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Fourier Optics and Computational Imaging
Beschreibung:
This book covers both the mathematics of inverse problems and optical systems design, and includes a review of the mathematical methods and Fourier optics. The first part of the book deals with the mathematical tools in detail with minimal assumption about prior knowledge on the part of the reader. The second part of the book discusses concepts in optics, particularly propagation of optical waves and coherence properties of optical fields that form the basis of the computational models used for image recovery. The third part provides a discussion of specific imaging systems that illustrate the power of the hybrid computational imaging model in enhancing imaging performance. A number of exercises are provided for readers to develop further understanding of computational imaging. While the focus of the book is largely on optical imaging systems, the key concepts are discussed in a fairly general manner so as to provide useful background for understanding the mechanisms of a diverse range of imaging modalities.
Preface 11
1 Introduction 13
1.1 Organization of the book 16
Part 1: Mathematical preliminaries
2 Fourier series and transform 21
2.1 Fourier Series 21
2.2 Gibbs phenomenon 23
2.3 Fourier transform as a limiting case of Fourier series 27
2.3.1 Fourier transform of the rectangle distribution 28
2.4 Sampling by averaging, distributions and delta function 30
2.5 Properties of delta function 32
2.6 Fourier transform of unit step and sign functions 33
2.7 Fourier transform of a train of delta functions 36
2.8 Fourier transform of a Gaussian 36
2.9 Fourier transform of chirp phase 37
2.10 Properties of Fourier transform 40
2.11 Fourier transform of the 2D circ function 42
2.12 Fourier slice theorem 43
2.13 Wigner distribution 45
3 Sampling Theorem 49
3.1 Poisson summation formula 50
3.2 Sampling theorem as a special case 51
3.3 Additional notes on the sampling formula 52
3.4 Sampling of carrier-frequency signals 53
3.5 Degrees of freedom in a signal: space bandwidth product 55
3.6 Slepian (prolate spheroidal) functions 56
3.6.1 Properties of matrix A(0) 59
3.7 Extrapolation of bandlimited functions 63
4 Operational introduction to Fast Fourier Transform 67
4.1 Definition 67
4.2 Usage of 2D Fast Fourier Transform for problems in Optics 69
5 Linear systems formalism and introduction to inverse problems in imaging 75
5.1 Space-invariant impulse response 77
5.2 Ill-posedness of inverse problems 78
5.3 Inverse filter 80
5.4 Wiener filter 82
6 Constrained optimization methods for image recovery 87
6.1 Image denoising 87
6.2 Image de-convolution by optimization 91
6.3 Blind image deconvolution 95
6.4 Compressive Imaging 97
6.4.1 Guidelines for sub-sampled data measurement and image recovery 99
6.4.2 System level implications of compressive imaging philosophy 103
6.5 Topics for further study 104
7 Random processes 107
7.1 Probability and random variables 107
7.1.1 Joint Probabilities 108
7.1.2 Baye's rule 108
7.1.3 Random Variables 109
7.1.4 Expectations and Moments 110
7.1.5 Characteristic function 112
7.1.6 Addition of two random variables 113
7.1.7 Transformation of random variables 113
7.1.8 Gaussian or Normal distribution 114
7.1.9 Central Limit Theorem 115
7.1.10 Gaussian moment theorem 116
7.2 Random Processes 117
7.2.1 Ergodic Process 118
7.2.2 Properties of auto-correlation function 119
7.2.3 Spectral Density: Wiener-Khintchine theorem 119
7.2.4 Orthogonal series representation of random processes 120
7.2.5 Complex Representation of random processes 121
7.2.6 Mandel's theorem on complex representation 123
Part 2: Concepts in optics
8 Geometrical Optics Essentials 127
8.1 Ray transfer matrix 127
8.2 Stops and pupils 130
9 Wave equation and introduction to diffraction of light 133
9.1 Introduction 133
9.2 Review of Maxwell equations 135
9.3 Integral theorem of Helmholtz and Kirchhoff 136
9.4 Diffraction from a planar screen 140
9.4.1 Kirchhoff Solution 141
9.4.2 Rayleigh-Sommerfeld Solution 141
10 The angular spectrum method 145
10.1 Angular spectrum method 145
11 Fresnel and Fraunhoffer diffraction 153
11.1 Fresnel diffraction 153
11.1.1 Computation of Fresnel diffraction patterns 155
11.1.2 Transport of Intensity Equation 156
