta. between that line (159) and the main axis of 
the prestored moire intensity profile gives the rotation angle error. The 
ratio between the period of that intensity signal (1510) and period 
P.sub.1 of the prestored moire intensity profile gives the scaling error 
a. 
The following paragraph describes the method of this example in more 
details. It describes how rotation angle error .delta. and scaling error 
care recovered, and also mentions conditions for rejecting or accepting a 
document. In the following explanation it is assumed that the scaling 
error .sigma. is larger than a certain fraction .sigma..sub.min (say, 0.5) 
and smaller than a certain number .sigma..sub.max (say, 2). The term 
"quasi-period" will mean in the following explanation a distance between 
two consecutive low-to-high (or high-to-low) intensity transitions of a 
possibly non-periodic one-dimensional signal. 
The rotation angle error 6 and the scaling error .sigma. between the 
prestored moire intensity profile and an acquired moire intensity profile 
can be determined, for example, by intersecting the acquired moire 
intensity profile with a straight line parallel to one of the two main 
axes, say the first axis, of the prestored moire intensity profile. A 
discrete straight line segment representing the intensity profile along 
this straight line is obtained by resampling the straight line at the 
acquired moire intensity profile resolution. The resulting discrete 
straight line segment (for example segment 155 in FIG. 15B, shown in the 
drawing as a continuous line) is subsequently analyzed and checked for a 
valid intensity variation along the line; a valid intensity variation is 
defined as an intensity variation with a quasi-period not smaller than 
.sigma..sub.min (for example, 0.5) times the smallest of the two periods 
P.sub.1, P.sub.2 of the prestored moire intensity profile and not larger 
than .sigma..sub.max (for example, 2) times the largest of the two periods 
P.sub.1, P.sub.2 of the prestored moire intensity profile. If such a valid 
intensity variation is not found, or if it is below a given intensity 
threshold, for example below half the maximal intensity difference, 
another discrete straight line segment is generated parallel to the 
previous discrete straight line segment (this new discrete straight line 
segment is called "a parallel instance" of the previous discrete straight 
line segment). This parallel discrete straight line segment is generated 
at a distance .gamma.(156 in FIG. 15B) apart from the previous discrete 
straight line segment (the distance .gamma. being, for example, 1/4 of 
period P.sub.2). Line segment 157 in FIG. 15B is an example of such a 
parallel discrete straight line segment. If again no valid intensity 
variation is detected, further parallel discrete straight line segments 
are generated as before at a distance .gamma. apart from each other and 
checked for valid intensity variations. If no valid intensity variation is 
detected after having generated discrete straight line segments across, 
for example, twice the full period P.sub.2, the document is rejected. In 
the case where a valid intensity variation is detected, it is checked if 
successive quasi-periods of the intensity variation along the discrete 
straight line segment are identical, i.e. if the one-dimensional intensity 
signal represented by the discrete straight line segment is periodic. In 
FIG. 15C, 1511 illustrates a non-periodic intensity signal with two 
non-identical successive quasi-periods, and 1510 illustrates a periodic 
intensity signal with two identical quasi-periods. If no periodicity is 
detected in the considered discrete straight line segment, a new rotated 
discrete straight line segment is generated whose orientation differs from 
the previous discrete straight line segment by a fraction (for example 
1/20) of .delta..sub.max, where .delta..sub.max is the maximal possible 
rotation angle error, for example .+-.10 degrees. An example of such a 
discrete straight line segment is shown by 159 in FIG. 15B. Further such 
rotated discrete straight line segments are generated, always rotated by a 
fraction of the maximal possible rotation angle, until one of them 
contains a periodic intensity signal with a period P not smaller than 
.sigma..sub.min (for example, 0.5) times the period P.sub.1 and not larger 
than .sigma..sub.max (for example, 2) times the period P.sub.1. (It should 
be understood that periodicity in a discrete signal is admitted up to a 
certain small precision error in pixels). If none of the successive 
rotated discrete straight line segments covering the angle range of 
.+-..delta..sub.max contains a periodic intensity signal with a period P 
not smaller than .sigma..sub.min (for example, 0.5) times the period of 
the prestored moire and not larger than .sigma..sub.max (for example, 2) 
times that period, the document with the basic screen is rejected. 
