Fourier Optics and Computational Imaging

Kedar Khare, Department of Physics, IIT Delhi, India Dr. Khare is currently an Assistant Professor in the Department of Physics at IIT Delhi, India. He received his Ph.D. in Optics from the Institute of Optics, University of Rochester, USA in 2004. He has held positions as a Research Associate at the University of Rochester, USA; and as a Scientist at the General Electric Global Research, NY, USA where he received the Chief Technologist's award for work on Compressive Imaging (2010).?He has made several original contributions in this area which have been published in the form of journal papers and patents.

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