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Springer, 2007. — 508 p.Since The Theory of the Moiré Phenomenon was published it became the main reference book in its field. It provided for the first time a complete, unified and coherent theoretical approach for the explanation of the moiré phenomenon, starting from the basics of the theory, but also going in depth into more advanced research results. However, it is clear that a single book cannot cover the full breadth of such a vast subject, and indeed, this original volume admittently concentrated on only some aspects of the moiré theory, while other interesting topics had to be left out.

Perhaps the most important area that remained beyond the scope of the original book consists of the moiré effects that occur between correlated random or aperiodic structures. These moiré effects are known as Glass patterns, after Leon Glass who described them in the late 1960s. However, this branch of the moiré theory remained for many years less widely known and less understood than its periodic or repetitive counterpart: Less widely known because moiré effects between aperiodic or random structures are less frequently encountered in everyday’s life, and less understood because these effects did not easily lend themselves to the same mathematical methods that so nicely explained the classical moiré effects between periodic or repetitive structures. Only recently has it been shown that in spite of their very different appearance and properties, moiré patterns between periodic or repetitive structures and Glass patterns between aperiodic or random structures are, in fact, particular cases of the same basic phenomenon, and all of them satisfy the same fundamental rules. These new research results have accumulated into a considerable volume that could no longer fit into an additional chapter in the original book. Rather, it became evident that this new material merits a new volume of its own, that should be entirely devoted to the aperiodic case.

And indeed, the study of the moiré phenomena which occur between random or aperiodic structures is not less interesting (and certainly not less surprising and fascinating) than the study of their periodic or repetitive counterparts. In a way, the diversity of the phenomena which occur in aperiodic superpositions is even more surprising than that of periodic cases, due to the unique interplay between the macroscopic and microscopic structures that appear in the superposition, and their very different properties. And yet, it turns out that using a suitable approach, earlier results from the periodic case can be extended to englobe the aperiodic case, too, although such an extension could a-priori seem hopeless. This applies to the main mathematical approaches already known from the periodic case, including the indicial equations method and the Fourier approach, both of which have been successfully extended to the aperiodic case. It should be noted, however, that just as in the periodic case, the indicial equations method as well as the Fourier approach are only applicable to the study of the macrostructures (the macroscopic phenomena in the superposition), but not to the study of the microstructures. And indeed, as it is clearly shown in the present volume, the study of the microstructures requires a different arsenal of mathematical tools. (Note that this is also true in the periodic case, and indeed, in the first volume, too, the microstructures had to be discussed in a separate chapter, Chapter 8 .)

The present book is intended to be a complementary, yet stand-alone companion to the original volume. Just like the first volume, it provides a full general purpose and application-independent exposition of the subject. It leads the reader through the various phenomena which occur in the superposition of correlated aperiodic layers, both in the image and in the spectral domain. And just like its predecessor, this volume favours a pictorial, intuitive approach which is supported by mathematics, and it includes a large number of figures and illustrative examples, some of which are visually striking and even spectacular.

Although this book has no pretentions to cover the entire moiré theory between aperiodic or random layers, it lays the foundations of this theory and gives for the first time an extended, comprehensive introduction to the subject. But it also goes in many aspects beyond the introductory level, and treats in depth some of the most interesting research results that have been recently obtained. Among other subjects, this book gives the full mathematical explanation of Glass patterns, describes their various properties, and explores their behaviour when the original layers undergo geometric transformations. It explains in detail the nature and the mathematical properties of the dot trajectories that often accompany Glass patterns in aperiodic screen superpositions, and it also investigates the one-dimensional counterparts of the classical Glass patterns that are generated in the superposition of correlated aperiodic line gratings. Most importantly, this book also provides the full Fourier-based approach, which allows us to explore Glass patterns and their intensity levels quantitatively and not only qualitatively. And finally, last but not the least, this book shows throughout the entire text and the accompanying figures the similarities as well as the differences between the moiré patterns which are obtained in periodic and in aperiodic cases, and it presents the theoretic results in a unified and coherent manner that clearly shows the relationship between the periodic and the aperiodic cases.