If such a periodic discrete straight line segment with a period P has been 
found, the scaling error .sigma. and the angle error .delta. of the 
acquired moire intensity profile are determined as follows: 
The scaling error .sigma. is obtained by .sigma.=P/P.sub.1, where P is the 
period of the so-obtained periodic intensity signal and P.sub.1 is the 
corresponding period of the prestored moire intensity profile. The angle 
error .delta. is the angle difference between this resulting periodic 
discrete straight line segment and the main axis of the prestored moire 
intensity profile (see angle .delta. in FIG. 15B). 
Having found the angle error .delta. and the scaling error .sigma. of the 
acquired moire intensity profile, a window of the acquired moire intensity 
profile containing at least one fill moire element given by its periods 
(.sigma.P.sub.1, .sigma.P.sub.2) in its two main directions is extracted, 
rotated and scaled by a linear transformation, where the rotation angle is 
-.delta. and the scaling factor is 1/.sigma., so as to obtain exactly the 
same periods and orientations as the periods (P.sub.1,P.sub.2) and 
orientations (.phi..sub.1, .phi..sub.2) of the prestored moire intensity 
profile. Regarding image extraction, affine transformation, scaling and 
rotation, see for example the book "Digital Image Processing", by W. K. 
Pratt, Chapter 14: "Geometrical image modification"). 
The geometrically corrected moire intensity profile thus obtained is then 
matched with the prestored moire intensity profile so as to produce a 
degree of proximity between the two. Matching a given image with a 
prestored image can be done, for example, by template matching, as 
described in the book "Digital Image Processing and Computer Vision", by 
R. J. Schalkoff, pp 279-286. For template matching, one may use the 
correlation techniques which give an intensity proximity value 
C(s.sub.x,s.sub.y) between the two images as a function of their relative 
shift (s.sub.x,s.sub.y). The largest intensity proximity value gives the 
translation error (.tau..sub.x,.tau..sub.y)=(s.sub.x,s.sub.y). If the 
so-computed largest intensity proximity value is higher than an 
experimentally determined intensity proximity threshold value the document 
is accepted, and otherwise the document is rejected. 
Accordingly, the method described in detail in the example above, where 
comparing a moire intensity profile with a prestored moire intensity 
profile is done by computer-based matching, which requires an acquisition 
of a moire intensity profile and a geometrical correction of a rotation 
angle error and of a scaling error in the acquired moire intensity 
profile, comprises the steps of: 
a) acquiring a moire intensity profile by an image acquisition means; 
b) intersecting the acquired moire intensity profile with a straigtht line 
parallel to a main axis of the prestored moire intensity profile; 
c) computing a discrete straight line segment representing the acquired 
moire intensity profile along the straight line by resampling the straight 
line intersecting the acquired moire intensity profile at the resolution 
of the acquired moire intensity profile; 
d) checking the considered discrete straight line segment as well as 
parallel instances of it for valid intensity variations defined as 
intensity variations with a quasi-period not smaller than .sigma..sub.min 
times the smallest of the two periods P.sub.1, P.sub.2 of the prestored 
moire intensity profile and not larger than .tau..sub.max times the 
largest of the two periods P.sub.1, P.sub.2 of the prestored moire 
intensity profile; 
e) rejecting the document in the case where no valid intensity variations 
occur in any of the parallel discrete straight line segment instances; 
f) in the case of valid intensity variations, rotating the discrete 
straight line segment showing valid intensity variations until an angle 
.delta. is reached in which the rotated discrete straight line segment 
comprises successive identical quasi-periods P of intensity variations; 
g) computing the scaling error .sigma.=P/P.sub.1 ; 
h) using angle .delta. and scaling error .sigma. to rotate by angle 
-.delta. and to scale by factor 1/.sigma. a window of the acquired moire 
intensity profile containing at least one period of said acquired moire 
intensity profile, thereby obtaining a geometrically corrected moire 
intensity profile; 
i) matching the so-obtained geometrically corrected moire intensity profile 
with the prestored moire intensity profile and obtaining a proximity value 
giving the proximity between the acquired moire intensity profile and the 
prestored moire intensity profile; and 
j) rejecting the document if the proximity value is lower than an 
experimentally determined threshold. 