11.1.3 Self imaging: Montgomery conditions and Talbott effect 160
11.1.4 Fractional Fourier transform 162
11.2 Fraunhoffer Diffraction 163
12 Coherence of light fields 167
12.1 Spatial and temporal coherence 167
12.1.1 Interference law 169
12.2 van Cittert and Zernike theorem 169
12.3 Space-frequency representation of the coherence function 171
12.4 Intensity interferometry: Hanbury Brown and Twiss effect 173
12.5 Photon counting formula 175
12.6 Speckle phenomenon 177
13 Polarization of light 183
13.1 The Jones matrix formalism 183
13.2 The QHQ geometric phase shifter 185
13.3 Degree of polarization 186
14 Analysis of optical systems 189
14.1 Transmission function for a thin lens 189
14.2 Fourier transforming property of thin lens 191
14.3 Canonical optical processor 193
14.4 Fourier plane filter examples 194
14.4.1 DC block or coronagraph 194
14.4.2 Zernike's phase contrast microscopy 195
14.4.3 Edge enhancement: vortex filter 197
14.4.4 Apodization filters 198
14.5 Frequency response of optical imaging systems: coherent and incoherent illumination 199
15 Imaging from information point of view 205
15.1 Eigenmodes of a canonical imaging system 206
15.1.1 Eigenfunctions and inverse problems 209
Part 3: Selected computational imaging systems
16 Digital Holography 217
16.1 Sampling considerations for recording of digital holograms 220
16.2 Complex field retrieval in hologram plane 221
16.2.1 Off-axis digital holography 222
16.2.2 Phase shifting digital holography 224
16.2.3 Optimization method for complex object wave recovery from digital holography 226
16.3 Digital holographic microscopy 229
16.4 Summary 230
17 Phase retrieval from intensity measurements 235
17.1 Gerchberg Saxton algorithm 237
17.2 Fienup's hybrid input-output algorithm 238
17.3 Phase retrieval with multiple intensity measurements 240
17.3.1 Phase retrieval with defocus diversity 240
17.3.2 Phase retrieval by spiral phase diversity 244
17.4 Gerchberg-Papoulis method for bandlimited extrapolation 247
18 Compact multi-lens imaging systems 253
18.1 Compact form factor computational camera 253
18.2 Lightfield cameras 256
18.2.1 The concept of lightfield 257
18.2.2 Recording the lightfield function with microlens array 259
19 PSF Engineering 267
19.1 Cubic phase mask 267
19.2 Log-asphere lens 271
19.3 Rotating point spread functions 273
20 Structural illumination imaging 277
20.1 Forward model and image reconstruction 279
21 Image reconstruction from projection data 285
21.1 X-ray projection data 286
21.2 Image reconstruction from projection data 287
22 Ghost Imaging 293
22.1 Schematic of a ghost imaging system 293
22.2 A signal processing viewpoint of ghost imaging 297
23 Appendix: Suggested Excercises 301
1 Introduction 13
1.1 Organization of the book 16
Part 1: Mathematical preliminaries
2 Fourier series and transform 21
2.1 Fourier Series 21
2.2 Gibbs phenomenon 23
2.3 Fourier transform as a limiting case of Fourier series 27
2.3.1 Fourier transform of the rectangle distribution 28
2.4 Sampling by averaging, distributions and delta function 30
2.5 Properties of delta function 32
2.6 Fourier transform of unit step and sign functions 33
2.7 Fourier transform of a train of delta functions 36
2.8 Fourier transform of a Gaussian 36
2.9 Fourier transform of chirp phase 37
2.10 Properties of Fourier transform 40
2.11 Fourier transform of the 2D circ function 42
2.12 Fourier slice theorem 43
2.13 Wigner distribution 45
3 Sampling Theorem 49
3.1 Poisson summation formula 50
3.2 Sampling theorem as a special case 51
3.3 Additional notes on the sampling formula 52
3.4 Sampling of carrier-frequency signals 53
3.5 Degrees of freedom in a signal: space bandwidth product 55
3.6 Slepian (prolate spheroidal) functions 56
3.6.1 Properties of matrix A(0) 59
3.7 Extrapolation of bandlimited functions 63
4 Operational introduction to Fast Fourier Transform 67
4.1 Definition 67
4.2 Usage of 2D Fast Fourier Transform for problems in Optics 69
5 Linear systems formalism and introduction to inverse problems in imaging 75
5.1 Space-invariant impulse response 77
5.2 Ill-posedness of inverse problems 78
5.3 Inverse filter 80
5.4 Wiener filter 82
6 Constrained optimization methods for image recovery 87
6.1 Image denoising 87
6.2 Image de-convolution by optimization 91
