Introduction to orthogonal transforms_With applications in data processing and analysis

Wang, Ruye Introduction to orthogonal transforms_With applications in data processing and analysis (CUP,2012)(ISBN 9780521797252)(600dpi)(591p) ......Page 3
Copyright ......Page 4
Contents vi ......Page 6
Preface xii ......Page 12
Acknowledgments xx ......Page 20
Notation xxi ......Page 21
1.1 Continuous and discrete signals 1 ......Page 23
1.2 Unit step and nascent delta functions 4 ......Page 26
1.3 Relationship between complex exponentials and delta functions 7 ......Page 29
1.4 Attributes of signals 9 ......Page 31
1.5 Signal arithmetics and transformations 11 ......Page 33
1.6 Linear and time-invariant systems 15 ......Page 37
1.7 Signals through continuous LTI systems 17 ......Page 39
1.8 Signals through discrete LTI systems 21 ......Page 43
1.9 Continuous and discrete convolutions 24 ......Page 46
1.10 Homework problems 29 ......Page 51
2.1.1 Vector space 34 ......Page 56
2.1.2 Inner product space 36 ......Page 58
2.1.3 Bases of vector space 43 ......Page 65
2.1.4 Signal representation by orthogonal bases 47 ......Page 69
2.1.5 Signal representation by standard bases 52 ......Page 74
2.1.6 An example: the Fourier transforms 55 ......Page 77
2.2.1 Linear transformation 57 ......Page 79
2.2.2 Eigenvalue problems 59 ......Page 81
2.2.3 Eigenvectors of D2 as Fourier basis 61 ......Page 83
2.2.4 Unitary transformations 64 ......Page 86
2.2.5 Unitary transformations in N-D space 66 ......Page 88
2.3.1 Projection theorem and pseudo-inverse 70 ......Page 92
2.3.2 Signal approximation 76 ......Page 98
2.4.1 Frames 81 ......Page 103
2.4.2 Signal expansion by frames and Riesz bases 82 ......Page 105
2.4.3 Frames in finite-dimensional space 90 ......Page 112
2.5 Kernel function and Mercer’s theorem 93 ......Page 115
2.6 Summary 99 ......Page 121
2.7 Homework problems 101 ......Page 123
3.1.1 Formulation of the Fourier expansion 105 ......Page 127
3.1.2 Physical interpretation 107 ......Page 129
3.1.3 Properties of the Fourier series expansion 109 ......Page 131
3.1.4 The Fourier expansion of typical functions 111 ......Page 133
3.2.1 Formulation of the CTFT 119 ......Page 141
3.2.2 Relation to the Fourier expansion 124 ......Page 146
3.2.3 Properties of the Fourier transform 125 ......Page 147
3.2.4 Fourier spectra of typical functions 132 ......Page 154
3.2.5 The uncertainty principle 140 ......Page 162
3.3 Homework problems 142 ......Page 164
4.1 Discrete-time Fourier transform 146 ......Page 168
4.1.1 Fourier transform of discrete signals 146 ......Page 165
4.1.2 Properties of the DTFT 151 ......Page 173
4.1.3 DTFT of typical functions 157 ......Page 179
4.1.4 The sampling theorem 160 ......Page 182
4.1.5 Reconstruction by interpolation 170 ......Page 192
4.2.1 Formulation of the DFT 173 ......Page 195
4.2.2 Array representation 179 ......Page 201
4.2.3 Properties of the DFT 183 ......Page 205
4.2.4 Four different forms of the Fourier transform 192 ......Page 214
4.2.5 DFT computation and fast Fourier transform 196 ......Page 218
4.3.1 Two-dimensional signals and their spectra 201 ......Page 223
4.3.2 Fourier transform of typical 2-D functions 204 ......Page 226
4.3.3 Four forms of 2-D Fourier transform 207 ......Page 229
4.3.4 Computation of the 2-D DFT 209 ......Page 231
4.4 Homework problems 215 ......Page 237
5.1 LTI systems in time and frequency domains 220 ......Page 242
5.2 Solving differential and difference equations 225 ......Page 247
5.3 Magnitude and phase filtering 232 ......Page 254
5.4 Implementation of 1-D filtering 238 ......Page 260
5.5 Implementation of 2-D filtering 249 ......Page 271
5.6 Hilbert transform and analytic signals 256 ......Page 278
5.7 Radon transform and image restoration from projections 261 ......Page 283
5.8 Orthogonal frequency-division modulation (OFDM) 269 ......Page 291
5.9 Homework problems 271 ......Page 293
6.1.1 From Fourier transform to Laplace transform 277 ......Page 299
6.1.2 The region of convergence 280 ......Page 302
6.1.3 Properties of the Laplace transform 281 ......Page 303
6.1.4 The Laplace transform of typical signals 284 ......Page 306
6.1.5 Analysis of continuous LTI systems by Laplace transform 286 ......Page 308
6.1.6 First-order system 292 ......Page 314
6.1.7 Second-order system 295 ......Page 317
6.1.8 The unilateral Laplace transform 307 ......Page 329
