E-book

# Fourier Optics and Computational Imaging

ISBN: 978-1-118-90036-9
352 pages
August 2015

## Description

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.

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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

19.2 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

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## Reviews

“This highly recommended text will also help readers understand how to integrate constraint optimization algorithms and stochastic methods into novel efficient algorithms for advanced imaging technology.”  (Optics & Photonics News, 18 December 2015)

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