Fourier Methods in Imaging (The Wiley-IS&T Series in Imaging Science and Technology)

Contents......Page 9
Series Editor’s Preface......Page 21
Preface......Page 25
1.1.1 The Imaging Chain......Page 27
1.2 The Three Imaging Tasks......Page 29
1.3.1 Ray Optics......Page 30
1.3.2 Wave Optics......Page 31
1.3.3 System Evaluation of Hubble Space Telescope......Page 33
1.4 Imaging Tasks in Medical Imaging......Page 34
1.4.1 Gamma-Ray Imaging......Page 35
1.4.2 Radiography......Page 37
1.4.3 Computed Tomographic Radiography......Page 39
2.1.1 Linearity......Page 41
2.2.1 Functions......Page 42
2.2.2 Functions with Continuous and Discrete Domains......Page 43
2.2.3 Continuous and Discrete Ranges......Page 45
2.2.4 Discrete Domain and Range – “Digitized” Functions......Page 46
2.2.5 Periodic, Aperiodic, and Harmonic Functions......Page 47
2.2.6 Symmetry Properties of Functions......Page 52
PROBLEMS......Page 53
3.1 Scalar Products......Page 55
3.1.1 Scalar Product of Distinct Vectors......Page 58
3.2.1 Simultaneous Evaluation of Multiple Scalar Products......Page 60
3.2.2 Matrix-Matrix Multiplication......Page 62
3.2.3 Square and Diagonal Matrices, Identity Matrix......Page 63
3.2.4 Matrix Transposes......Page 64
3.2.5 Matrix Inverses......Page 65
3.3 Vector Spaces......Page 67
3.3.1 Basis Vectors......Page 69
3.3.2 Vector Subspaces Associated with a System......Page 70
PROBLEMS......Page 74
4 Complex Numbers and Functions......Page 77
4.1.2 Sum and Difference of Two Complex Numbers......Page 78
4.2 Graphical Representation of Complex Numbers......Page 79
4.3 Complex Functions......Page 82
4.5 Argand Diagrams of Complex-Valued Functions......Page 88
PROBLEMS......Page 89
5.1 Vectors with Complex-Valued Components......Page 91
5.1.1 Inner Product......Page 92
5.2 Matrix Analogues of Shift-Invariant Systems......Page 93
5.2.1 Eigenvectors and Eigenvalues......Page 96
5.2.2 Projections onto Eigenvectors......Page 98
5.2.3 Diagonalization of a Circulant Matrix......Page 101
5.2.4 Matrix Operators for Shift-Invariant Systems......Page 106
5.3.1 Inverse Imaging Problem......Page 110
5.3.2 Solution of Inverse Problems via Diagonalization......Page 112
5.3.3 Matrix–Vector Formulation of System Analysis......Page 113
5.4.1 Inner Product of Continuous Functions......Page 114
5.4.2 Complete Sets of Basis Functions......Page 117
5.4.3 Orthonormal Basis Functions......Page 118
5.4.5 Eigenfunctions of Continuous Operators......Page 119
PROBLEMS......Page 120
6 1-D Special Functions......Page 123
6.1 Definitions of 1-D Special Functions......Page 124
6.1.2 Rectangle Function......Page 125
6.1.4 Signum Function......Page 127
6.1.5 Step Function......Page 128
6.1.6 Exponential Function......Page 130
6.1.7 Sinusoid......Page 131
6.1.8 SINC Function......Page 135
6.1.9 SINC2 Function......Page 137
6.1.10 Gamma Function......Page 138
6.1.11 Quadratic-Phase Sinusoid – “Chirp” Function......Page 141
6.1.12 Gaussian Function......Page 143
6.1.13 “SuperGaussian” Function......Page 145
6.1.14 Bessel Functions......Page 147
6.1.15 Lorentzian Function......Page 150
6.1.16 Thresholded Functions......Page 151
6.2 1-D Dirac Delta Function......Page 152
6.2.1 1-D Dirac Delta Function Raised to a Power......Page 157
6.2.2 Sifting Property of 1-D Dirac Delta Function......Page 158
6.2.3 Symmetric (Even) Pair of 1-D Dirac Delta Functions......Page 159
