Dot trajectories in the superposition of random screens: analysis and synthesis

Transcription

1 1472 J. Opt. Soc. Am. A/ Vol. 21, No. 8/ August 2004 Isaac Amiror Dot trajectories in the superposition of ranom screens: analysis an synthesis Isaac Amiror Laboratoire e Systèmes Périphériques, Ecole Polytechnique Féérale e Lausanne, 1015 Lausanne, Switzerlan Receive September 4, 2003; revise manuscript receive January 15, 2004; accepte February 9, 2004 Moiré effects that occur in the superposition of aperioic layers such as correlate ranom ot screens are known as Glass patterns. One of the most interesting properties of such moiré effects, which clearly istinguish them from their perioic counterparts, is unoubtely the appearence in the superposition of intriguing microstructure ot alignments, also known as ot trajectories. These ot trajectories may have ifferent geometric shapes, epening on the transformations unergone by the superpose layers. In the case of simple linear transformations such as layer rotations or layer scalings, the resulting ot trajectories are rather simple (circular, raial, spiral, elliptic, hyperbolic, linear, etc.); but in more complex layer transformations the ot trajectories can have much more interesting an surprising shapes. A full mathematical analysis of the ot trajectories, their morphology, an their various properties is provie. Furthermore, it is shown how the approach also allows us to synthesize correlate ranom screens that give in their superposition ot trajectories having any esire geometric shapes. Finally, it is also explaine why such ot trajectories are visible only in superpositions of aperioic screens but not in superpositions of perioic screens Optical Society of America OCIS coes: INTRODUCTION It is a well-known fact that the superposition of perioic layers (such as line gratings or ot screens) may give rise to new perioic structures that o not exist in any of the iniviual layers [see, for example, Fig. 1()]. These perioic structures, calle moiré effects, have been thoroughly stuie in the past, 1 3 an their mathematical founations are toay fully unerstoo. 4 The same is also true for moiré effects between repetitive layers (i.e., between geometric transformations of perioic layers) It is also known that the superposition of aperioic layers, such as ranom ot screens, may give rise to a ifferent type of moiré pattern, which consists of a single structure resembling a top-viewe funnel or a istant galaxy in the night sky [see, for example, Figs. 1(c) an 1(e)]. This phenomenon is known in literature as a Glass pattern, after Leon Glass, who escribe it in the late 1960s. 11,12 These Glass patterns are often surroune by typical microstructure ot arrangements, known as ot trajectories, that have various shapes such as circles, raial lines, spirals, hyperbolas, an ellipses, epening on the case (see Fig. 1). Glass patterns in the superposition of ranom ot screens remaine for a long perio much less unerstoo than moiré effects that occur between perioic or repetitive layers, partly because they i not easily len themselves to the same mathematical tools that so nicely explaine the classical moiré effects between perioic or repetitive layers (geometric consierations, inicial equations, Fourier series evelopments, etc.). Only recently has it been shown that in spite of their ifferent appearance, moiré patterns between perioic or repetitive layers an Glass patterns between aperioic layers are, in fact, particular cases of the same basic phenomenon, an all of them satisfy the same funamental rules These recent results basically concern the macroscopic phenomena that occur in the layer superpositions (the Glass or moiré patterns, their behavior uner layer transformations, an their intensity profiles as etermine by the Fourier theory). Nevertheless, several questions concerning the microstructure ot arrangements (the ot trajectories) that accompany these phenomena in the superposition still remain open. These questions concern, in particular, the mathematical interpretation of the ot trajectories, their morphology, an the reasons that they are visible only in superpositions of aperioic screens an not in superpositions of perioic screens. In the present paper we focus our attention on these ot trajectories, an a mathematical explanation is provie regaring their origin, their nature, their morphology, an their various properties. Furthermore, it will be shown how our mathematical unerstaning of these phenomena allows us not only to analyze the ot trajectories that appear in any given superposition but also to synthesize ot screens that give in their superposition ot trajectories having any esire morphology. Finally, on the basis of our results it will also be explaine why such ot trajectories are not visible in superpositions of perioic layers. It shoul be note that in the present stuy we are not intereste in the ot shapes of the iniviual layers an in their influence on the macroscopic moiré intensity profiles, as escribe in Ref. 15, an we concentrate only on the ot trajectories that accompany these phenomena. These two approaches are therefore complementary, an they can be use either inepenently of each other or combine (for example, for synthesizing a Glass pattern /2004/ $ Optical Society of America

2 Isaac Amiror Vol. 21, No. 8 / August 2004 / J. Opt. Soc. Am. A 1473 Fig. 1. (continues on next page). Glass patterns between aperioic ot screens an their moire counterparts between perioic ot screens. The aperioic ot screen (a) an its perioic counterpart (b) were use for generating all the superpositions shown in this figure. (c) The superposition of two ientical copies of aperioic ot screen (a) with a small angle ifference gives a Glass pattern about the center of rotation. Note the Glass pattern s typical microstructure consisting of concentric circular ot trajectories. () When the superpose layers are perioic, a perioic moire pattern is generate instea of the Glass pattern. (e) Same as (c), but with a small scaling ifference rather than a small angle ifference between the superpose layers. In this case the microstructure consists of raial ot trajectories. (f) The perioic counterpart of (e). having a specifie intensity profile, which is surroune by ot trajectories with a given morphology). The present paper is structure as follows: Section 2 introuces the terminology an the basic notions that will

