The book is designed to serve as a textbook for advanced undergraduate and graduate students enrolled in physics and electronics and communication engineering and mathematics. The book provides an introduction to Fourier optics in light of new developments in the area of computational imaging over the last couple of decades. There is an in-depth discussion of mathematical methods such as Fourier analysis, linear systems theory, random processes, and optimization-based image reconstruction techniques. These techniques are very much essential for a better understanding of the working of computational imaging systems. It discusses topics in Fourier optics, e.g., diffraction phenomena, coherent and incoherent imaging systems, and some aspects of coherence theory. These concepts are then used to describe several system ideas that combine optical hardware design and image reconstruction algorithms, such as digital holography, iterative phase retrieval, super-resolution imaging, point spread function engineering for enhanced depth-of-focus, projection-based imaging, single-pixel or ghost imaging, etc. The topics covered in this book can provide an elementary introduction to the exciting area of computational imaging for students who may wish to work with imaging systems in their future careers.
Author(s): Kedar Khare, Mansi Butola, Sunaina Rajora
Edition: 2
Publisher: Springer
Year: 2023
Language: English
Pages: 294
City: Cham
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