6.2.4 Antisymmetric (Odd) Pair of 1-D Dirac Delta Functions......Page 160
6.2.5 COMB Function......Page 161
6.2.6 Derivatives of 1-D Dirac Delta Function......Page 163
6.2.7 Dirac Delta Function with Functional Argument......Page 165
6.3 1-D Complex-Valued Special Functions......Page 168
6.3.2 Complex Quadratic-Phase Exponential Function......Page 169
6.3.3 “Superchirp” Function......Page 171
6.3.4 Complex-Valued Lorentzian Function......Page 173
6.4 1-D Stochastic Functions – Noise......Page 175
6.4.1 Moments of Probability Distributions......Page 177
6.4.2 Discrete Probability Laws......Page 178
6.4.3 Continuous Probability Distributions......Page 182
6.4.4 Signal-to-Noise Ratio......Page 186
6.4.7 Approximations to SNR......Page 187
6.5 Appendix A: Area of SINC[x] and SINC2[x]......Page 188
6.6 Appendix B: Series Solutions for Bessel Functions J0[x] and J1[x]......Page 192
PROBLEMS......Page 195
7.1 2-D Separable Functions......Page 197
7.1.2 Rotated Coordinates as Scalar Products......Page 198
7.2.1 2-D Constant Function......Page 200
7.2.2 Rectangle Function......Page 201
7.2.4 2-D Signum and STEP Functions......Page 202
7.2.6 SINC2 Function......Page 204
7.2.8 2-D Sinusoid......Page 206
7.3 2-D Dirac Delta Function and its Relatives......Page 208
7.3.1 2-D Dirac Delta Function in Cartesian Coordinates......Page 209
7.3.2 2-D Dirac Delta Function in Polar Coordinates......Page 210
7.3.3 2-D Separable COMB Function......Page 212
7.3.4 2-D Line Delta Function......Page 213
7.3.5 2-D “Cross” Function......Page 220
7.4 2-D Functions with Circular Symmetry......Page 221
7.4.1 Cylinder (Circle) Function......Page 222
7.4.3 Circularly Symmetric Bessel Function of Zero Order......Page 223
7.4.4 Besinc or Sombrero Function......Page 226
7.4.5 Circular Triangle Function......Page 227
7.4.6 Ring Delta Function......Page 228
7.5.1 Complex 2-D Sinusoid......Page 230
7.6 Special Functions of Three (or More) Variables......Page 231
PROBLEMS......Page 232
8 Linear Operators......Page 233
8.1 Linear Operators......Page 234
8.2 Shift-Invariant Operators......Page 239
8.3 Linear Shift-Invariant (LSI) Operators......Page 242
8.3.1 Linear Shift-Variant Operators......Page 247
8.4 Calculating Convolutions......Page 248
8.5 Properties of Convolutions......Page 249
8.5.2 Area of a Convolution......Page 251
8.6 Autocorrelation......Page 252
8.6.1 Autocorrelation of Stochastic Functions......Page 254
8.7 Crosscorrelation......Page 255
8.8 2-D LSI Operations......Page 258
8.8.1 Line-Spread and Edge-Spread Functions......Page 259
8.9 Crosscorrelations of 2-D Functions......Page 260
8.10 Autocorrelations of 2-D Functions......Page 261
PROBLEMS......Page 262
9.1 Transforms of Continuous-Domain Functions......Page 265
9.1.1 Example 1: Input and Reference Functions are Even Sinusoids......Page 268
9.1.2 Example 2: Even Sinusoid Input, Odd Sinusoid Reference......Page 271
9.1.3 Example 3: Odd Sinusoid Input, Even Sinusoid Reference......Page 272
9.1.4 Example 4: Odd Sinusoid Input and Reference......Page 273
9.2 Linear Combinations of Reference Functions......Page 276
9.2.2 Examples of the Hartley Transform......Page 277
9.2.3 Inverse of the Hartley Transform......Page 278
9.3 Complex-Valued Reference Functions......Page 280
9.4 Transforms of Complex-Valued Functions......Page 282
9.5 Fourier Analysis of Dirac Delta Functions......Page 285
9.6 Inverse Fourier Transform......Page 287
