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Preface | xv |
Colour Plates | xix |
1. Introduction | 1 |
1.1 The moiré effect | 1 |
1.2 A brief historical background | 2 |
1.3 The scope of the present book | 3 |
1.4 Overview of the following chapters | 4 |
1.5 About the exercises and the moiré demonstration samples | 7 |
2. Background and basic notions | 9 |
2.1 Introduction | 9 |
2.2 The spectral approach; images and their spectra | 10 |
2.3 Superposition of two cosinusoidal gratings | 15 |
2.4 Superposition of three or more cosinusoidal gratings | 18 |
2.5 Binary square waves and their spectra | 21 |
2.6 Superposition of binary gratings; higher order moirés | 23 |
2.7 The impulse indexing notation | 30 |
2.8 The notational system for superposition moirés | 33 |
2.9 Singular moiré states; stable vs. unstable moiré-free superpositions | 35 |
2.10 The intensity profile of the moiré and its perceptual contrast | 38 |
2.11 Square grids and their superpositions | 40 |
2.12 Dot-screens and their superpositions | 44 |
2.13 Sampling moirés; moirés as aliasing phenomena | 48 |
2.14 Advantages of the spectral approach | 51 |
Problems | 52 |
3. Moiré minimization | 59 |
3.1 Introduction | 59 |
3.2 Colour separation and halftoning | 60 |
3.3 The challenge of moiré minimization in colour printing | 62 |
3.4 Navigation in the moiré parameter space | 64 |
3.4.1 The case of two superposed screens | 65 |
3.4.2 The case of three superposed screens | 68 |
3.5 Finding moiré-free screen combinations for colour printing | 71 |
3.6 Results and discussion | 75 |
Problems | 77 |
4. The moiré profile form and intensity levels | 81 |
4.1 Introduction | 81 |
4.2 Extraction of the profile of a moiré between superposed line-gratings | 82 |
4.3 Extension of the moiré extraction to the 2D case of superposed screens | 89 |
4.4 The special case of the (1,0,-1,0)-moiré | 96 |
4.4.1 The forms of the moiré cells | 97 |
4.4.2 The orientation of the moiré cells | 101 |
4.5 The case of more complex and higher order moirés | 102 |
Problems | 103 |
5. The algebraic foundation of the spectrum properties | 109 |
5.1 Introduction | 109 |
5.2 The support of a spectrum; lattices and modules | 109 |
5.2.1 Lattices and modules in Rn | 110 |
5.2.2 Application to the frequency spectrum | 113 |
5.3 The mapping between the impulse indices and their geometric locations | 114 |
5.4 A short reminder from linear algebra | 115 |
5.4.1 The image and the kernel of a linear transformation | 115 |
5.4.2 Partition of a vector space into equivalence classes | 116 |
5.4.3 The partition of V into equivalence classes induced by Phi | 117 |
5.4.4 The application of these results to our continuous case | 118 |
5.5 The discrete mapping Psi vs. the continuous mapping Phi | 118 |
5.6 The algebraic interpretation of the impulse locations
in the spectrum support |
121 |
5.6.1 The global spectrum support | 121 |
5.6.2 The individual impulse-clusters | 123 |
5.6.3 The spread-out clusters slightly off the singular state | 125 |
5.7 Examples | 126 |
5.8 Concluding remarks | 143 |
Problems | 146 |
6. Fourier-based interpretation of the algebraic spectrum properties | 149 |
6.1 Introduction | 149 |
6.2 Image domain interpretation of the algebraic structure
of the spectrum support |
149 |
6.3 Image domain interpretation of the impulse-clusters in the spectrum | 151 |
6.4 The amplitude of the collapsed impulse-clusters in a sigular state | 152 |
6.5 The exponential Fourier expression for two-grating superpositions | 153 |
6.6 Two-grating superpositions and their singular states | 155 |
6.6.1 Two gratings with identical frequencies | 155 |
6.6.2 Two gratings with different frequencies | 157 |
6.7 Two-screen superpositions and their singular states | 158 |