In the case of a color basic screen, a prestored color moire image can be 
obtained in the same way as in the case of a black-and-white basic screen 
and compared with a color moire image acquired by a color image 
acquisition device. The computation of rotation angle error .delta. and 
scaling error a can be done as in the case of a black-and-white basic 
screen, by computing from the Red Green Blue (RGB) pixel values of the 
acquired color moire image the corresponding Y I Q values, where Y 
represents the achromatic intensity values and I and Q represent the 
chromaticity values of the color moire image (for a detailed description 
of the R G B to Y I Q coordinate transformation, see for example the book 
"Computer Graphics: Principles and Practice", by J. D. Foley, A. Van Dam, 
S. K. Feiner and J. F. Hughes, Section 13.3.3, p. 589). 
Matching a prestored color moire image with an acquired color moire image 
(after it has been geometrically corrected) can be done in a similar 
manner as in the black-and-white case, using the Y coordinate as the 
achromatic moire intensity profile. As in the black-and-white case, the 
largest intensity proximity value and the translation error (m,x) (i.e. 
the phase differences in the two main directions between the prestored and 
the acquired moire images) can be found, for example, by template 
matching. Here, too, if the largest intensity proximity value is lower 
than an experimentally determined intensity proximity threshold value, the 
document is rejected. But if the intensity proximity value is higher than 
the experimentally determined proximity threshold value, the document 
undergoes an additional test using the chromaticity acceptance criterion, 
which is based on a chromatic Euclidian distance. 
Using the same phase differences (.tau..sub.x,.tau..sub.y), a chromatic 
Euclidian distance in the IQ colorimetric plane is computed for each pixel 
between the geometrically corrected acquired moire image and the prestored 
moire image. The average chromatic Euclidian distance is a measure of a 
chromatic proximity between the acquired moire image and the prestored 
moire image: a small average chromatic Euclidian distance indicates a high 
degree of proximity, and vice versa. Using this criterion, a document is 
accepted if the average chromatic Euclidian distance is lower than an 
experimentally determined chromatic Euclidian distance threshold, and 
rejected if the average chromatic Euclidian distance is higher than an 
experimentally determined chromatic Euclidian distance threshold. 
The maximal possible rotation angle error .delta..sub.max can be 
experimentally determined by acquiring the moire image obtained when a 
document is fed by the document handling device with the greatest possible 
rotational feeding error, and by comparing the orientation of the 
so-acquired moire image with the orientation of the prestored moire image. 
Furthermore, various instances of the original document as well as 
reproductions of it (simulating counterfeited documents) may be acquired 
according to the method described above. The different intensity proximity 
values obtained for the original documents on the one hand, and for the 
reproductions on the other hand, enable the setting of the experimentally 
determined intensity proximity threshold value, so that the intensity 
proximity values of the original documents are above the threshold and the 
intensity proximity values of the reproduced documents are below the 
threshold. The same technique is also applied for setting the 
experimentally determined chromatic Euclidian distance threshold, so that 
for original documents the average chromatic Euclidian distances are below 
the chromatic Euclidian distance threshold and for reproduced documents 
the chromatic Euclidian distances are above this threshold. 
As mentioned above in the section "The multichromatic case", when a color 
document is printed at high resolution, color registration problems occur. 
Counterfeiters trying to falsify the color document by printing it using a 
standard printing process will also have, in addition to the problems of 
creating the basic screen, problems of color registration. Without correct 
color registration, the basic screen will incorporate distorted screen 
dots. Therefore, the intensity profile of the moire acquired with the 
master screen applied to a counterfeited document will clearly distinguish 
itself, in terms of form and intensity as well as in terms of color, from 
the moire profile obtained when applying the master screen to the 
non-counterfeited document. The measures of proximity with respect to both 
intensity and chromaticity, as described above, will clearly distinguish 
between a falsified document and a genuine one and allow the rejection of 
counterfeited documents by the apparatus described below. Since 
counterfeiters will always have color printers with less accuracy than the 
official bodies responsible for printing the original valuable documents 
(banknotes, checks, etc.), the disclosed authentication method remains 
valid even with the quality improvement of color reproduction 
technologies. 