9.7 Fourier Transforms of 1-D Special Functions......Page 289
9.7.2 Fourier Transform of Rectangle......Page 290
9.7.3 Fourier Transforms of Sinusoids......Page 292
9.7.4 Fourier Transform of Signum and Step......Page 294
9.7.5 Fourier Transform of Exponential......Page 296
9.7.6 Fourier Transform of Gaussian......Page 301
9.7.7 Fourier Transforms of Chirp Functions......Page 302
9.7.8 Fourier Transform of COMB Function......Page 305
9.8 Theorems of the Fourier Transform......Page 306
9.8.3 Fourier Transform of a Fourier Transform......Page 307
9.8.5 Scaling Theorem......Page 310
9.8.6 Shift Theorem......Page 313
9.8.7 Filter Theorem......Page 315
9.8.8 Modulation Theorem......Page 321
9.8.9 Derivative Theorem......Page 323
9.8.10 Fourier Transform of Complex Conjugate......Page 324
9.8.11 Fourier Transform of Crosscorrelation......Page 325
9.8.13 Rayleigh’s Theorem......Page 328
9.8.14 Parseval’s Theorem......Page 330
9.8.15 Fourier Transform of Periodic Function......Page 332
9.8.16 Spectrum of Sampled Function......Page 333
9.8.18 Spectra of Stochastic Signals......Page 334
9.8.19 Effect of Nonlinear Operations of Spectra......Page 336
9.9 Appendix: Spectrum of Gaussian via Path Integral......Page 346
PROBLEMS......Page 347
10.1 2-D Fourier Transforms......Page 351
10.1.1 2-D Fourier Synthesis......Page 352
10.2 Spectra of Separable 2-D Functions......Page 353
10.2.2 Fourier Transform of δ[x, y]......Page 354
10.2.3 Fourier Transform of δ[x - x0, y - y0]......Page 356
10.2.6 Fourier Transform of GAUS[x, y]......Page 358
10.2.8 Theorems of Spectra of Separable Functions......Page 360
10.3 Theorems of 2-D Fourier Transforms......Page 361
10.3.3 2-D Shift Theorem......Page 362
10.3.4 2-D Filter Theorem......Page 363
10.3.5 2-D Derivative Theorem......Page 364
10.3.6 Spectra of Rotated 2-D Functions......Page 366
10.3.7 Transforms of 2-D Line Delta and Cross Functions......Page 367
PROBLEMS......Page 371
11.1 The Hankel Transform......Page 373
11.1.1 Hankel Transform of Dirac Delta Function......Page 377
11.2 Inverse Hankel Transform......Page 379
11.3.3 Central-Ordinate Theorem......Page 380
11.3.6 Derivative Theorem......Page 381
11.4.1 Hankel Transform of J0(2π r ρ 0)......Page 382
11.4.2 Hankel Transform of CYL(r)......Page 384
11.4.3 Hankel Transform of r-1......Page 386
11.4.4 Hankel Transforms from 2-D Fourier Transforms......Page 387
11.4.5 Hankel Transform of r2 GAUS(r)......Page 389
11.4.6 Hankel Transform of CTRI(r)......Page 390
11.5 Appendix: Derivations of Equations (11.12) and (11.14)......Page 391
PROBLEMS......Page 395
12.1 Line-Integral Projections onto Radial Axes......Page 397
12.1.1 Radon Transform of Dirac Delta Function......Page 403
12.1.2 Radon Transform of Arbitrary Function......Page 405
12.2.1 Cylinder Function CYL(r)......Page 406
12.2.2 Ring Delta Function δ(r - r0)......Page 408
12.2.3 Rectangle Function RECT[x, y]......Page 410
12.2.4 Corral Function COR[x, y]......Page 411
12.3.1 Radon Transform of a Superposition......Page 413
12.3.2 Radon Transform of Scaled Function......Page 414
12.3.4 Central-Slice Theorem......Page 415
12.3.5 Filter Theorem of the Radon Transform......Page 416
12.4 Inverse Radon Transform......Page 417
12.4.1 Recovery of Dirac Delta Function from Projections......Page 418
12.4.2 Summation of Projections over Azimuths......Page 424
12.5.1 Radial “Slices” of f [x, y]......Page 428
