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| Preface | xiii |
| 1. Introduction | 1 |
| 1.1 The moiré effect between aperiodic structures | 1 |
| 1.2 A brief historical background and main applications | 4 |
| 1.3 The scope of the present book | 5 |
| 1.4 Overview of the following chapters | 7 |
| 1.5 About the exercises and the moiré demonstration samples | 9 |
| 2. Background and basic notions | 11 |
| 2.1 Introduction | 11 |
| 2.2 Periodic, repetitive and aperiodic layers | 15 |
| 2.3 Superposition of aperiodic layers | 19 |
| 2.3.1 Glass patterns and correlation | 19 |
| 2.3.2 Stable vs. singular moiré-free superpositions | 24 |
| 2.3.3 Macrostructures and microstructures in the superposition | 26 |
| 2.4 The element distribution in the original layers and its influence | 28 |
| 2.5 Multilayer superpositions | 30 |
| Problems | 32 |
| 3. Glass patterns and fixed loci | 47 |
| 3.1 Introduction | 47 |
| 3.2 The fixed point theorem | 48 |
| 3.3 Behaviour of Glass patterns and periodic moirés under affine mappings | 51 |
| 3.3.1 Behaviour under layer rotations | 51 |
| 3.3.2 Behaviour under layer scalings | 53 |
| 3.3.3 Behaviour under layer shifts | 54 |
| 3.3.4 Behaviour under a general affine transformation | 59 |
| 3.4 Behaviour of Glass patterns under general layer transformations | 63 |
| 3.4.1 Examples with non-linear layer mappings | 65 |
| 3.5 Mappings in both layers; mutual fixed loci | 72 |
| 3.6 Synthesis of fixed loci in the superposition | 83 |
| 3.7 Almost fixed points | 87 |
| Problems | 96 |
| 4. Microstructures: dot trajectories and their morphology | 105 |
| 4.1 Introduction | 105 |
| 4.2 Morphology of the microstructures; dot trajectories | 106 |
| 4.3 Dot trajectories as solution curves of a system of differential equations | 107 |
| 4.4 Dot trajectories as a vector field | 109 |
| 4.4.1 The curve equations of the dot trajectories | 114 |
| 4.5 The dot trajectories when both layers undergo transformations | 124 |
| 4.6 Synthesis of dot trajectories | 134 |
| 4.7 Dot trajectories in periodic and in repetitive cases | 137 |
| 4.8 The microstructures under different superposition rules | 139 |
| 4.9 The visual interpretation of microstructures | 140 |
| Problems | 148 |
| 5. Moiré phenomena between periodic or aperiodic screens | 157 |
| 5.1 Introduction | 157 |
| 5.2 Brief review: moiré patterns, Glass patterns and dot trajectories | 158 |
| 5.3 A few detailed examples to illustrate the formal results | 161 |
| 5.4 Invariance properties of moiré patterns, Glass patterns and dot trajectories | 175 |
| Problems | 181 |
| 6. Glass patterns in the superposition of aperiodic line gratings | 187 |
| 6.1 Introduction | 187 |
| 6.2 Glass patterns in the superposition of straight line gratings | 188 |
| 6.2.1 Superposition of 1D vs. 2D aperiodic layers | 189 |
| 6.2.2 Superposition of periodic vs. aperiodic line gratings | 192 |
| 6.3 Mathematical derivations: generalization of the indicial equations method | 193 |
| 6.4 Examples of Glass patterns in the superposition of curved line gratings | 196 |
| 6.5 The effect of adding constraints to the original layers | 208 |
| 6.6 A first step towards the intensity profiles of Glass and moiré patterns | 214 |
| Problems | 221 |
| 7. Quantitative analysis and synthesis of Glass patterns | 225 |
| 7.1 Introduction | 225 |
| 7.2 Brief review of the Fourier approach in the periodic case | 226 |
| 7.2.1 Spectra of periodic and aperiodic layers | 228 |
| 7.2.2 Moiré effects in the superposition of periodic gratings | 230 |
| 7.2.3 Moiré effects in the superposition of periodic dot screens | 233 |
| 7.2.4 Shape of the intensity profile of the moiré pattern | 236 |
| 7.2.5 Orientation and size of the moiré cells | 238 |
| 7.3 Intensity profile of Glass patterns in the superposition of aperiodic gratings | 238 |
| 7.3.1 Superposition of correlated gratings | 240 |
| 7.3.2 Superposition of uncorrelated gratings | 246 |
| 7.4 Intensity profile of Glass patterns in the superposition of aperiofic screens | 248 |
| 7.4.1 Superposition of correlated screens | 249 |
| 7.4.2 Shape of the intensity profile of the Glass pattern | 252 |
| 7.4.3 Orientation and size of the Glass pattern | 253 |
| 7.4.4 Cases with several fixed points or with continuous fixed lines | 254 |
| 7.4.5 Superposition of uncorrelated screens | 258 |
| 7.4.6 Discussion | 258 |
| 7.5 Higher order moirés | 259 |
| 7.6 Intermediate, partly periodic cases | 260 |
| 7.7 Intermediate, partly correlated cases | 262 |
| 7.8 Glass patterns and cross correlation | 264 |
| Problems | 270 |
Appendices
| A. Fixed point theorems for first- and second-order polynomial mappings | 281 |