6.2.1 From Fourier transform to z-transform 311 ......Page 333
6.2.2 Region of convergence 314 ......Page 336
6.2.3 Properties of the z-transform 316 ......Page 338
6.2.4 The z-transform of typical signals 321 ......Page 343
6.2.5 Analysis of discrete LTI systems by z-transform 322 ......Page 344
6.2.6 First- and second-order systems 327 ......Page 349
6.2.7 The unilateral z-transform 332 ......Page 354
6.3 Homework problems 335 ......Page 357
7.1.1 Continuous Hartley transform 339 ......Page 361
7.1.2 Properties of the Hartley transform 341 ......Page 363
7.1.3 Hartley transform of typical signals 343 ......Page 365
7.1.4 Discrete Hartley transform 345 ......Page 367
7.1.5 The 2-D Hartley transform 348 ......Page 370
7.2.1 The continuous cosine and sine transforms 353 ......Page 375
7.2.2 From DFT to DCT and DST 355 ......Page 377
7.2.3 Matrix forms of DCT and DST 360 ......Page 382
7.2.4 Fast algorithms for the DCT and DST 366 ......Page 388
7.2.5 DCT and DST filtering 370 ......Page 392
7.2.6 The 2-D DCT and DST 373 ......Page 395
7.3 Homework problems 377 ......Page 399
8.1.1 Hadamard matrix 379 ......Page 401
8.1.2 Hadamard-ordered Walsh-Hadamard transform (WHTh) 381 ......Page 403
8.1.3 Fast Walsh-Hadamard transform algorithm 382 ......Page 404
8.1.4 Sequency-ordered Walsh-Hadamard matrix (WHTw) 384 ......Page 406
8.1.5 Fast Walsh-Hadamard transform (sequency ordered) 386 ......Page 408
8.2.1 Slant matrix 392 ......Page 414
8.2.2 Slant transform and its fast algorithm 395 ......Page 417
8.3.1 Continuous Haar transform 398 ......Page 420
8.3.2 Discrete Haar transform 400 ......Page 422
8.3.3 Computation of the discrete Haar transform 403 ......Page 425
8.3.4 Filter bank implementation 405 ......Page 427
8.4 Two-dimensional transforms 408 ......Page 430
8.5 Homework problems 411 ......Page 433
9.1.1 Signals as stochastic processes 412 ......Page 434
9.1.2 Signal correlation 415 ......Page 437
9.2.1 Continuous KLT 417 ......Page 439
9.2.2 Discrete KLT 418 ......Page 440
9.2.3 Optimalities of the KLT 419 ......Page 441
9.2.4 Geometric interpretation of the KLT 423 ......Page 445
9.2.5 Principal component analysis (PCA) 426 ......Page 448
9.2.6 Comparison with other orthogonal transforms 427 ......Page 449
9.2.7 Approximation of the KLT by the DCT 432 ......Page 454
9.3.1 Image processing and analysis 438 ......Page 460
9.3.2 Feature extraction for pattern classification 444 ......Page 466
9.4.1 Singular value decomposition 449 ......Page 471
9.4.2 Application in image compression 454 ......Page 479
9.5 Homework problems 456 ......Page 478
10.1.1 Short-time Fourier transform and Gabor transform 461 ......Page 483
10.1.2 The Heisenberg uncertainty 462 ......Page 484
10.2.1 Mother and daughter wavelets 464 ......Page 486
10.2.2 The forward and inverse wavelet transforms 466 ......Page 488
10.3 Properties of the CTWT 468 ......Page 490
10.4 Typical mother wavelet functions 471 ......Page 493
10.5.1 Discretization of wavelet functions 474 ......Page 496
10.5.2 The forward and inverse transform 476 ......Page 498
10.5.3 A fast inverse transform algorithm 478 ......Page 500
10.6 Wavelet transform computation 481 ......Page 503
10.7 Filtering based on wavelet transform 484 ......Page 506
10.8 Homework problems 490 ......Page 512
11.1.1 Scale spaces 492 ......Page 514
11.1.2 Wavelet spaces 498 ......Page 520
11.1.3 Properties of the scaling and wavelet filters 501 ......Page 523
11.1.4 Relationship between scaling and wavelet filters 504 ......Page 526
11.1.5 Wavelet series expansion 506 ......Page 528
11.1.6 Construction of scaling and wavelet functions 508 ......Page 530
11.2.1 Discrete wavelet transform (DWT) 518 ......Page 540
11.2.2 Fast wavelet transform (FWT) 521 ......Page 543
11.3.1 Two-channel filter bank and inverse DWT 523 ......Page 545
11.3.2 Two-dimensional DWT 530 ......Page 552
11.4 Applications in filtering and compression 535 ......Page 557
11.5 Homework problems 542 ......Page 564
Appendices 545 ......Page 567
A.l Basic definitions 546 ......Page 568
A.2 Eigenvalues and eigenvectors 551 ......Page 573
A.3 Hermitian matrix and unitary matrix 552 ......Page 574
A.5 Vector and matrix differentiation 554 ......Page 576
B.l Random variables 556 ......Page 578
B.2 Multivariate random variables 558 ......Page 580
B.3 Stochastic models 562 ......Page 584
Bibliography 565 ......Page 587
Index 566 ......Page 588
cover......Page 1
back cover 569 ......Page 591