3 1474 J. Opt. Soc. Am. A / Vol. 21, No. 8 / August 2004 Isaac Amiror Fig. 1. (continue). (g) Same as (c) an (e), but with both a small angle an a small scaling ifference between the superpose layers. In this case the microstructure consists of spiral ot trajectories. (h) The perioic counterpart of (g). (i) Same as (e), but here the scale layer unergoes unequal scaling rates in the x an y irections (0 s x 1, s y 1). Note the hyperbolic ot trajectories in the superposition. (j) The perioic counterpart of (i). (k) Same as (i), but with a small angle ifference, too, between the superpose layers. In this case the microstructure consists of elliptic ot trajectories. (l) The perioic counterpart of (k). be neee for the rest of the paper. Section 3 escribes the superposition of aperioic layers an explains its main properties. Then, Sections 4 an 5 explain the mathematical meaning of the ot trajectories an show how to analyze an how to synthesize them. Finally, Section 6 explains why ot trajectories are observe only in

4 Isaac Amiror Vol. 21, No. 8/August 2004/J. Opt. Soc. Am. A 1475 aperioic superpositions an not in their perioic counterparts, an Section 7 presents the main conclusions. Remark: The PostScript files that generate the ot screens use in the figures of this paper are available on the internet. 16 They can be ownloae an printe on transparencies by using any stanar PostScript printer. Superposing these transparencies manually with varying orientations, shifts, etc., can give a vivi emonstration of the Glass patterns an their ynamic behavior in the superposition, beyon the few static figures that illustrate this paper. 2. BACKGROUND AND BASIC NOTIONS In this introuctory section we briefly review the basic notions an terminology that will be use later. Since we will be ealing throughout this paper with layers an layer superpositions, let us start by explaining these notions an their main properties. In fact, a layer (or image ) is the most general term we use to cover anything in the image omain. It can be perioic or not, continuous or binary, etc. However, we still nee to make some basic assumptions regaring our layers. First of all, we limit ourselves here to monochrome, black-an-white images. This means that each image can be represente by a reflectance function r(x, y), which assigns to any point (x, y) of the image a value between 0 an 1 representing its light reflectance: 0 for black (i.e., no reflecte light), 1 for white (i.e., full light reflectance), an intermeiate values for in-between shaes. In the case of transparencies, the reflectance function is replace by a transmittance function efine in a similar way. A superposition of such images can be obtaine by overprinting or by laying printe transparencies on top of each other. Since the superposition of black an any other shae always gives black, this suggests a multiplicative moel for the superposition of monochrome images. Thus, when two monochrome images are superpose, the reflectance of the resulting image is given by the prouct of the reflectance functions of the iniviual images: r(x, y) r 1 (x, y)r 2 (x, y). Let us now explain what we mean by perioic an by aperioic or stochastic layers. A one-imensional function f(x) is sai to be perioic if there exists a nonzero number p such that for any x R, f(x p) f(x). Similarly, a two-imensional (2D) layer r(x, y) is sai to be perioic if there exists a nonzero vector p ( p 1, p 2 ) such that for any (x, y) R 2, r(x p 1, y p 2 ) r(x, y). If there exist two inepenent vectors having this property, r(x, y) is sai to be two-fol perioic. A layer r(x, y) is sai to be aperioic if it is not perioic. For example, the image of a human portrait or a natural lanscape is aperioic. A ranom ot screen consisting of ranomly positione black ots is also aperioic. Note, however, that this ranom ot screen may also be regare as a stochastic layer, from a more statistical point of view, if we consier the screen in question as just one possible realization of a stochastic process that has some given statistical istribution. In the context of ranom ot screens we often use the terms aperioic layer, stochastic layer, an ranom layer as synonyms. Next, let us introuce the notions of macrostructure an microstructure. It is well known that when perioic layers (line gratings, ot screens, etc.) are superpose, new structures of two istinct levels may appear in the superposition that o not exist in any of the original layers: macrostructures an microstructures (Ref. 4, Chap. 8). The macrostructures, also known as moiré patterns, are much coarser than the etail of the original layers, an they are clearly visible even when observe from a istance. The microstructures, on the contrary, are almost as small as the perios of the original layers (typically, just 2 5 times larger), an therefore they are visible only when one is examining the superposition from a close istance or through a magnifying glass. These tiny structures are also calle rosettes, owing to the various flowerlike shapes they often form in the superposition of perioic ot screens. Similar phenomena exist also in the superposition of aperioic or stochastic layers, where both macrostructures an microstructures may become visible. Here, too, macrostructures can be clearly seen when observe from a istance, whereas microstructure ot arrangements (such as ot trajectories; see Section 3 below) are visible only when one is examining the superposition from a close istance. Note that the istinction between macrostructures an microstructures may seem at first quite artificial an subjective. But in reality, macrostructures an microstructures are simply two ifferent facets of the same phenomenon in the layer superposition. In fact, they only represent two ifferent scales at which we consier the same phenomenon: In the macroscopic scale we consier the global, average behavior of the phenomenon (much like the rules of classical physics), whereas in the microscopic scale we stuy the behavior of the same phenomenon from the point of view of the iniviual screen elements an the interactions between them (much like the stuy of the same physical rules through the behavior of molecules an atoms). 17 Finally, for the sake of simplicity we consier here only layers having a uniform istribution of their microstructure elements (an hence a constant mean gray level), although our results hol also in more complex structures such as halftone graations an halftone images with varying gray levels. 3. SUPERPOSITION OF APERIODIC LAYERS While the superposition of two similar perioic layers generates moiré effects that are themselves perioic, the superposition of two similar aperioic layers generates an aperioic moiré effect known as a Glass pattern (compare the right- an the left-han-sie images in each row of Fig. 1). This moiré pattern is typically concentrate aroun a certain point in the superposition (which is the fixe point of the unerlying layer transformation 13 ), an unlike perioic moirés, it graually isappears as we go farther away from this point. Depening on whether it was obtaine by rotation of one of the superpose layers, by a scaling transformation, or by a combination of the two, it gives rise to an intriguing orering of the microstructure elements in the superposition in ot trajectories

