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The starting point of the work on which this book is based was, indeed, in the research of moiré phenomena in the context of the colour printing process. The initial aim of this research was to understand the nature and the causes of the superposition moiré patterns between regular screens in order to find how to avoid, or at least minimize, their adverse effect on colour printing. This interesting research led us, after all, to a much more far-reaching mathematical understanding of the moiré phenomenon, whose interest stands in its own right, independently of any particular application. Based on these results, the present book offers a profound insight into the moiré phenomenon and a solid theoretical basis for its full understanding. In addition to the question of moiré minimization between regular screens, the book covers many interesting and important subjects such as the navigation in the moiré parameter space, the intensity profile forms of the moiré, its singular states, its periodic or almost-periodic properties, the phase of the superposed layers and of each of the eventual moirés, the relations between macro- and microstructures in the superposition, polychromatic moirés between colour layers, etc. All this is done in the most general way for any number of superposed layers having any desired forms (line-gratings, dot-screens with any dot shape, etc.). The main aim of this book is, therefore, to present all this material in the form of a single, unified and coherent text, starting from the basics of the theory, but also going in depth into recent research results and showing the new insight they offer in the understanding of the moiré phenomenon.
Fourier-based tools are but a natural choice when dealing with periodic phenomena; and, indeed, our approach is largely based on the Fourier theory. We consider each of the superposed layers as a function (reflectance or transmittance function) having values in the range between 0 and 1. We study the original layers, their superpositions, and their moiré effects by analyzing their properties both in the image domain and in the spectral, frequency domain using the Fourier theory. Further results are obtained by investigating the spectrum using concepts from geometry of numbers and linear algebra, and by interpreting the corresponding image-domain properties by means of the theories of periodic and almost-periodic functions. However, no prior knowledge of these fields of mathematics is assumed, and the required background is fully introduced in the text (in Chapter 5 and in Appendices A and B, respectively). The only prerequisite mathematical background is limited to undergraduate mathematics and an elementary familiarity with the Fourier theory (Fourier series, Fourier transforms, convolutions, Dirac impulses, etc.).
This book presents a comprehensive approach that provides a full explanation of the various phenomena which occur in the superposition, both in the image and in the spectral domains. This includes not only a quantitative and qualitative analysis of the moiré effect, but also the synthesis of moiré effects having any desired geometric forms and intensity profiles. In the first chapters we present the basic theory which covers the most fundamental case, namely: the superposition of monochrome, periodic layers. In later chapters of the book we extend the theory to the even more fascinating cases of polychromatic moirés and moirés between repetitive, non-periodic layers. Throughout the whole text we favour a pictorial, intuitive approach supported by mathematics, and the discussion is accompanied by a large number of figures and illustrative examples, some of which are visually striking and even spectacular.
This book is intended for students, scientists, and engineers wishing to widen their knowledge of the moiré effect; on the other hand it also offers a beautiful demonstration of the Fourier theory and its relationship with other fields of mathematics and science. Teachers and students of imaging science will find moiré phenomena to be an excellent didactic tool for illustrating the Fourier theory and its practical applications in one or more dimensions (Fourier transforms, Fourier series, convolutions, etc.). People interested in the various moiré applications and moiré-based technologies will find in this book a theoretical explanation of the moiré phenomenon and its properties. Readers interested in mathematics will find in the book a novel approach combining Fourier theory and geometry of numbers; physicists and crystallographers may be interested in the intricate relationship between the macro- and microstructures in the superposition and their relation to the theories of periodic and almost-periodic functions; and colour scientists and students will find in the polychromatic moirés a vivid demonstration of the additive and subtractive principles of colour theory. Finally, the occasional reader will enjoy the beauty of the effects demonstrated throughout this book, and - it is our hope - may be tempted to learn more about their nature and their properties.
The material in this book is based on the author's personal research at the Swiss Federal Institute of Technology of Lausanne (EPFL: Ecole Polytechnique Fédérale de Lausanne), and on his Ph.D. thesis (thesis No. 1341 entitled: Analysis of Moiré Patterns in Multi-Layer Superpositions) which won the best EPFL thesis award in 1995.
This work would have never been possible without the support and the excellent research environment provided by the EPFL. In particular, the author wishes to express his gratitude to Prof. Roger D. Hersch, the head of the Peripheral Systems Laboratory of the EPFL, for his encouragement throughout the different stages of this project. Many thanks are due to Dr. Patrick Emmel, a friend and colleague, for reviewing parts of the text and for his many helpful suggestions. And last but not least, many thanks go to Dr. Liesbeth Mol from Kluwer Academic Publishers for her helpfulness and availability throughout the publishing cycle.
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