12.5.2 Central-Slice Transforms of Special Functions......Page 429
12.5.3 Inverse Central-Slice Transform......Page 435
12.6 Three Transforms of Four Functions......Page 436
12.7 Fourier and Radon Transforms of Images......Page 445
PROBLEMS......Page 446
13.1 Moment Theorem......Page 447
13.1.2 Second Moment – Moment of Inertia......Page 450
13.1.3 Central Moments – Variance......Page 451
13.1.4 Evaluation of 1-D Spectra from Moments......Page 453
13.1.5 Spectra of 1-D Superchirps via Moments......Page 457
13.1.6 2-D Moment Theorem......Page 459
13.1.7 Moments of Circularly Symmetric Functions......Page 461
13.2 1-D Spectra via Method of Stationary Phase......Page 462
13.2.1 Examples of Spectra via Stationary Phase......Page 466
13.3 Central-Limit Theorem......Page 478
13.4.1 Equivalent Width......Page 480
13.4.3 Variance as a Measure of Width......Page 481
PROBLEMS......Page 483
14 Discrete Systems, Sampling, and Quantization......Page 485
14.1 Ideal Sampling......Page 486
14.1.1 Ideal Sampling of 2-D Functions......Page 487
14.1.3 Is the Sampling Operation Shift Invariant?......Page 488
14.1.4 Aliasing Artifacts......Page 491
14.2 Ideal Sampling of Special Functions......Page 493
14.2.1 Ideal Sampling of δ[x] and COMB[x]......Page 496
14.3 Interpolation of Sampled Functions......Page 498
14.3.1 Examples of Interpolation......Page 504
14.4 Whittaker–Shannon Sampling Theorem......Page 505
14.5.1 Frequency Recovered from Aliased Samples......Page 506
14.5.2 “Unwrapping” the Phase of Sampled Functions......Page 508
14.6 “Prefiltering” to Prevent Aliasing......Page 509
14.6.1 Prefiltered Images Recovered from Samples......Page 510
14.6.2 Sampling and Reconstruction of Audio Signals......Page 511
14.7 Realistic Sampling......Page 512
14.8.2 Finite-Support Interpolators in Space Domain......Page 517
14.8.3 Realistic Frequency-Domain Interpolators......Page 521
14.9 Quantization......Page 526
14.9.1 Quantization “Noise”......Page 529
14.9.2 SNR of Quantization......Page 531
14.10 Discrete Convolution......Page 533
PROBLEMS......Page 535
15 Discrete Fourier Transforms......Page 537
15.1 Inverse of the Infinite-Support DFT......Page 539
15.2 DFT over Finite Interval......Page 540
15.2.1 Finite DFT of f [x] = 1[x]......Page 548
15.2.2 Scale Factor in DFT......Page 550
15.2.4 Summary of Finite DFT......Page 552
15.3 Fourier Series Derived from Fourier Transform......Page 553
15.4 Efficient Evaluation of the Finite DFT......Page 555
15.4.1 DFT of Two Samples – The “Butterfly”......Page 556
15.4.2 DFT of Three Samples......Page 557
15.4.4 DFT of Six Samples......Page 558
15.4.5 DFT of Eight Samples......Page 559
15.5.1 Computational Intensity......Page 560
15.5.2 “Centered” versus “Uncentered” Arrays......Page 562
15.5.3 Units of Measure in the Two Domains......Page 564
15.5.4 Ensuring Periodicity of Arrays – Data “Windows”......Page 565
15.5.5 A Garden of 1-D FFT Windows......Page 571
15.5.6 Undersampling and Aliasing......Page 577
15.5.8 Zero Padding......Page 580
15.5.9 Discrete Convolution and the Filter Theorem......Page 581
15.5.10 Discrete Transforms of Quantized Functions......Page 585
15.5.11 Parseval’s Theorem for DFT......Page 586
15.5.12 Scaling Theorem for Sampled Functions......Page 588
15.6 FFTs of 2-D Arrays......Page 589
15.6.1 Interpretation of 2-D FFTs......Page 590
15.7 Discrete Cosine Transform......Page 593