5 1476 J. Opt. Soc. Am. A/ Vol. 21, No. 8/ August 2004 Isaac Amiror having a circular, raial, or spiral shape [see Figs. 1(c), 1(e), an 1(g)]. 12 Other layer transformations may give rise to Glass patterns having elliptic, hyperbolic, or other geometrically shape trajectories [Figs. 1(i) an 1(k)]. 12 However, when we rotate one of the aperioic layers by 180 [see Figs. 2(a) an 2(b)], the Glass pattern isappears. As alreay explaine by Glass, this phenomenon occurs thanks to the local correlation between the structures of the two superpose layers; in fact, its intensity can be use as a visual inication of the egree of correlation between the two layers at each point of the superposition. Thus, when two ientical layers having the same arbitrary structure are slightly rotate on top of each other [see Fig. 1(c)], a visible Glass pattern is generate aroun the center of rotation (the fixe point), inicating the high correlation between the two layers in this area. Within the center of this Glass pattern the corresponing elements from the two layers fall almost exactly on top of each other, but slightly away from the center they fall just next to each other, generating circular trajectories of point pairs. Farther away from the center the correlation between the two layers becomes smaller an smaller, an the elements from both layers fall in an arbitrary, noncorrelate manner; in this area the Glass pattern is no longer visible. This explains why the Glass pattern graually ecays an isappears as we go away from its center. Note, however, that when the two superpose layers are not at all correlate, no Glass pattern appears in the superposition [this is, inee, what happens when we rotate one of the aperioic transparencies by 180 on top of its ientical copy, as shown in Fig. 2(a)]. In intermeiate cases, where the two superpose layers are only partially correlate, the Glass pattern becomes weaker an less perceptible, epening on the egree of the correlation that still remains between the superpose layers. As we can see, the explanation above is base on an observation of the iniviual elements of the original layers an their behavior in the superposition. We say, therefore, that this explanation is base on the microstructure. To obtain the point of view of the macrostructure, we have to look at the layers an their superposition from a greater istance, where the iniviual elements of the layers are no longer iscerne by the eye an what we see is only an integrate gray-level average of the microstructure in each area of the superposition. From the point of view of the macrostructure, the center of the Glass pattern consists of a brighter gray level than areas farther away (owing to the partial overlapping of the microstructure elements of the two layers in this area); farther away, the macroscopic gray level is arker (because elements from the two layers are more likely to fall sie by sie, thus increasing the covering rate an the macroscopic gray level). This means that the Glass pattern is not just an optical illusion, an it correspons, inee, to the physical reality. In fact, just like in the perioic case (see Proposition 8.1 in Ref. 4), moiré patterns are simply the macroscopic interpretation of the variations in the microstructures throughout the superposition. On the other han, the orering of the microstructure elements within a Glass pattern into circular, raial, or spiral ot trajectories is no longer visible from far away (try to observe the images in the left column of Fig. 1 from a istance of 3 4 m, where the iniviual elements of the layers are no longer iscerne by the eye). These ot trajectories are not part of the macrostructure escription, an they belong to the microstructure of the superposition, just like rosettes in the perioic case. An inee, from the point of view of the macrostructure there is no istinction between gray levels obtaine when the neighboring elements in the superposition are locate on circular trajectories, owing to rotation, or on raial trajectories, owing to a scaling transformation: What counts in both cases is the resulting mean coverage rate, which etermines the overall gray level, an not the specific geometric arrangement of the ots. In the present paper we focus our attention on the microstructure of the Glass patterns an on the arrangement of the iniviual ots into ot trajectories. It is important to note that Fourier analysis methos are not applicable here, since they treat only the global aspects of the superposition but o not go own to the level of the Fig. 2. (a) an (b) Same as (c) an (e) in Fig. 1, but with one of the layers being rotate by 180 ; in this case the Glass patterns isappear completely.