6.3 Blind image deconvolution 95
6.4 Compressive Imaging 97
6.4.1 Guidelines for sub-sampled data measurement and image recovery 99
6.4.2 System level implications of compressive imaging philosophy 103
6.5 Topics for further study 104
7 Random processes 107
7.1 Probability and random variables 107
7.1.1 Joint Probabilities 108
7.1.2 Baye's rule 108
7.1.3 Random Variables 109
7.1.4 Expectations and Moments 110
7.1.5 Characteristic function 112
7.1.6 Addition of two random variables 113
7.1.7 Transformation of random variables 113
7.1.8 Gaussian or Normal distribution 114
7.1.9 Central Limit Theorem 115
7.1.10 Gaussian moment theorem 116
7.2 Random Processes 117
7.2.1 Ergodic Process 118
7.2.2 Properties of auto-correlation function 119
7.2.3 Spectral Density: Wiener-Khintchine theorem 119
7.2.4 Orthogonal series representation of random processes 120
7.2.5 Complex Representation of random processes 121
7.2.6 Mandel's theorem on complex representation 123
Part 2: Concepts in optics
8 Geometrical Optics Essentials 127
8.1 Ray transfer matrix 127
8.2 Stops and pupils 130
9 Wave equation and introduction to diffraction of light 133
9.1 Introduction 133
9.2 Review of Maxwell equations 135
9.3 Integral theorem of Helmholtz and Kirchhoff 136
9.4 Diffraction from a planar screen 140
9.4.1 Kirchhoff Solution 141
9.4.2 Rayleigh-Sommerfeld Solution 141
10 The angular spectrum method 145
10.1 Angular spectrum method 145
11 Fresnel and Fraunhoffer diffraction 153
11.1 Fresnel diffraction 153
11.1.1 Computation of Fresnel diffraction patterns 155
11.1.2 Transport of Intensity Equation 156
11.1.3 Self imaging: Montgomery conditions and Talbott effect 160
11.1.4 Fractional Fourier transform 162
11.2 Fraunhoffer Diffraction 163
12 Coherence of light fields 167
12.1 Spatial and temporal coherence 167
12.1.1 Interference law 169
12.2 van Cittert and Zernike theorem 169
12.3 Space-frequency representation of the coherence function 171
12.4 Intensity interferometry: Hanbury Brown and Twiss effect 173
12.5 Photon counting formula 175
12.6 Speckle phenomenon 177
13 Polarization of light 183
13.1 The Jones matrix formalism 183
13.2 The QHQ geometric phase shifter 185
13.3 Degree of polarization 186
14 Analysis of optical systems 189
14.1 Transmission function for a thin lens 189
14.2 Fourier transforming property of thin lens 191
14.3 Canonical optical processor 193
14.4 Fourier plane filter examples 194
14.4.1 DC block or coronagraph 194
14.4.2 Zernike's phase contrast microscopy 195
14.4.3 Edge enhancement: vortex filter 197
14.4.4 Apodization filters 198
14.5 Frequency response of optical imaging systems: coherent and incoherent illumination 199
15 Imaging from information point of view 205
15.1 Eigenmodes of a canonical imaging system 206
15.1.1 Eigenfunctions and inverse problems 209
Part 3: Selected computational imaging systems
16 Digital Holography 217
16.1 Sampling considerations for recording of digital holograms 220
16.2 Complex field retrieval in hologram plane 221
16.2.1 Off-axis digital holography 222
16.2.2 Phase shifting digital holography 224
16.2.3 Optimization method for complex object wave recovery from digital holography 226
16.3 Digital holographic microscopy 229
16.4 Summary 230
17 Phase retrieval from intensity measurements 235
17.1 Gerchberg Saxton algorithm 237
17.2 Fienup's hybrid input-output algorithm 238
17.3 Phase retrieval with multiple intensity measurements 240
17.3.1 Phase retrieval with defocus diversity 240
17.3.2 Phase retrieval by spiral phase diversity 244
17.4 Gerchberg-Papoulis method for bandlimited extrapolation 247
18 Compact multi-lens imaging systems 253
18.1 Compact form factor computational camera 253
18.2 Lightfield cameras 256
18.2.1 The concept of lightfield 257
18.2.2 Recording the lightfield function with microlens array 259
19 PSF Engineering 267
19.1 Cubic phase mask 267
19.2 Log-asphere lens 271
19.3 Rotating point spread functions 273
20 Structural illumination imaging 277
20.1 Forward model and image reconstruction 279
21 Image reconstruction from projection data 285
21.1 X-ray projection data 286
21.2 Image reconstruction from projection data 287
22 Ghost Imaging 293
22.1 Schematic of a ghost imaging system 293
22.2 A signal processing viewpoint of ghost imaging 297
23 Appendix: Suggested Excercises 301