PROBLEMS......Page 597
16 Magnitude Filtering......Page 599
16.1.1 Magnitude Filters......Page 600
16.1.2 Phase (“Allpass”) Filters......Page 601
16.2 Eigenfunctions of Convolution......Page 602
16.3 Power Transmission of Filters......Page 603
16.4 Lowpass Filters......Page 605
16.4.3 1-D Uniform Averager......Page 607
16.4.4 2-D Lowpass Filters......Page 609
16.5.1 Ideal 1-D Highpass Filter......Page 611
16.5.2 1-D Differentiators......Page 612
16.5.3 2-D Differentiators......Page 613
16.5.4 High-Frequency Boost Filters – Image Sharpeners......Page 614
16.6 Bandpass Filters......Page 615
16.7 Fourier Transform as a Bandpass Filter......Page 620
16.8 Bandboost and Bandstop Filters......Page 622
16.9 Wavelet Transform......Page 625
16.9.1 Tiling of Frequency Domain with Orthogonal Wavelets......Page 626
PROBLEMS......Page 628
17 Allpass (Phase) Filters......Page 629
17.1 Power-Series Expansion for Allpass Filters......Page 630
17.2 Constant-Phase Allpass Filter......Page 631
17.3 Linear-Phase Allpass Filter......Page 632
17.4.1 Impulse Response and Transfer Function......Page 634
17.4.2 Scaling of Quadratic-Phase Transfer Function......Page 638
17.5 Allpass Filters with Higher-Order Phase......Page 641
17.5.1 Odd-Order Allpass Filters with n = 3......Page 644
17.6 Allpass Random-Phase Filter......Page 645
17.7 Relative Importance of Magnitude and Phase......Page 652
17.8 Imaging of Phase Objects......Page 654
17.9.1 1-D “M–C–M” Chirp Fourier Transform......Page 658
17.9.2 1-D “C–M–C” Chirp Fourier Transform......Page 660
17.9.3 M–C–M and C–M–C with Opposite-Sign Chirps......Page 663
17.9.5 Optical Correlator......Page 664
17.9.6 Optical Chirp Fourier Transformer......Page 667
PROBLEMS......Page 671
18 Magnitude–Phase Filters......Page 673
18.1.2 Differentiation......Page 674
18.1.3 Integration......Page 676
18.2 Fourier Transform of Ramp Function......Page 679
18.3 Causal Filters......Page 680
18.4 Damped Harmonic Oscillator......Page 684
18.6 Mixed Filter with Quadratic Phase......Page 687
PROBLEMS......Page 692
19.1 Linear Filters for the Imaging Tasks......Page 693
19.2 Deconvolution – “Inverse Filtering”......Page 695
19.2.1 Conditions for Exact Recovery via Inverse Filtering......Page 697
19.2.2 Inverse Filter for Uniform Averager......Page 698
19.2.3 Inverse Filter for Ideal Lowpass Filter......Page 701
19.2.4 Inverse Filter for Decaying Exponential......Page 704
19.3 Optimum Estimators for Signals in Noise......Page 705
19.3.1 Wiener Filter......Page 706
19.3.2 Wiener Filter Example......Page 714
19.3.3 Wiener–Helstrom Filter......Page 715
19.3.4 Wiener–Helstrom Filter Example......Page 719
19.3.5 Constrained Least-Squares Filter......Page 721
19.4 Detection of Known Signals – Matched Filter......Page 722
19.4.1 Inputs for Matched Filters......Page 727
19.5 Analogies of Inverse and Matched Filters......Page 729
19.5.1 Wiener and Wiener–Helstrom “Matched” Filter......Page 732
19.6 Approximations to Reciprocal Filters......Page 734
19.6.1 Small-Order Approximations of Reciprocal Filters......Page 737
19.6.2 Examples of Approximate Reciprocal Filters......Page 739
19.7 Inverse Filtering of Shift-Variant Blur......Page 745
PROBLEMS......Page 746
20 Filtering in Discrete Systems......Page 749
20.1.1 1-D Translation......Page 750
20.1.2 2-D Translation......Page 752
20.2.1 1-D Averagers......Page 754
20.2.2 2-D Averagers......Page 756