6 Isaac Amiror Vol. 21, No. 8/August 2004/J. Opt. Soc. Am. A 1477 iniviual screen ots an their local behavior. 18 In the following section we will see what mathematical tools can be use to investigate these microscopic phenomena. 4. MORPHOLOGY OF THE MICROSTRUCTURES: ANALYSIS OF DOT TRAJECTORIES One of the most interesting aspects of aperioic moiré effects, first mentione by Glass, 12 concerns the morphology of the microstructures that are generate in the superposition of aperioic screens. As we can see in many of the figures throughout the present paper, fixe points in the superposition of aperioic screens are often surroune by ot trajectories having various geometric shapes, such as circles, ellipses, hyperbolas, or spirals, etc., epening on the case. For example, a fixe point that occurs as a result of a rotation of one of the superpose layers is typically surroune by circular ot trajectories [Fig. 1(c)], while a fixe point ue to a uniform scaling transformation is typically surroune by raial ot trajectories [Fig. 1(e)]. Spiral ot trajectories, on their part, are typical of intermeiate cases involving both rotation an scaling [Fig. 1(g)], an hyperbolic ot trajectories typically occur aroun the fixe point in scaling transformations involving expansion along one irection an contraction along a ifferent irection [Fig. 1(i)]. Some interesting ot trajectories that are generate by nonlinear transformations of the original aperioic layers are shown in Fig. 3. Note that ot trajectories may appear even in superpositions where no fixe points exist [see, for example, Fig. 3()]. This large versatility in the ot trajectories an in their shapes certainly eserves investigation. It is therefore our aim here to stuy the ot trajectories, their morphology, an their various properties. Suppose, to begin with, that we superpose two ientical aperioic screens exactly on top of each other an that we apply to one of the superpose screens a transformation g(x, y) given by x g 1 x, y, y g 2 x, y. (1) (Note that x an y o not represent erivatives but rather a ifferent coorinate system in the plane.) As a simple example, we may think of the linear transformation: x a 1 x b 1 y, y a 2 x b 2 y. (2) The ot trajectories that are generate in these superpositions are physically mae of pairs of ots, one ot from each layer. Each such pair in the superposition represents, in fact, two successive locations of the same ot, namely, the ot s location before an after the layer transformation g(x, y) has been applie. Let us try to see how these ot trajectories can be analyze mathematically. A. Dot Trajectories As Solution Curves of a System of Differential Equations A first attempt to interpret such ot trajectories has alreay been presente in Ref. 12. Accoring to this approach, we consier the finite transformation g(x, y) as an iteration of an infinitesimal transformation. This reasoning leas one from Eqs. (1) to a pair of first-orer ifferential equations that correspon to our infinitesimal transformation: t x t g 1(x t, y t ), t y t g 2(x t, y t ). (3) For example, in the particular case of linear transformation (2), we get t x t a 1x t b 1 y t, t y t a 2x t b 2 y t. (4) The solution of system (3) consists of a family of curves in the x, y plane, whose members iffer from each other by some constants c (Ref. 19, pp ). The parametric representation of each of these curves is x(t), y(t), where the parameter t may be thought of as time. Each of these solution curves is calle a trajectory since it traces out the evolution of the curve as t is being varie. Accoring to this reasoning, 12 the ot trajectories observe in the superposition of aperioic ot screens simply represent the trajectories (i.e., the solution curves) of this unerlying pair of ifferential equations. 20 An, inee, this explanation agrees with experimental evience in various cases. However, a eeper examination of the question shows that this reasoning is not always true. This is clearly emonstrate by the following examples. Example 1. Consier the ientity transformation g(x, y) (x, y), which is a particular case of Eqs. (2) with a 1 b 2 1 an a 2 b 1 0. Obviously, in this case the two layers perfectly coincie on top of each other, an no ot trajectories are generate. However, the solution of the corresponing system of ifferential equations (4) with a 1 b 2 1 an a 2 b 1 0 gives the family of straight lines y cx for any constant c, an its solution curves (trajectories) consist of raial lines emanating from the origin (see Ref. 19, p. 168). While such raial ot trajectories can be expecte in the layer superposition when the mapping being applie to one of the screens is a scaling by s 1 [see Fig. 1(e)], it is clear that in our present case, where s 1, no ot trajectories will appear in the superposition. Example 2. Suppose that we apply to one of the two superpose layers a rotation by a small angle, as illustrate in Fig. 1(c). This linear transformation is clearly a particular case of Eqs. (2), with a 1 cos, b 1 sin, a 2 sin, an b 2 cos. Accoring to Fig. 1(c) we woul expect the solution curves of ifferential equations (4) to consist of circular trajectories about the center of rotation. However, a short verification shows that this is not always the case. Although for 90, where a 1 b 2 0, a 2 1 an b 1 1, the ifferential equations o have circular trajectories as expecte, it turns out that for 0 (the ientity transformation, where a 1 b 2 1 an a 2 b 1 0) the solution curves of the ifferential equations consist of a family of raial lines emanating from the origin. Furthermore, for all intermeiate rotation angles 0 90 the solution curves consist of a family of spirals

7 1478 J. Opt. Soc. Am. A / Vol. 21, No. 8 / August 2004 Isaac Amiror Fig. 3. (a) Aperioic ot screen that has unergone the parabolic transformation g(x, y) (x ay 2, y). (b) The superposition of two ientical aperioic ot screens, one of which has unergone the parabolic transformation g(x, y). Since this transformation oes not involve layer shifts, the two layers clearly coincie along the x axis. (c) The superposition of two ientical aperioic ot screens, one of which has unergone the parabolic transformation g(x, y) (x ay 2, y), while the untransforme screen has been slightly shifte by x 0. A pair of linear Glass patterns is generate in their superposition. () Same as in (c) but with a slight layer shift of (x 0, y 0 ) rather than x 0. (e) The superposition of two ientical aperioic ot screens, one of which has unergone the parabolic transformation g(x, y) (x ay 2, y), while the untransforme layer has been slightly scale up. (f) An example with four fixe points: the superposition of two ientical aperioic ot screens, one of which has unergone a vertical parabolic transformation plus a slight vertical shift of y 0 ownwar, while the other has unergone a horizontal parabolic transformation plus a slight horizontal shift of x 0 to the left. Note the two circular an the two hyperbolic Glass patterns that are clearly visible about the fixe points in the superposition.