As already mentioned, this book is intended to be independent, as far as possible, from its predecessor. For this reason some fundamental notions from the original volume that are necessary for the comprehension of the new material are shortly reviewed here in Chapter 2, and some figures and results from the original volume are also reminded (or slightly adapted) in later chapters. Nevertheless, in order to avoid excessive repetitions, reference is occasionally made to some specific points in the original volume. On the other hand, it is interesting to note that in many ways the present volume also sheds new light on topics that were already explained in its predecessor, for example, by showing how these topics can be interpreted within a larger, unified framework that covers both periodic and aperiodic cases. And indeed, the wider point of view offered by the present volume often deepens our understanding of classical results from the periodic moiré theory.

This book is intended for students, scientists and engineers wishing to widen their knowledge of the moiré effect and its relationship with random or aperiodic structures. In particular, it will be very useful for people interested in the various moiré applications and in moiré-based technologies. The reader will find in this book not only a theoretical explanation of the moiré phenomena in question, but also practical recipes accompanied by many examples for the synthesis of aperiodic layers that give in their superposition moiré patterns having desired shapes, intensity profiles or microstructures. The prerequisite mathematical background is limited to an elementary familiarity with calculus and with the Fourier theory. But even occasional readers with no mathematical background will certainly enjoy the beauty of the effects illustrated throughout this book, and — it is our hope — may be tempted to learn more about their nature and their properties.Introduction

Background and basic notions

Glass patterns and fixed loci

Microstructures: dot trajectories and their morphology

Moiré phenomena between periodic or aperiodic screens

Glass patterns in the superposition of aperiodic line gratings

Quantitative analysis and synthesis of Glass patterns

A. Fixed point theorems for first- and second-order polynomial mappings

B. The various interpretations of a 2D transformation

C. The Jacobian of a 2D transformation and its significance

D. Direct and inverse spatial transformations

E. Convolution and cross correlation

F. The Fourier treatment of random images and of their superpositions

G. Integral transforms

H. Miscellaneous issues and derivations

Glossary of the main terms

Perhaps the most important area that remained beyond the scope of the original book consists of the moiré effects that occur between correlated random or aperiodic structures. These moiré effects are known as Glass patterns, after Leon Glass who described them in the late 1960s. However, this branch of the moiré theory remained for many years less widely known and less understood than its periodic or repetitive counterpart: Less widely known because moiré effects between aperiodic or random structures are less frequently encountered in everyday’s life, and less understood because these effects did not easily lend themselves to the same mathematical methods that so nicely explained the classical moiré effects between periodic or repetitive structures. Only recently has it been shown that in spite of their very different appearance and properties, moiré patterns between periodic or repetitive structures and Glass patterns between aperiodic or random structures are, in fact, particular cases of the same basic phenomenon, and all of them satisfy the same fundamental rules. These new research results have accumulated into a considerable volume that could no longer fit into an additional chapter in the original book. Rather, it became evident that this new material merits a new volume of its own, that should be entirely devoted to the aperiodic case.

And indeed, the study of the moiré phenomena which occur between random or aperiodic structures is not less interesting (and certainly not less surprising and fascinating) than the study of their periodic or repetitive counterparts. In a way, the diversity of the phenomena which occur in aperiodic superpositions is even more surprising than that of periodic cases, due to the unique interplay between the macroscopic and microscopic structures that appear in the superposition, and their very different properties. And yet, it turns out that using a suitable approach, earlier results from the periodic case can be extended to englobe the aperiodic case, too, although such an extension could a-priori seem hopeless. This applies to the main mathematical approaches already known from the periodic case, including the indicial equations method and the Fourier approach, both of which have been successfully extended to the aperiodic case. It should be noted, however, that just as in the periodic case, the indicial equations method as well as the Fourier approach are only applicable to the study of the macrostructures (the macroscopic phenomena in the superposition), but not to the study of the microstructures. And indeed, as it is clearly shown in the present volume, the study of the microstructures requires a different arsenal of mathematical tools. (Note that this is also true in the periodic case, and indeed, in the first volume, too, the microstructures had to be discussed in a separate chapter, Chapter 8 .)