| A.1 Introduction | 281 |
| A.2 The fixed point theorem for linear or affine mappings | 281 |
| A.3 The fixed point theorem for second-order polynomial mappings | 284 |
| A.4 Mutual fixed points between two mappings; application to the moiré theory | 288 |
| B. The various interpretations of a 2D transformation | 289 |
| B.1 Introduction | 289 |
| B.2 Interpretation as two surfaces over the plane or as two sets of level lines | 289 |
| B.3 Interpretation as a mapping from the plane into itself | 290 |
| B.4 Interpretation as a domain transformation r(g(x,y)) | 294 |
| B.5 Interpretation as a coordinate change | 295 |
| B.6 Interpretation as a 2D vector field | 298 |
| B.7 Relationship between the different representations of g(x,y) | 301 |
| B.8 Remark on the local reflection of a 2D transformation | 306 |
| C. The Jacobian of a 2D transformation and its significance | 309 |
| C.1 Introduction | 309 |
| C.2 Geometric interpretation of the Jacobian | 309 |
| C.3 Properties of the transformation g(x,y) that can be deduced from its Jacobian | 311 |
| C.4 The local orientation properties of a transformation g(x,y) | 319 |
| C.5 Other properties of g(x,y) that can be deduced from its Jacobian matrix | 322 |
| D. Direct and inverse spatial transformations | 327 |
| D.1 Introduction | 327 |
| D.2 Background and basic notions | 327 |
| D.3 A deeper look into the domain and range planes of the mapping (u,v) = g(x,y) | 332 |
| D.4 2D transformations and their inverse | 336 |
|
D.4.1 The image of the standard Cartesian grid under the
transformations g and g-1 |
337 |
| D.4.2 The image of a general curve under the transformations g and g-1 | 340 |
| D.5 The active and passive interpretations of a transformation | 343 |
| D.6 Domain and range transformations of a function | 347 |
| D.6.1 The 1D case | 349 |
| D.6.2 The 2D case | 351 |
|
D.6.3 The effect of transformation g on objects and on their
characteristic functions |
355 |
| D.7 The relative point of view: object deformations vs. coordinate deformations | 356 |
| D.8 Examples | 357 |
| D.9 Other possible sources of confusion | 386 |
| D.9.1 Forward and backward mapping algorithms in digital imaging | 386 |
| D.9.2 Pre-multiplication and post-multiplication based notations | 389 |
| D.10 Implications to the moiré theory: issues related to the figures | 390 |
| D.11 Fixed points of a superposition in terms of direct or inverse transformations | 399 |
| D.11.1 Fixed points when only one layer is transformed | 399 |
| D.11.2 Fixed points when both layers undergo transformations | 401 |
| D.12 Useful approximations | 405 |
| E. Convolution and cross correlation | 411 |
| E.1 Introduction | 411 |
| E.2 Convolution | 411 |
| E.3 Cross correlation | 413 |
| E.4 Extension to more general cases | 415 |
| E.5 The Fourier transform of convolution and cross correlation | 416 |
| E.6 Methods for quantifying the correlation; similarity measures | 418 |
| F. The Fourier treatment of random images and of their superpositions | 421 |
| F.1 Introduction | 421 |
| F.2 Stochastic processes and their power spectra | 421 |
| F.3 Possible stochastic modelizations of random screens and gratings | 425 |
| F.3.1 Point processes | 425 |
| F.3.2 Shot noise | 426 |
| F.3.3 Random fields | 429 |
| F.4 Stochastic modelization of layer superpositions | 430 |
| F.5 Evaluation of the stochastic vs. deterministic approaches for our application | 430 |
| G. Integral transforms | 433 |
| G.1 Introduction | 433 |
| G.2 Fourier decomposition of periodic and aperiodic structures | 433 |
| G.3 Generalized Fourier decomposition of geometrically transformed structures | 434 |
| G.4 Integral transforms and their kernels | 435 |
| G.5 The use of generalized Fourier transforms in the moiré theory | 439 |
| H. Miscellaneous issues and derivations | 443 |
| H.1 Classification of the dot trajectories | 443 |
| H.1.1 Classification of the dot trajectories in the linear case | 444 |
| H.1.2 Classification of the dot trajectories in the non-linear case | 447 |
| H.2 The connection between the vector fields h1(x,y) and h2(x,y) in Sec. 4.5 | 451 |
| H.3 Hybrid (1,-1)-moiré effects whose moiré bands have 2D intensity profiles | 452 |
| I. Glossary of the main terms | 457 |
| I.1 About the glossary | 457 |
| I.2 Terms in the image domain | 457 |
| I.3 Terms in the spectral domain | 461 |
| I.4 Terms related to moiré | 463 |
| I.5 Terms related to light and colour | 466 |
| I.6 Miscellaneous terms | 467 |
| List of notations and symbols | 473 |
| List of abbreviations | 475 |
| References | 477 |
| Index | 485 |
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