8 Isaac Amiror Vol. 21, No. 8/August 2004/J. Opt. Soc. Am. A 1479 (see, for example, Ref. 21, p. 144); as 90 these spirals graually approach circles, but as 0, i.e., for small rotation angles, the spirals straighten out an graually approach raial straight lines. This result oes not agree with our expectations base on Fig. 1(c). These examples clearly show that the traitional approach base on the system of ifferential equations (4) oes not correctly correspon to the ot trajectories that appear in our layer superposition when the mapping (2) is applie to one of the superpose layers. The reasons for this failure as well as ways to overcome it will become clear shortly. B. Dot Trajectories As a Vector Fiel A better unerstaning of this problem can be obtaine by consiering the situation from another point of view. As explaine in Appenix A, any two-imensional transformation g(x, y) can be also interprete as a vector fiel that assigns to each point (x, y) in the x, y plane the vector g(x, y). This vector fiel can be illustrate graphically by rawing starting from each point (x, y) an arrow having the length an the orientation of the vector g(x, y). This interpretation has the important avantage of clearly showing in a visual way the effect of the transformation g(x, y) on any point of the x, y plane. It shoul be note that the system of ifferential equations (3) is simply a ifferent representation of the same vector fiel g(x, y), since its solution curves x(t), y(t) express the fiel lines (trajectories) of the vector fiel g(x, y) (see Appenix A). Now, remember that the ot trajectories in our superposition of two aperioic ot screens consist of pairs of ots, which represent the location of a screen ot before an after the layer transformation g(x, y) has been applie. These ot pairs can be represente, therefore, as a vector fiel, which assigns to each point (x, y) R 2 a vector that connects (x, y) to its new location g(x, y) R 2 uner the transformation g. 22 It is important to note, however, that the vector fiel of the transformation g(x, y) itself oes not have this property; that is, the vector it assigns to (x, y) oes not connect (x, y) to its estination g(x, y) but rather to the point (x, y) g(x, y). For example, if we consier the ientity transformation g(x, y) (x, y) (see example 1), it is clear that in this case the vector attache to each point (x, y) is the vector (x, y) itself, which points, therefore, to the point (2x, 2y) an not to the estination point uner g, which is the point (x, y). Therefore, in orer to obtain a vector fiel that correctly represents our ot trajectories, we must consier, instea of the transformation g(x, y) itself (the transformation that has been applie to one of the superpose layers), the relative transformation between the two layers, which is given by h x, y g x, y x, y. (5) An, inee, if we raw the vector fiel representation of this transformation, we obtain exactly what we esire: The vector fiel of h(x, y) assigns to each point (x, y) the vector g(x, y) (x, y), which connects the original point (x, y) to its estination uner the layer transformation g, the point g(x, y). If we take one step further an allow both of the superpose layers to be transforme, one by a mapping g 1 (x, y) an the other by a mapping g 2 (x, y), then the relative transformation between the two layers becomes h x, y g 1 x, y g 2 x, y. (6) Note, however, that in this case the ot pairs that make up the ot trajectories in the superposition no longer represent a ot s location before an after the layer transformation has been applie but rather the new locations of the same original ot uner the transformation g 1 (x, y) an uner the transformation g 2 (x, y). It shoul be also mentione that since the two ots that compose each ot pair in the layer superposition are ientical, the ot pairs (an hence the ot trajectories) remain unchange when we interchange the transformations unergone by the two layers. In other wors, the ot trajectories in the superposition o not show the irection (the positive or negative sense) of the ifference vectors. These results can be therefore summarize as follows: Proposition 1. Suppose that we are given two ientical aperioic ot screens that are superpose on top of each other in full coincience, ot on ot. When we apply to one of the layers a transformation g(x, y), we obtain in the superposition ot trajectories that correspon to the vector fiel h(x, y) g(x, y) (x, y) [or, equivalently, to the vector fiel h(x, y) (x, y) g(x, y)]. Similarly, if we apply g 1 (x, y) to one of the layers an g 2 (x, y) to the other layer, where both g 1 (x, y) an g 2 (x, y) are weak transformations, we obtain in the superposition ot trajectories that correspon to the vector fiel h(x, y) g 1 (x, y) g 2 (x, y) [or, equivalently, to the vector fiel h(x, y) g 2 (x, y) g 1 (x, y)]. It shoul be unerstoo, however, that although such ot trajectories are theoretically generate in the superposition in all cases, they can be clearly visible [an correspon to the vector fiel h(x, y)] only if the layer transformations g(x, y), g 1 (x, y), an g 2 (x, y) are not too violent. Otherwise, the correlation between the superpose layers is strongly reuce, an the visual effect in the layer superposition may be lost an no longer correspon to the vector fiel. Let us now see a few examples to illustrate how this approach explains the ot trajectories obtaine in our aperioic screen superpositions. We start with the two cases that cause us trouble in examples 1 an 2. Example 3. Suppose that one of the superpose screens, say, the first one, is scale by s 0. In orer not to lose the correlation between the two screens, we assume that s 1. The relative transformation between the two layers is given in this case by h x, y sx, sy x, y s 1 x, y. Regaring this linear transformation as a vector fiel, we see that it assigns to any point (x, y) in the x, y plane a vector (s 1)(x, y). Clearly, if s 1 this vector fiel consists of raial trajectories emanating from the origin [see Fig. 4(b)], whereas if 0 s 1 it consists of raial trajectories pointing to the origin [Fig. 4()]. The origin itself is a fixe point of this vector fiel. An, inee, this fully agrees with the raial ot trajectories that we observe in the screen superposition in both of these cases.

9 1480 J. Opt. Soc. Am. A/ Vol. 21, No. 8/ August 2004 Isaac Amiror Fig. 4. Vector fiel representation of the relative transformation h(x, y) g(x, y) (x, y) where g(x, y) (the transformation unergone by the first layer) is (a) a slight rotation; (b) a slight scaling (sx, sy) with s 1; (c) both a slight rotation an a slight scaling with s 1; () a slight scaling with 0 s 1; (e) a slight scaling with 0 s x 1, s y 1; (f) both a slight scaling with 0 s x 1, s y 1 an a small rotation. Compare with Figs. 1(c), 1(e), 1(g), 1(i), an 1(k). Note, however, that the ot trajectories in the screen superposition [Fig. 1(e)] o not show the irection along the trajectories [which is inicate by the arrowheas of Figs. 4(b) an 4()]. Finally, let us examine the case of s 1, which cause us trouble in example 1. In this case (scaling by 1) our vector fiel h(x, y) assigns to each point (x, y) in the x, y plane the zero vector; hence this transformation has no