20.3.1 1-D Derivative......Page 757
20.3.2 2-D Derivative Operators......Page 758
20.3.4 Second Derivative......Page 760
20.3.5 2-D Second Derivative......Page 762
20.3.6 Laplacian......Page 763
20.4.1 1-D Sharpeners......Page 766
20.4.2 2-D Sharpening Operators......Page 768
20.5 2-D Gradient......Page 769
20.6 Pattern Matching......Page 770
20.6.1 Normalization of Contrast of Detected Features......Page 773
20.6.2 Amplified Discrete Matched Filters......Page 774
20.7.1 Derivative......Page 775
PROBLEMS......Page 777
21 Optical Imaging in Monochromatic Light......Page 779
21.1.1 Seemingly “Plausible” Models of Light in Imaging......Page 780
21.1.2 Imaging Systems Based on Ray “Selection” by Absorption......Page 784
21.1.3 Imaging System that Selects and Reflects Rays......Page 786
21.1.5 Model of Imaging Systems......Page 787
21.2.1 Wave Description of Light......Page 788
21.2.3 Propagation of Light......Page 791
21.2.4 Examples of Fresnel Diffraction......Page 798
21.3 Fraunhofer Diffraction......Page 809
21.3.1 Examples of Fraunhofer Diffraction......Page 811
21.4 Imaging System based on Fraunhofer Diffraction......Page 816
21.5 Transmissive Optical Elements......Page 818
21.5.1 Optical Elements with Constant or Linear Phase......Page 819
21.5.2 Lenses with Spherical Surfaces......Page 820
21.6.1 Single Positive Lens with z1 >> 0......Page 822
21.6.2 Single-Lens System, Fresnel Description of Both Propagations......Page 825
21.6.3 Amplitude Distribution at Image Point......Page 829
21.6.4 Shift-Invariant Description of Optical Imaging......Page 832
21.6.5 Examples of Single-Lens Imaging Systems......Page 833
21.7.1 Response of System at “Nonimage” Point......Page 837
21.7.2 Chirp Fourier Transform and Fraunhofer Diffraction......Page 842
PROBLEMS......Page 845
22.1.1 Optical Interference......Page 849
22.1.2 Spatial Coherence......Page 854
22.2 Polychromatic Source – Temporal Coherence......Page 864
22.3 Imaging in Incoherent Light......Page 868
22.4 System Function in Incoherent Light......Page 871
22.4.1 Incoherent MTF......Page 872
22.4.2 Comparison of Coherent and Incoherent Imaging......Page 873
PROBLEMS......Page 879
23 Holography......Page 881
23.1.1 Two Points: Object and Reference......Page 882
23.1.2 Multiple Object Points......Page 888
23.1.3 Fraunhofer Hologram of Extended Object......Page 890
23.1.4 Nonlinear Fraunhofer Hologram of Extended Object......Page 892
23.2 Holography in Fresnel Diffraction Region......Page 893
23.2.1 Object and Reference Sources in Same Plane......Page 894
23.2.3 Reconstruction of Real Image: z2 > 0......Page 898
23.2.4 Object and Reference Sources in Different Planes......Page 899
23.2.5 Reconstruction of Point Object......Page 904
23.2.6 Extended Object and Planar Reference Wave......Page 908
23.2.7 Interpretation of Fresnel Hologram as Lens......Page 909
23.3 Computer-Generated Holography......Page 911
23.3.1 CGH in the Fraunhofer Diffraction Region......Page 912
23.3.2 Examples of Cell CGHs......Page 916
23.3.3 2-D Lohmann Holograms......Page 920
23.3.4 Error-Diffused Quantization......Page 921
23.4 Matched Filtering with Cell-Type CGH......Page 924
23.5 Synthetic-Aperture Radar (SAR)......Page 926
23.5.1 Range Resolution......Page 930
23.5.2 Azimuthal Resolution......Page 932
23.5.3 SAR System Architecture......Page 933
PROBLEMS......Page 940
References......Page 943
Index......Page 947