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 111 Introduction 131.1 Organization of the book 16Part 1: Mathematical preliminaries2 Fourier series and transform 212.1 Fourier Series 212.2 Gibbs phenomenon 232.3 Fourier transform as a limiting case of Fourier series 272.3.1 Fourier transform of the rectangle distribution 282.4 Sampling by averaging, distributions and delta function 302.5 Properties of delta function 322.6 Fourier transform of unit step and sign functions 332.7 Fourier transform of a train of delta functions 362.8 Fourier transform of a Gaussian 362.9 Fourier transform of chirp phase 372.10 Properties of Fourier transform 402.11 Fourier transform of the 2D circ function 422.12 Fourier slice theorem 432.13 Wigner distribution 453 Sampling Theorem 493.1 Poisson summation formula 503.2 Sampling theorem as a special case 513.3 Additional notes on the sampling formula 523.4 Sampling of carrier-frequency signals 533.5 Degrees of freedom in a signal: space bandwidth product 553.6 Slepian (prolate spheroidal) functions 563.6.1 Properties of matrix A(0) 593.7 Extrapolation of bandlimited functions 634 Operational introduction to Fast Fourier Transform 674.1 Definition 674.2 Usage of 2D Fast Fourier Transform for problems in Optics 695 Linear systems formalism and introduction to inverse problems in imaging 755.1 Space-invariant impulse response 775.2 Ill-posedness of inverse problems 785.3 Inverse filter 805.4 Wiener filter 826 Constrained optimization methods for image recovery 876.1 Image denoising 876.2 Image de-convolution by optimization 916.3 Blind image deconvolution 956.4 Compressive Imaging 976.4.1 Guidelines for sub-sampled data measurement and image recovery 996.4.2 System level implications of compressive imaging philosophy 1036.5 Topics for further study 1047 Random processes 1077.1 Probability and random variables 1077.1.1 Joint Probabilities 1087.1.2 Baye's rule 1087.1.3 Random Variables 1097.1.4 Expectations and Moments 1107.1.5 Characteristic function 1127.1.6 Addition of two random variables 1137.1.7 Transformation of random variables 1137.1.8 Gaussian or Normal distribution 1147.1.9 Central Limit Theorem 1157.1.10 Gaussian moment theorem 1167.2 Random Processes 1177.2.1 Ergodic Process 1187.2.2 Properties of auto-correlation function 1197.2.3 Spectral Density: Wiener-Khintchine theorem 1197.2.4 Orthogonal series representation of random processes 1207.2.5 Complex Representation of random processes 1217.2.6 Mandel's theorem on complex representation 123Part 2: Concepts in optics8 Geometrical Optics Essentials 1278.1 Ray transfer matrix 1278.2 Stops and pupils 1309 Wave equation and introduction to diffraction of light 1339.1 Introduction 1339.2 Review of Maxwell equations 1359.3 Integral theorem of Helmholtz and Kirchhoff 1369.4 Diffraction from a planar screen 1409.4.1 Kirchhoff Solution 1419.4.2 Rayleigh-Sommerfeld Solution 14110 The angular spectrum method 14510.1 Angular spectrum method 14511 Fresnel and Fraunhoffer diffraction 15311.1 Fresnel diffraction 15311.1.1 Computation of Fresnel diffraction patterns 15511.1.2 Transport of Intensity Equation 15611.1.3 Self imaging: Montgomery conditions and Talbott effect 16011.1.4 Fractional Fourier transform 16211.2 Fraunhoffer Diffraction 16312 Coherence of light fields 16712.1 Spatial and temporal coherence 16712.1.1 Interference law 16912.2 van Cittert and Zernike theorem 16912.3 Space-frequency representation of the coherence function 17112.4 Intensity interferometry: Hanbury Brown and Twiss effect 17312.5 Photon counting formula 17512.6 Speckle phenomenon 17713 Polarization of light 18313.1 The Jones matrix formalism 18313.2 The QHQ geometric phase shifter 18513.3 Degree of polarization 18614 Analysis of optical systems 18914.1 Transmission function for a thin lens 18914.2 Fourier transforming property of thin lens 19114.3 Canonical optical processor 19314.4 Fourier plane filter examples 19414.4.1 DC block or coronagraph 19414.4.2 Zernike's phase contrast microscopy 19514.4.3 Edge enhancement: vortex filter 19714.4.4 Apodization filters 19814.5 Frequency response of optical imaging systems: coherent and incoherent illumination 19915 Imaging from information point of view 20515.1 Eigenmodes of a canonical imaging system 20615.1.1 Eigenfunctions and inverse problems 209Part 3: Selected computational imaging systems16 Digital Holography 21716.1 Sampling considerations for recording of digital holograms 22016.2 Complex field retrieval in hologram plane 22116.2.1 Off-axis digital holography 22216.2.2 Phase shifting digital holography 22416.2.3 Optimization method for complex object wave recovery from digital holography 22616.3 Digital holographic microscopy 22916.4 Summary 23017 Phase retrieval from intensity measurements 23517.1 Gerchberg Saxton algorithm 23717.2 Fienup's hybrid input-output algorithm 23817.3 Phase retrieval with multiple intensity measurements 24017.3.1 Phase retrieval with defocus diversity 24017.3.2 Phase retrieval by spiral phase diversity 24417.4 Gerchberg-Papoulis method for bandlimited extrapolation 24718 Compact multi-lens imaging systems 25318.1 Compact form factor computational camera 25318.2 Lightfield cameras 25618.2.1 The concept of lightfield 25718.2.2 Recording the lightfield function with microlens array 25919 PSF Engineering 26719.1 Cubic phase mask 26719.2 Log-asphere lens 27119.3 Rotating point spread functions 27320 Structural illumination imaging 27720.1 Forward model and image reconstruction 27921 Image reconstruction from projection data 28521.1 X-ray projection data 28621.2 Image reconstruction from projection data 28722 Ghost Imaging 29322.1 Schematic of a ghost imaging system 29322.2 A signal processing viewpoint of ghost imaging 29723 Appendix: Suggested Excercises 301