10 Isaac Amiror Vol. 21, No. 8/August 2004/J. Opt. Soc. Am. A 1481 fixe points an no trajectories exactly as we woul expect when the two ientical screens fully coincie on top of each other. Thus by consiering the vector fiel of the relative layer transformation h(x, y) we have overcome our troubles from example 1. Example 4. Let us now consier the linear mapping that correspons to a rotation of the first layer about the origin by a small angle. In this case the vector fiel h(x, y) g(x, y) (x, y) assigns to each point (x, y) in the x, y plane a vector that connects it to its new location after the rotation. This vector fiel consists of a circular arrangement of arrows (ot pairs) about the origin, as we can see in Fig. 4(a). This agrees, inee, with the ot trajectories that are generate in the screen superposition, as shown in Fig. 1(c). These examples show clearly that by consiering the vector fiel of the relative layer transformation h(x, y), we have overcome our troubles from examples 1 an 2. Now, remember that the system of ifferential equations (3) is simply a ifferent representation of the vector fiel g(x, y), since its solution curves x(t), y(t) express the fiel lines (trajectories) of the vector fiel g(x, y). It can be expecte, therefore, that by using the relative layer transformation h(x, y) instea of g(x, y) in Eqs. (3) we can also cure the ifferential-equation approach of Subsection 4.A an make it work properly. A short investigation of this point shows inee that in general the trajectories of the moifie ifferential equations, t x t h 1(x t, y t ), t y t h 2(x t, y t ), correspon well to our ot trajectories in the superposition; an yet, in some cases there still exist some minor iscrepancies between them. It turns out that the arrow representation of the vector fiel h(x, y) fully correspons to our ot trajectories, because each of the arrows connects the location of the same screen ot in the first an in the secon transforme layers. But the solution curves (trajectories) of the moifie system of ifferential equations o not correspon to the motion of a screen ot uner the given layer transformation: Although the solution curves of this system of ifferential equations express mathematically the fiel lines of the relative layer transformation h(x, y), these fiel lines are not always precisely what we are looking for but only a close approximation. This subject is iscusse further in Appenix B. It shoul be mentione, however, that since we in any case must restrict ourselves to weak layer transformations (in orer not to estroy the correlation in the superposition), this iscrepancy is marginal an it can be neglecte for all practical nees (as we will see, for instance, in example 6 below). To further illustrate our mathematical interpretation of the ot trajectories in the superposition as a vector fiel, Figs. 4 an 5 provie a graphical representation of the vector fiel h(x, y) for some of the most interesting superpositions shown in Figs. 1 an 3. These cases inclue both linear an nonlinear layer transformations, an they illustrate ot trajectories that occur aroun one or more fixe points, as well as ot trajectories that are generate when no fixe points exist in the superposition. Note the perfect agreement between the ot trajectories in each of the superpositions an the arrows showing the ot isplacements in the corresponing vector fiel h(x, y). These examples clearly show how the vector fiel approach presente above explains the ot trajectories that are generate in the superposition of aperioic screens. It shoul be remembere, however, that the ot trajectories are visible in the layer superposition owing to the correlation between the superpose layers. Hence if the original layers are not sufficiently correlate, or if the transformations g(x, y) are too violent an estroy the correlation between the layers, no ot trajectories will be seen in the superposition (even if the corresponing vector fiel still inclues visible solution curves). 5. SYNTHESIS OF DOT TRAJECTORIES Having succeee in analyzing the ot trajectories that occur in the superposition of aperioic screens, we may ask now whether we can also synthesize such effects. In other wors, can we synthesize, for any given vector fiel h(x, y), two aperioic ot screens that generate in their superposition ot trajectories corresponing to h(x, y)? Suppose that we are given a vector fiel h(x, y) or, equivalently, a system of ifferential equations t x t h 1(x t, y t ), t y t h 2(x t, y t ). We wish to synthesize an aperioic screen superposition whose ot trajectories visually illustrate our given vector fiel (or system of ifferential equations). We start by superposing two ientical aperioic ot screens on top of each other in full coincience. As we alreay know from the first part of proposition 1, if we apply a transformation g(x, y) to one of the layers we obtain in the superposition the ot trajectories that correspon to h(x, y) g(x, y) (x, y) [or equivalently to h(x, y) (x, y) g(x, y)]. Therefore, in orer to obtain in the superposition ot trajectories which correspon to h(x, y), we may apply to one of the superpose layers the transformation g(x, y) (x, y) h(x, y) [or equivalently, g(x, y) (x, y) h(x, y)], while the other layer remains unchange. Similarly, it follows from the secon part of proposition 1 that by applying a transformation g 1 (x, y) to one of the superpose layers an a transformation g 2 (x, y) to the other layer, where both g 1 (x, y) an g 2 (x, y) are weak transformations, we obtain in the superposition the ot trajectories of h(x, y) g 1 (x, y) g 2 (x, y). This gives us an alternative way to synthesize ot trajectories that correspon to h(x, y), by applying to both of the superpose layers transformations g 1 (x, y) an g 2 (x, y) whose ifference gives h(x, y). For example, we may istribute the eformation in equal parts between the two layers as follows: g 1 x, y x, y 1 2 h x, y, g 2 x, y x, y 1 2 h x, y, which gives, again, g 1 (x, y) g 2 (x, y) h(x, y).