Apparatus for the authentication of documents using the intensity profile 
of moire patterns 
An apparatus for the visual authentication of documents comprising a basic 
screen may comprise a master screen (either a dot-screen or a microlens 
array) prepared in accordance with the present disclosure, which is to be 
placed on the basic screen of the document, while the document itself is 
placed on the top of a box containing a diffuse light source (or possibly 
under a source of diffuse light, in case the master screen is a microlens 
array and the moire intensity profile is observed by reflection). If the 
authentication is made by visualization, i.e. by a human operator, human 
biosystems (a human eye and brain) are used as a means for the acquisition 
of the moire intensity profile produced by the superposition of the basic 
screen and the master screen, and as a means for comparing the acquired 
moire intensity profile with a prestored moire intensity profile. 
An apparatus for the automatic authentication of documents, whose block 
diagram is shown in FIG. 10, comprises a master screen 101 (either a 
dot-screen or a microlens array), an image acquisition means (102) such as 
a CCD camera, a source of light (not shown in the drawing), and a 
comparing processor (103) for comparing the acquired moire intensity 
profile with a prestored moire intensity profile. In case the match fails, 
the document will not be authenticated and the document handling device of 
the apparatus (104) will reject the document. The comparing processor 103 
can be realized by a microcomputer comprising a processor, memory and 
input-output ports. An integrated one-chip microcomputer can be used for 
that purpose. For automatic authentication, the image acquisition means 
102 needs to be connected to the microprocessor (the comparing processor 
103), which in turn controls a document handling device 104 for accepting 
or rejecting a document to be authenticated, according to the comparison 
operated by the microprocessor. 
The prestored moire intensity profile can be obtained either by image 
acquisition, for example by means of a CCD camera, of the superposition of 
a sample basic screen and the master screen, or it can be obtained by 
precalculation. The precalculation can be done either in the image domain 
or in the spectral domain, as explained earlier in the present disclosure. 
The comparing processor makes the image comparison by matching a given 
image with a prestored image; examples of ways of carrying out this 
comparison have been presented in detail in the previous section. This 
comparison produces at least one proximity value giving the degree of 
proximity between the acquired moire intensity profile and the prestored 
moire intensity profile. These proximity values are then used as criteria 
for making the document handling device accept or reject the document. 
Advantages of the present invention 
The present invention completely differs from methods previously known in 
the art which use moire effects for the authentication of documents. In 
such existing methods, the original document is provided with special 
patterns or elements which when counterfeited by means of halftone 
reproduction show a moire pattern of high contrast. Similar methods are 
also used for the prevention of digital photocopying or digital scanning 
of documents. In all these previously known methods, the presence of moire 
patterns indicates that the document in question is counterfeit. However, 
the present invention is unique inasmuch as it takes advantage of the 
intentional generation of a moire pattern having a particular intensity 
profile, whose existence and whose shape are used as a means of 
authentication of the document, and all this without having any latent 
image predesigned on the document. The approach on which the present 
invention is based further differs from that of prior art in that it not 
only provides fill mastering of the qualitative geometric properties of 
the generated moire (such as its period and its orientation), but it also 
permits to determine quantitatively the intensity levels of the generated 
moire. 
The fact that moire effects generated between superposed dot-screens are 
very sensitive to any microscopic variations in the screened layers makes 
any document protected according to the present invention practically 
impossible to counterfeit, and serves as a means to easily distinguish 
between a real document and a falsified one. 
Furthermore, unlike previously known moire-based anticounterfeiting 
methods, which are only effective against counterfeiting by digital 
equipment (digital scanners or photocopiers), the present invention is 
equally effective in the cases of analog or digital equipment. 
A further important advantage of the present invention is that it can be 
used for authenticating documents printed on any kind of support, 
including paper, plastic materials, etc., which may be transparent or 
opaque. Furthermore, the present invented method can be incorporated into 
the standard document printing process, so that it offers high security at 
the same cost as standard state of the art document production. 
Yet a further advantage of the present invention is that it can be used, 
depending on the needs, either as an overt means of document protection 
which is intended for the general public; or as a covert means of 
protection which is only detectable by the competent authorities or by 
automatic authentication devices; or even as a combination of the two, 
thereby permitting various levels of protection. The covert methods 
disclosed in the present invention also have the additional advantage of 
being extremely difficult to re-engineer, thus further enhancing document 
security.