6.8 The general superposition of m layers and its singular states | 161 |
Problems | 163 |
7. The superposition phase | 165 |
7.1 Introduction | 165 |
7.2 The phase of a periodic function | 166 |
7.3 The phase terminology for periodic functions in the 1D case | 168 |
7.4 The phase terminology for 1-fold periodic functions in the 2D case | 169 |
7.5 The phase terminology for the general 2D case: 2-fold periodic functions | 171 |
7.5.1 Using the period-vector notation | 172 |
7.5.2 Using the step-vector notation | 173 |
7.6 Moiré phases in the superposition of periodic layers | 176 |
7.7 The influence of layer shifts on the overall superposition | 179 |
Problems | 186 |
8. Macro- and microstructures in the superposition | 191 |
8.1 Introduction | 191 |
8.2 Rosettes in singular states | 194 |
8.2.1 Rosettes in periodic singular states | 194 |
8.2.2 Rosettes in almost-periodic singular states | 195 |
8.3 The influence of layer shifts on the rosettes in singular states | 198 |
8.4 The microstructure slightly off the singular state; the relationship
between macro- and microstructures |
200 |
8.5 The microstructure in stable moiré-free superpositions | 201 |
8.6 Rational vs. irrational screen superpositions; rational approximants | 204 |
8.7 Algebraic formalization | 210 |
8.8 The microstructure of the conventional 3-screen superposition | 218 |
8.9 Variance or invariance of the microstructure under layer shifts | 223 |
8.10 Period-coordinates and period-shifts in the Fourier decomposition | 226 |
Problems | 231 |
9. Polychromatic moiré effects | 233 |
9.1 Introduction | 233 |
9.2 Some basic notions from colour theory | 234 |
9.2.1 Physical aspects of colour | 234 |
9.2.2 Physiological aspects of colour | 235 |
9.3 Extension of the spectral approach to the polychromatic case | 236 |
9.3.1 The representation of images and image superpositions | 236 |
9.3.2 The influence of the human visual system | 240 |
9.3.3 The Fourier-spectrum convolution and the superposition moirés | 241 |
9.4 Extraction of the moiré intensity profiles | 241 |
9.5 The (1,-1)-moiré between two colour line-gratings | 242 |
9.6 The (1,0,-1,0)-moiré between two colour dot-screens | 245 |
9.7 The case of more complex and higher-order moirés | 246 |
Problems | 246 |
10. Moirés between repetitive, non-periodic layers | 249 |
10.1 Introduction | 249 |
10.2 Repetitive, non-periodic layers | 250 |
10.3 The influence of a coordinate change on the spectrum | 258 |
10.4 Curvilinear cosinusoidal gratings and their different types of spectra | 264 |
10.4.1 Gradual transitions between cosinusoidal gratings
of different types |
268 |
10.5 The Fourier decomposition of curved, repetitive structures | 272 |
10.5.1 The Fourier decomposition of curvilinear gratings | 272 |
10.5.2 The Fourier decomposition of curved line-grids and dot-screens | 274 |
10.6 The spectrum of curved, repetitive structures | 275 |
10.6.1 The spectrum of curvilinear gratings | 275 |
10.6.2 The spectrum of curved line-grids and dot-screens | 278 |
10.7 The superposition of curved, repetitive layers | 279 |
10.7.1 Moirés in the superposition of curved, repetitive layers | 279 |
10.7.2 Image domain vs. spectral domain investigation of
the superposition |
282 |
10.7.3 The superposition of a parabolic grating and a periodic
straight grating |
283 |
10.7.4 The superposition of two parabolic gratings | 290 |
10.7.5 The superposition of a circular grating and a periodic
straight grating |
297 |
10.7.6 The superposition of two circular gratings | 306 |
10.7.7 The superposition of a zone grating and a periodic
straight grating |
311 |
10.7.8 The superposition of two circular zone gratings | 319 |
10.8 Periodic moirés in the superposition of non-periodic layers | 323 |