The present book is intended to be a complementary, yet stand-alone companion to the original volume. Just like the first volume, it provides a full general purpose and application-independent exposition of the subject. It leads the reader through the various phenomena which occur in the superposition of correlated aperiodic layers, both in the image and in the spectral domain. And just like its predecessor, this volume favours a pictorial, intuitive approach which is supported by mathematics, and it includes a large number of figures and illustrative examples, some of which are visually striking and even spectacular.

Although this book has no pretentions to cover the entire moiré theory between aperiodic or random layers, it lays the foundations of this theory and gives for the first time an extended, comprehensive introduction to the subject. But it also goes in many aspects beyond the introductory level, and treats in depth some of the most interesting research results that have been recently obtained. Among other subjects, this book gives the full mathematical explanation of Glass patterns, describes their various properties, and explores their behaviour when the original layers undergo geometric transformations. It explains in detail the nature and the mathematical properties of the dot trajectories that often accompany Glass patterns in aperiodic screen superpositions, and it also investigates the one-dimensional counterparts of the classical Glass patterns that are generated in the superposition of correlated aperiodic line gratings. Most importantly, this book also provides the full Fourier-based approach, which allows us to explore Glass patterns and their intensity levels quantitatively and not only qualitatively. And finally, last but not the least, this book shows throughout the entire text and the accompanying figures the similarities as well as the differences between the moiré patterns which are obtained in periodic and in aperiodic cases, and it presents the theoretic results in a unified and coherent manner that clearly shows the relationship between the periodic and the aperiodic cases.

As already mentioned, this book is intended to be independent, as far as possible, from its predecessor. For this reason some fundamental notions from the original volume that are necessary for the comprehension of the new material are shortly reviewed here in Chapter 2, and some figures and results from the original volume are also reminded (or slightly adapted) in later chapters. Nevertheless, in order to avoid excessive repetitions, reference is occasionally made to some specific points in the original volume. On the other hand, it is interesting to note that in many ways the present volume also sheds new light on topics that were already explained in its predecessor, for example, by showing how these topics can be interpreted within a larger, unified framework that covers both periodic and aperiodic cases. And indeed, the wider point of view offered by the present volume often deepens our understanding of classical results from the periodic moiré theory.

This book is intended for students, scientists and engineers wishing to widen their knowledge of the moiré effect and its relationship with random or aperiodic structures. In particular, it will be very useful for people interested in the various moiré applications and in moiré-based technologies. The reader will find in this book not only a theoretical explanation of the moiré phenomena in question, but also practical recipes accompanied by many examples for the synthesis of aperiodic layers that give in their superposition moiré patterns having desired shapes, intensity profiles or microstructures. The prerequisite mathematical background is limited to an elementary familiarity with calculus and with the Fourier theory. But even occasional readers with no mathematical background will certainly enjoy the beauty of the effects illustrated throughout this book, and — it is our hope — may be tempted to learn more about their nature and their properties.Introduction

Background and basic notions

Glass patterns and fixed loci

Microstructures: dot trajectories and their morphology

Moiré phenomena between periodic or aperiodic screens

Glass patterns in the superposition of aperiodic line gratings

Quantitative analysis and synthesis of Glass patterns

A. Fixed point theorems for first- and second-order polynomial mappings

B. The various interpretations of a 2D transformation

C. The Jacobian of a 2D transformation and its significance

D. Direct and inverse spatial transformations

E. Convolution and cross correlation

F. The Fourier treatment of random images and of their superpositions

G. Integral transforms

H. Miscellaneous issues and derivations

Glossary of the main terms

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