11 1482 J. Opt. Soc. Am. A/ Vol. 21, No. 8/August 2004 Isaac Amiror Having unerstoo these basic principles, let us now illustrate the synthesis of ot trajectories by a few examples. Example 5. Suppose we wish to synthesize an aperioic screen superposition whose ot trajectories visually illustrate the vector fiel h(x, y) efine by h x, y 2xy, y 2 x 2 [see Fig. 6(a)]. As we have just seen, this can be one, for example, by applying to the layers the following transformations g 1 (x, y) an g 2 (x, y), which istribute the eformation h(x, y) in equal parts between the two layers: g 1 x, y x, y 1 2 h x, y x xy, y 1 2 y x 2, g 2 x, y x, y 1 2 h x, y x xy, y 1 2 y x 2. An inee, if we apply the transformations g 1 (x, y) an g 2 (x, y), respectively, to the two superpose layers, we obtain in the superposition ot trajectories that correspon to the given vector fiel h(x, y). This is clearly illustrate in Fig. 7(a). To improve the visual effect, the transformations g 1 (x, y) an g 2 (x, y) can be softene by replacing h(x, y) in both equations by h(x, y), where is a small positive fraction; this guarantees that the correlation between the two superpose layers remains strong enough to give clearly visible ot trajectories. Example 6. As a secon example, let us choose a system of two ifferential equations that is well known in physics, such as the nonlinear system escribing the behavior of a free unampe penulum (see, for example, Ref. 19, pp or Ref. 23, pp ). The system of ifferential equations is given in this case by x t y t, t y t ksin x t, t where k is a positive constant. The solution curves x(t), y(t) of this system are illustrate in Figs. 6(b) an 6(c). Since the solution curves of this system are the fiel lines of the vector fiel h(x, y) ( y, k sin x) (see Appenix A), we consier the two layer transformations: Fig. 5. Vector fiel representation of the relative transformation h(x, y) g 1 (x, y) g 2 (x, y), where (a) g 1 (x, y) is a slight parabolic transformation an g 2 (x, y) is a slight horizontal shift of x 0,(b)g 1 (x, y) is a slight parabolic transformation an g 2 (x, y) is a slight shift of (x 0, y 0 ), (c) g 1 (x, y) is a slight parabolic transformation an g 2 (x, y) is a slight linear scaling, () g 1 (x, y) is a slight vertical parabolic transformation with a vertical shift of y 0 ownwar, an g 2 (x, y) is a slight horizontal parabolic transformation with a horizontal shift of x 0 to the left Compare with Figs. 3(c), 3(), 3(e) an 3(f).

12 Isaac Amiror Vol. 21, No. 8/August 2004/J. Opt. Soc. Am. A 1483 Fig. 6. (a) Vector fiel representation of the nonlinear transformation h(x, y) (2xy, y 2 x 2 ). (b) Vector fiel representation of the nonlinear system of ifferential equations escribing the behavior of a free unampe penulum. (c) Solution curves of the system of ifferential equations of (b), rawn in a larger scale for clarity. Fig. 7. Examples of synthesize aperioic layers that give in their superposition ot trajectories having a given esire shape. (a) Dot trajectories that show the flow lines of the vector fiel h(x, y) (2xy, y 2 x 2 ), (b) ot trajectories that illustrate the nonlinear system of ifferential equations escribing the behavior of a free unampe penulum. The transformations applie to the original layers in (a) are g 1 (x, y) (x, y) 1 2 h(x, y) an g 2(x, y) (x, y) 1 2 h(x, y), an in (b) they are g 1(x, y) (x, y) 1 2 ( y, k sin x) an g 2 (x, y) (x, y) 1 2 ( y, k sin x). Note the agreement with the vector fiels shown in Fig. 6. g 1 x, y x, y 1 2 h x, y x 1 2 y, y 1 2 k sin x, g 2 x, y x, y 1 2 h x, y x 1 2 y, y 1 2 k sin x. An inee, as shown in Fig. 7(b), if we apply these transformations to our superpose aperioic screens, we obtain in the superposition ot trajectories that correspon to Fig. 6(b). These examples illustrate, inee, how our results can be use to synthesize aperioic screen superpositions having any esire ot trajectories. It shoul be repeate,

