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Fourier Optics
and
Computational Imaging
Fourier Optics
and
Computational Imaging
Kedar Khare
Department of Physics
Indian Institute of Technology Delhi
New Delhi, India
John Wiley & Sons Ltd.
Ane Books Pvt. Ltd.
Fourier Optics and Computational Imaging
© 2016 Kedar Khare
This edition published by
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Contents
Preface 11
1 Introduction 131.1 Organization of the book . . . . . . . . . . . . . . . 16
Part 1: Mathematical preliminaries
2 Fourier series and transform 212.1 Fourier Series . . . . . . . . . . . . . . . . . . . . . 212.2 Gibbs phenomenon . . . . . . . . . . . . . . . . . . 232.3 Fourier transform as a limiting case of Fourier series 27
2.3.1 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 . . . . . . . . . . . . . . . . . . 45
6 Fourier Optics and Computational Imaging
3 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 . . . . . . . . 56
3.6.1 Properties of matrix A(0) . . . . . . . . . . . 593.7 Extrapolation of bandlimited functions . . . . . . . . 63
4 Operational introduction to Fast Fourier Transform 674.1 Definition . . . . . . . . . . . . . . . . . . . . . . . 674.2 Usage of 2D Fast Fourier Transform for problems in
Optics . . . . . . . . . . . . . . . . . . . . . . . . . 69
5 Linear systems formalism and introduction to inverse prob-lems in imaging 755.1 Space-invariant impulse response . . . . . . . . . . . 775.2 Ill-posedness of inverse problems . . . . . . . . . . . 785.3 Inverse filter . . . . . . . . . . . . . . . . . . . . . . 805.4 Wiener filter . . . . . . . . . . . . . . . . . . . . . . 82
6 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 . . . . . . . . . . . . . . . . 97
6.4.1 Guidelines for sub-sampled data measure-ment and image recovery . . . . . . . . . . . 99
6.4.2 System level implications of compressiveimaging philosophy . . . . . . . . . . . . . . 103
6.5 Topics for further study . . . . . . . . . . . . . . . . 104
7 Random processes 1077.1 Probability and random variables . . . . . . . . . . . 107
7.1.1 Joint Probabilities . . . . . . . . . . . . . . 108
Contents 7
7.1.2 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 . . . . . . . . . . 116
7.2 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 123
Part 2: Concepts in optics
8 Geometrical Optics Essentials 1278.1 Ray transfer matrix . . . . . . . . . . . . . . . . . . 1278.2 Stops and pupils . . . . . . . . . . . . . . . . . . . . 130
9 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 . . . . . . . . . . . 140
9.4.1 Kirchhoff Solution . . . . . . . . . . . . . . 1419.4.2 Rayleigh-Sommerfeld Solution . . . . . . . 141
10 The angular spectrum method 14510.1 Angular spectrum method . . . . . . . . . . . . . . . 145
8 Fourier Optics and Computational Imaging
11 Fresnel and Fraunhoffer diffraction 15311.1 Fresnel diffraction . . . . . . . . . . . . . . . . . . 153
11.1.1 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 . . . . . . . . . 162
11.2 Fraunhoffer Diffraction . . . . . . . . . . . . . . . . 163
12 Coherence of light fields 16712.1 Spatial and temporal coherence . . . . . . . . . . . 167
12.1.1 Interference law . . . . . . . . . . . . . . . 16912.2 van Cittert and Zernike theorem . . . . . . . . . . . 16912.3 Space-frequency representation of the coherence func-
tion . . . . . . . . . . . . . . . . . . . . . . . . . . 17112.4 Intensity interferometry: Hanbury
Brown and Twiss effect . . . . . . . . . . . . . . . . 17312.5 Photon counting formula . . . . . . . . . . . . . . . 17512.6 Speckle phenomenon . . . . . . . . . . . . . . . . . 177
13 Polarization of light 18313.1 The Jones matrix formalism . . . . . . . . . . . . . 18313.2 The QHQ geometric phase shifter . . . . . . . . . . 18513.3 Degree of polarization . . . . . . . . . . . . . . . . 186
14 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 . . . . . . . . . . . . . 194
14.4.1 DC block or coronagraph . . . . . . . . . . 19414.4.2 Zernike’s phase contrast microscopy . . . . 19514.4.3 Edge enhancement: vortex filter . . . . . . . 19714.4.4 Apodization filters . . . . . . . . . . . . . . 198
14.5 Frequency response of optical imaging systems: co-herent and incoherent illumination . . . . . . . . . . 199
Contents 9
15 Imaging from information point of view 20515.1 Eigenmodes of a canonical imaging
system . . . . . . . . . . . . . . . . . . . . . . . . . 20615.1.1 Eigenfunctions and inverse problems . . . . 209
Part 3: Selected computational imaging systems
16 Digital Holography 21716.1 Sampling considerations for recording of digital holo-
grams . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . 230
17 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 . . . 244
17.4 Gerchberg-Papoulis method for bandlimited extrap-olation . . . . . . . . . . . . . . . . . . . . . . . . 247
18 Compact multi-lens imaging systems 25318.1 Compact form factor computational
camera . . . . . . . . . . . . . . . . . . . . . . . . . 25318.2 Lightfield cameras . . . . . . . . . . . . . . . . . . 256