10.9 Moiré analysis and synthesis in the superposition of curved,
repetitive layers |
329 |
10.9.1 The case of curvilinear gratings | 329 |
10.9.2 The case of curved dot-screens | 337 |
10.10 Local frequencies and singular states in curved, repetitive cases | 343 |
10.11 Moirés in the superposition of screen gradations | 347 |
10.12 Concluding remarks | 348 |
Problems | 349 |
11. Other possible approaches for moiré analysis | 353 |
11.1 Introduction | 353 |
11.2 The indicial equations method | 353 |
11.2.1 Evaluation of the method | 358 |
11.2.2 Comparison with the spectral approach | 359 |
11.3 Approximation using the first harmonic | 360 |
11.3.1 Evaluation of the method | 362 |
11.4 The local frequency method | 363 |
11.4.1 Evaluation of the method | 368 |
11.4.2 Comparison with the spectral approach | 369 |
11.5 Concluding remarks | 369 |
Problems | 370 |
Appendices
A. Periodic functions and their spectra | 375 |
A.1 Introduction | 375 |
A.2 Periodic functions, their Fourier series and their spectra in the 1D case | 375 |
A.3 Periodic functions, their Fourier series and their spectra in the 2D case | 378 |
A.3.1 1-fold periodic functions in the x or y direction | 378 |
A.3.2 2-fold periodic functions in the x and y directions | 378 |
A.3.3 1-fold periodic functions in an arbitrary direction | 380 |
A.3.4 2-fold periodic functions in arbitrary directions
(skew-periodic functions) |
381 |
A.4 The period-lattice and the frequency-lattice (= spectrum support) | 386 |
A.5 The matrix notation, its appeal, and its limitations for our needs | 389 |
A.6 The period-vectors Pi vs. the step-vectors Ti | 392 |
B. Almost-periodic functions and their spectra | 395 |
B.1 Introduction | 395 |
B.2 A simple illustrative example | 395 |
B.3 Definitions and main properties | 396 |
B.4 The spectrum of almost-periodic functions | 399 |
B.5 The different classes of almost-periodic functions and their spectra | 401 |
B.6 Characterization of functions according to their spectrum support | 404 |
B.7 Almost-periodic functions in two variables | 406 |
C. Miscellaneous issues and derivations | 409 |
C.1 Derivation of the classical moiré formula (2.9) of Sec. 2.4 | 409 |
C.2 Derivation of the first part of Proposition 2.1 of Sec. 2.5 | 410 |
C.3 Invariance of the impulse amplitudes under rotations and x,y scalings | 411 |
C.3.1 Invariance of the 2D Fourier transform under rotations | 411 |
C.3.2 Invariance of the impulse amplitudes under x, y scalings | 411 |
C.4 Shift and phase | 412 |
C.4.1 The shift theorem | 412 |
C.4.2 The particular case of periodic functions | 414 |
C.4.3 The phase of a periodic function: the phi1 and the phi2 notations | 415 |
C.5 The function Rc(u) converges to delta(u) as alpha-->0 | 417 |
C.6 The 2D spectrum of a cosinusoidal zone grating | 418 |
C.7 The convolution of two orthogonal line-impulses | 419 |
C.8 The compound line-impulse of the singular (k1,k2)-line-impulse cluster | 420 |
C.9 The 1D Fourier transform of the chirp cos(ax2 + b) | 423 |
C.10 The 2D Fourier transform of the 2D chirp cos(ax2 + by2 + c) | 424 |
C.11 The spectrum of screen gradations | 425 |
C.12 Convergence issues related to Fourier series | 429 |
C.12.1 On the convergence of Fourier series | 429 |
C.12.2 Multiplication of infinite series | 430 |
C.13 Moiré effects in image reproduction | 432 |
D. Glossary of the main terms | 433 |
D.1 About the glossary | 433 |
D.2 Terms in the image domain | 434 |
D.3 Terms in the spectral domain | 438 |
D.4 Terms related to moiré | 442 |
D.5 Terms related to light and colour | 444 |
D.6 Miscellaneous terms | 446 |
List of notations and symbols | 449 |
List of abbreviations | 453 |
References | 455 |
Index | 465 |
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