13 1484 J. Opt. Soc. Am. A/ Vol. 21, No. 8/ August 2004 Isaac Amiror however, that the ot trajectories can be visible in the superposition only when the transformations that we apply to the two layers are not too violent ; otherwise, the correlation between the layers is strongly reuce, an the visual effect is lost. Finally, it shoul be note that this synthesis of ot trajectories is completely inepenent of the synthesis of Glass patterns having a preetermine intensity profile (gray level istribution), which was escribe earlier in Ref. 15. That previous contribution explaine how to esign two ranom ot screens having an ientical ot istribution but ifferent ot shapes (for example, 1 -shape ots in one layer an tiny pinholes on a black backgroun in the other layer), which generate in their superposition a Glass pattern having a given macroscopic intensity profile (in this example, a Glass pattern having the shape of a consierably magnifie 1 ). In the present paper, on the contrary, we are not intereste in the shape of the iniviual ots in the superpose layers nor in the macroscopic intensity profile of the resulting Glass pattern in the superposition. Rather, we are intereste in the geometric transformations that are applie to each of the superpose layers an in their influence on the resulting shapes of the microstructures (ot trajectories) that are generate aroun the Glass pattern. Obviously, the two approaches can be use inepenently of each other an can even be combine in orer to synthesize in the superposition a Glass pattern having a given intensity profile that is surroune by any esire ot trajectories. 6. DOT TRAJECTORIES IN PERIODIC, REPETITIVE, AND APERIODIC CASES We now return to the intriguing question that we have alreay mentione earlier: Why o we observe ot trajectories only in aperioic superpositions an not in their corresponing perioic or repetitive superpositions (compare, for example, the left an right cases in each row of Fig. 1)? A careful examination, uner a magnifying glass, of the microstructure in a perioic or a repetitive superposition [for example, in Fig. 1()] will reveal that similar circular, elliptical, or hyperbolic arrangements of ot pairs o exist about the fixe points in such cases, too. However, in perioic or repetitive cases the ot locations are strongly constraine by an impose orering, which limits the egrees of freeom in the microstructure an makes the ot trajectories much less conspicuous than in aperioic superpositions. Furthermore, this impose orering also generates in the superposition a highly visible moiré structure whose presence obscurs the arrangements of the correlate ot pairs in the microstructure. In aperioic cases the microstructure orering is not perturbe by any other impose structures, an therefore the ot trajectories can manifest themselves freely in the superposition. Note, however, that the ot trajectories are less visible when the correlation between the superpose layers is lower (for example, when a certain amount of ranom noise is ae to the original layers) or when each of the superpose layers consists of elements of a ifferent shape (for example, 1 -shape elements in one layer an tiny pinholes in the other layer, as in Ref. 15). 7. CONCLUSIONS In the present paper we concentrate on the microstructure ot alignments, known as ot trajectories, that appear in the superpositions of correlate aperioic ot screens. We show that these ot trajectories visually express the vector fiel representation of the relative layer transformation h(x, y) between the two superpose layers, an we show why the explanations base on the absolute layer transformations o not work. Having unerstoo this phenomenon, we then go the other way aroun an show how one can synthesize aperioic ot screens that give in their superposition ot trajectories that correspon to any esire vector fiel h(x, y) or system of ifferential equations t x t h 1(x t, y t ), t y t h 2(x t, y t ). Finally, we also explain why ot trajectories are clearly visible in superpositions of aperioic screens but not in superpositions of perioic screens. APPENDIX A: MATHEMATICAL INTERPRETATIONS OF A TWO-DIMENSIONAL MAPPING g(x, y) Consier a system of two equations in two inepenent variables x an y, u g 1 x, y, v g 2 x, y, (A1) or, in vector notation, u g x, (A2) where x (x, y), u (u, v), an g(x, y) g 1 (x, y), g 2 (x, y). Clearly, g(x, y) is a vector function, i.e., a function that returns for each point (x, y) R 2 a new point (u, v) R 2. The mathematical relationship efine by Eqs. (A1) [or alternatively by Eq. (A2)] can be interprete in several ifferent yet completely equivalent ways, as explaine below. Because all of these interpretations are mathematically equivalent, we are free in each case to use any of them, accoring to our best jugment. In a first interpretation, we may consier the vector function g(x, y) of Eq. (A2) a mapping from the x, y plane onto itself. Alternatively, we may interpret system (A1) or its vector representation (A2) as a change of coorinates in the plane, whereby a new curvilinear coorinate system u, v is efine instea of the original Cartesian coorinate system x, y. Finally, another useful interpretation is obtaine by consiering the vector function g(x, y) of (A2) as a vector fiel. A vector fiel in R 2 is a function g(x, y) that assigns to each point (x, y) in the x, y plane a vector (u, v) g(x, y). Well-known examples in physics are electric or magnetic fiels an the gravitation fiel of the earth, all of which are vector fiels (that are efine in the three-imensional space R 3 ). A vector fiel in R 2 can be illustrate graphically by rawing an arrow emanating from each point (x, y) of the

14 Isaac Amiror Vol. 21, No. 8/August 2004/J. Opt. Soc. Am. A 1485 x, y plane (or, more practically, from some representative points efine on a given gri within the x, y plane), where the length an the orientation of each arrow inicate the length an the orientation of the vector that has been assigne by g(x, y) to the point (x, y) [see Fig. 8(a)]. It is important to note, however, that this arrow oes not connect the point (x, y) to its estination g(x, y) uner the transformation g but rather to the point (x, y) g(x, y). To better visualize a vector fiel, one may raw its trajectories (also known as fiel lines). Loosely speaking, these are the curves obtaine by following the arrows in Fig. 8(a) an joining them into continuous curves in the x, y plane [see Fig. 8(b)]. More precisely, trajectories (or fiel lines) are curves for which the tangent vector to the curve at each point (x, y) is exactly g(x, y). Thus, at every point (x, y), the irection in which the trajectory runs is etermine by the vector g(x, y). Note that except for points where g(x, y) is not efine or where g(x, y) 0, every point in the plane belongs to one an only one trajectory. In Cartesian coorinates, the trajectories of a vector fiel g(x, y) are given in parametric form by a family of curves x(t), y(t) that are solutions of the system of ifferential equations (see Ref. 24, p. 526), t x t g 1(x t, y t ), t y t g 2(x t, y t ), (A3) where t is the parameter of each of the curves. Ways of solving systems of ifferential equations such as Eqs. (A3) can be foun, along with many illustrative examples an figures showing their trajectories, in Chap. 4 of Ref. 19. A complete classification of the ifferent trajectory shapes for linear ifferential equation systems, incluing noes, sale points, center points, an spirals, can be foun in Ref. 19, pp ; Ref. 25, pp ; an Sec. 1.4 of Ref. 23. A similar classification for nonlinear ifferential equation systems can be foun in Ref. 25, pp Fig. 8. (a) Vector fiel representation of the nonlinear transformation h(x, y) (x ay 2, y) (sx, sy) with a an s (b) Some fiel lines (trajectories) of this vector fiel. Fig. 9. (a) The arrow (vector) connecting point (x, y) to its new location g(x, y) after the application of the rotation transformation g is not tangential to the circle but rather a chor. Therefore the fiel line emanating from (x, y) slightly spirals inwar, an it is not purely circular. (b) Clearly, the smaller the rotation angle, the closer the fiel line follows the circle. The rotation angles use in (a) an (b) are, respectively, 45 an 5.