Fourier Optics and Computational Imaging

Description

Preface to the Second Edition Preface to the First Edition Contents About the Authors 1 Introduction 1.1 Scope of Imaging Research 1.2 Computational Imaging Model 1.3 Organization of the Book References Part I Mathematical Preliminaries 2 Fourier Series and Transform 2.1 Fourier Series 2.2 Gibbs Phenomenon 2.3 Fourier Transform as a Limiting Case of Fourier Series 2.4 Sampling by Averaging, Distributions and Delta Function 2.4.1 Properties of Delta Function 2.5 Fourier Transform of Unit Step and Sign Functions 2.6 Fourier Transform of a Train of Delta Functions 2.7 Fourier Transform of a Gaussian Function 2.8 Fourier Transform of the Chirp Phase Function 2.9 Properties of Fourier Transform 2.10 Fourier Transform of the 2D Circ Function 2.11 Fourier Slice Theorem 2.12 Wigner Distribution References 3 Sampling Theorem 3.1 Sampling Theorem via Poisson Summation Formula 3.1.1 Poisson Summation Formula for Bandlimited Signal 3.1.2 Additional Notes on the Sampling Formula 3.2 Sampling of Carrier-Frequency Signals 3.3 Degrees of Freedom in the Signal—Space-Bandwidth Product 3.4 Slepian (Prolate Spheroidal) Functions 3.4.1 Properties of Matrix A(0) 3.4.2 Extrapolation of Bandlimited Functions 3.5 Band-Pass Analogues of Prolate Spheroidal Functions References 4 Operational Introduction to Fast Fourier Transform 4.1 Definition of Discrete Fourier Transform 4.2 Usage of 2D Fast Fourier Transform for Problems in Optics References 5 Linear System Formalism and Introduction to Inverse Problems in Imaging 5.1 Space-Invariant Impulse Response 5.2 Ill-Posedness of Inverse Problems 5.3 Inverse Filter 5.4 Wiener Filter 5.5 Generalized Wiener Filter References 6 Optimization Approach to Image Reconstruction 6.1 Image Denoising 6.1.1 Euler-Lagrange Problem in Variational Calculus 6.2 Functional Gradient for Complex-Valued Solutions 6.3 Image Deconvolution by Optimization 6.4 Compressive Imaging 6.4.1 Guidelines for Sub-sampled Data Measurement and Image Recovery 6.5 Optimization-Based Image Recovery Without a Free Parameter 6.6 Topics for Further Study References 7 Random Processes 7.1 Probability and Random Variables 7.1.1 Random Variables 7.1.2 Characteristic Function 7.1.3 Gaussian or Normal Distribution 7.2 Random Processes 7.2.1 Spectral Density: Wiener-Khintchine Theorem 7.2.2 Orthogonal Series Representation of Random Processes 7.3 Complex Representation of Random Processes 7.3.1 Mandel's Theorem on Complex Representation References Part II Concepts in Optics 8 Geometrical Optics Essentials 8.1 Ray Transfer Matrix 8.2 Stops and Pupils References 9 Wave Equation and Diffraction of Light 9.1 Review of Maxwell Equations 9.2 Weyl Representation of Spherical Waves 9.3 Angular Spectrum Method 9.4 Numerical Computation of Diffraction Fields Using Angular Spectrum Method 9.5 Fresnel and Fraunhofer Approximations 9.5.1 Transport of Intensity Equation (TIE) 9.5.2 Self-Imaging: Montgomery Conditions and Talbot Effect 9.5.3 Fractional Fourier Transform 9.6 Fraunhofer Diffraction References 10 Coherence of Light Fields 10.1 Spatial and Temporal Coherence 10.1.1 Interference Law 10.2 van Cittert and Zernike Theorem 10.3 Space-frequency Representation of the Coherence Function 10.4 Intensity Interferometry: Hanbury Brown and Twiss Effect 10.5 Photon Counting Formula 10.6 Speckle Phenomenon References 11 Polarization of Light 11.1 The Jones Matrix Formalism 11.2 The QHQ Geometric Phase Shifter 11.3 Degree of Polarization 11.4 Non-uniformly Polarized Light References 12 Analysis of Optical Systems 12.1 Transmission Function for a Thin Lens 12.2 Fourier Transforming Property of a Thin Lens 12.3 Canonical Optical Processor 12.3.1 DC Block or Dark Field Imaging 12.3.2 Zernike's Phase Contrast Microscopy 12.3.3 Highlighting of Edges with Vortex Filter 12.3.4 Apodization Filters 12.4 Frequency Response of Optical Imaging Systems: Coherent and Incoherent Illumination References 13 Imaging from Information Point of View 13.1 Eigenmodes of a Canonical Imaging System 13.1.1 Eigenfunctions and Inverse Problems References Part III Computational Imaging Systems 14 Digital Holography 14.1 Some Practical Aspects of Digital Holography Systems 14.1.1 In-line and Off-axis Configurations 14.1.2 Sampling of Digital Holograms 14.1.3 Numerical Aperture of the Hologram Recording System 14.2 Complex Object Wave Recovery in the Hologram Plane 14.2.1 Off-axis Digital Holography 14.2.2 Phase Shifting Digital Holography 14.2.3 Optimization Method for Complex Object Wave Recovery from Digital Holography 14.2.4 Noise Advantage Offered by the Optimization Method 14.3 Digital Holographic Microscopy 14.4 In-line Digital Holography of Particulates or Weak Scattering Objects 14.5 True 3D Image Reconstruction in Digital Holography References 15 Non-interferometric Phase Retrieval 15.1 Lensless Coherent X-ray Diffraction Imaging 15.2 Error Reduction Algorithm 15.3 Hybrid Input-Output (HIO) Algorithm 15.3.1 The Relaxed-Reflect-Reflect (RRR) Algorithm for Phase Retrieval 15.3.2 The Relaxed Average Alternating Reflection (RAAR) Algorithm for Phase Retrieval 15.4 Phase Retrieval Using Complexity Guidance 15.5 Phase Retrieval with Multiple Intensity Measurements: Fourier Ptychography 15.6 Phase Retrieval as an Optimization Problem References 16 Compact Multi-lens Imaging Systems 16.1 Compact Form Factor Computational Camera 16.2 Lightfield Cameras 16.2.1 The Concept of Lightfield 16.2.2 Recording the Lightfield Function with Microlens Array References 17 PSF Engineering 17.1 Extending Depth of Focus from a Focal Stack 17.2 Extended Depth of Field with Cubic Phase Mask 17.3 Extended Depth of Focus Using the Log-Asphere Lens 17.4 Rotating Point Spread Functions References 18 Structured Illumination Imaging 18.1 Forward Model and Image Reconstruction 18.2 Other Super-Resolution Microscopy Techniques 18.2.1 Simulated Emission Depletion Microscopy (STED) 18.2.2 Stochastic Optical Reconstruction Microscopy (STORM) References 19 Image Reconstruction from Projections 19.1 X-ray Projection Data 19.2 Image Reconstruction from Projection Data 19.3 Holographic Tomography References 20 Correlation Imaging 20.1 Single-Pixel Ghost Imaging 20.1.1 A Signal-Processing Viewpoint of Ghost Imaging 20.2 Imaging Through a Scattering Medium Using Speckle Memory Effect References Index