18.2.1 The concept of lightfield . . . . . . . . . . . 25718.2.2 Recording the lightfield function with
microlens array . . . . . . . . . . . . . . . . 259
10 Fourier Optics and Computational Imaging
19 PSF Engineering 26719.1 Cubic phase mask . . . . . . . . . . . . . . . . . . 26719.2 Log-asphere lens . . . . . . . . . . . . . . . . . . . 27119.3 Rotating point spread functions . . . . . . . . . . . 273
20 Structural illumination imaging 27720.1 Forward model and image
reconstruction . . . . . . . . . . . . . . . . . . . . . 279
21 Image reconstruction from projection data 28521.1 X-ray projection data . . . . . . . . . . . . . . . . . 28621.2 Image reconstruction from projection data . . . . . . 287
22 Ghost Imaging 29322.1 Schematic of a ghost imaging system . . . . . . . . 29322.2 A signal processing viewpoint of ghost imaging . . . 297
23 Appendix: Suggested Excercises 301
Preface
This book is an outgrowth of a series of lectures on Fourier Opticsand Imaging that I have been teaching to the students at Indian Insti-tute of Technology Delhi for the past few years. There are excellentbooks and other reference resources on various aspects of Optics,Photonics, Mathematics and Image Processing. However, when Istarted teaching students from various backgrounds and disciplinesat IIT Delhi, I strongly felt the need of a single self-contained vol-ume that covered the fundamental ideas required to appreciate therecent developments in Computational Imaging in a concise man-ner. The material in this book has been written at a level suitable foradvanced undergraduate or post-graduate students who may wish topursue research and development activities involving imaging sys-tems for their future careers.
I would like to thank the excellent mentors I had at IIT Kharag-pur during my undergraduate years, who encouraged me to take upa career in research and academics. I especially thank my Ph.D. su-pervisor Prof. Nicholas George at Institute of Optics, University ofRochester, USA for introducing me to the exciting world of imag-ing research. Over the years he has been much more than just aPh.D. adviser to me and I am sure he will be very happy to see thisbook getting published. Several years spent at the General ElectricGlobal Research center in Niskayuna, NY, USA further broadenedmy appreciation and expertise in relation to a wider range of prac-tical imaging systems and I am thankful to my colleagues there forproviding a great real-world learning opportunity. During the lastfew years I have been grateful to have company of several wonderfulfaculty colleagues at IIT Delhi who have helped me to get started asan independent faculty member. Many thanks are also due to several
12 Fourier Optics and Computational Imaging
enthusiastic students at IIT Delhi who make me think of new waysto understand the material discussed in this book and several alliedtopics. Finally I wholeheartedly thank my wife Sarita, my daughtersRasika and Sanika, and all other family members in appreciation oftheir patience, encouragement and constant support over the last fewof years while this manuscript was being prepared.
Kedar KhareNew DelhiJune 2015
1. Introduction
Imaging systems form an integral part of our lives. Research and de-velopment efforts in diverse areas of science and technology over thelast few decades have resulted in practical imaging systems that weoften take for granted today. From cell phone cameras to advanceddiagnostic imaging systems, from weather prediction based on satel-lite imagery to night vision devices meant for soldiers guarding ournational frontiers, there are numerous examples of imaging systemsthat touch all aspects of our lives. Further, in current science andtechnological investigations dealing with sub-nano to astronomicallength scales, imaging systems allow researchers to visualize naturalphenomena or objects of interest and in effect, directly contribute tonew discoveries. Imaging is thus an active interdisciplinary researcharea that is relevant to a wide range of fundamental and applied prob-lems.
The chart in Fig. 1.1 shows a wide variety of topics where imag-ing systems have become indispensable. It may not be too much ofan exaggeration to say that every current Science and Technologystudent is very likely to encounter at least one of the topics listed inthis chart during his/her career. While it is difficult to describe allthe topics listed in Fig. 1.1 in detail in a single book, luckily for usthe various imaging modalities share many common principles andmathematical ideas that are useful for design and analysis of imag-ing systems. The goal of this book is to present these basic toolsfollowed by a discussion of some specific imaging systems so as toprovide the reader sufficient background to enter the area of imagingresearch.
It is possible to describe a variety of imaging systems schemati-cally by a simple model as shown in Fig. 1.2. A source of radiation
Fourier Optics and Computational Imaging, First Edition. Kedar Khare.© Kedar Khare 2016. Published by Athena Academic Ltd and John Wiley & Sons Ltd.
