FAST ALGORITHMS FOR THE DIGITAL COMPUTATION OF LINEAR CANONICAL TRANSFORMS A DISSERTATION SUBMITTED TO THE DEPARTMENT OF ELECTRICAL ENGINEERING AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Aykut Koc¸ March 2011 © 2011 by Aykut Koc. All Rights Reserved. Re-distributed by Stanford University under license with the author. This dissertation is online at: http://purl.stanford.edu/fq782pt6225 ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Lambertus Hesselink, Primary Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Shanhui Fan I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. R Pease Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives. iii iv Abstract LTHOUGH IT IS straightforward to determine the relationship between the in-focus A image and the object of a simple optical system such as a lens, it is far more challenging to compute the input/output relationships of general first-order and astigmatic optical systems. Such optical systems are known as quadratic-phase systems (QPS) and they include the Fresnel propagation in free space, propagation in graded-index media, passage through thin lenses, and arbitrary concatenations of any number of these, including anamorphic, astigmatic, nonorthogonal elements. Such computation is accomplished by representing the physical system with a general mathematical framework of integrations against kernels and then distilling the entire system into one input-output relationship that can be represented by a linear integral transform. The underlying mathematical integral transforms can be applied to a wider field of signal processing where they are known as the linear canonical transform (LCT) of a signal. Conventional numerical integration methods have a computational complexity of O(N 2) where N is the space-bandwidth product of the sampling scheme, e.g. the number of pixels in the field for an optical system. The algorithms described here yield a complexity of only O(N log N). The key is the use of different decompositions (or factorizations) of a given input/output relationship into simpler ones. Instead of following the general physical subparts in cascaded systems and computing input-output relations separately, these algorithms use the simplest possible decompositions to represent the entire system in terms of least possible number of steps. The algorithms are Fast Fourier Transform (FFT) based methods and the only essential v deviation from exactness arises from approximating a continuous Fourier transform (FT) with the discrete Fourier transform (DFT). Thus the algorithms work with a performance similar to that of the fast Fourier transform algorithm in computing the Fourier transform, both in terms of speed and accuracy. Unlike conventional techniques these algorithms also track and control the space-bandwidth products, in order to achieve information that is theoretically sufficient but not wastefully redundant. vi to my late grandmother, Fatma vii viii Acknowledgments WAITED FOR writing the acknowledgments section until I got all three signatures and I hence made my Ph.D thesis de facto approved. I did not realize that this very small part of the thesis would end up being by far the hardest one to write, even after the relief of knowing that my Ph.D story has come to a happy ending. Five and a half years have passed at Stanford with their joys and sorrows. During these years, I worked on my Ph.D, a study which is in general deceptively seen as an individual accomplishment. However, after all these years, I believe that I did not do it alone but with the contributions from several people. Therefore, now I will do my best to deliver the rights to those who collectively made this thesis possible. I am proud of having them in my life more than I am proud of earning a Ph.D at Stanford. First of all, I want to send my thanks to my advisor Professor Lambertus ‘Bert’ Hes- selink. I am indebted most of my Stanford experience to him and I cannot express the full extent of my gratitude to him. I sincerely think that he is a truly ideal advisor in all aspects. Professionally, academically and personally, he has thought me lots. His guidance, expe- rience, support and vast knowledge were always there for me. Lastly, a final and special thank you goes to Bert for the hospitality he showed during the great weekend retreats at his estate in Tahoe that we enjoyed as a research group. Those were wonderful times that I will always remember. My oral defense and thesis reading committee members, Professor Shanhui Fan and Professor R. Fabian Pease helped me through the hardships of my orals exam and thesis ix writing stages. I thank them for their feedback, insights and inputs that contributed to my thesis. I would also like to thank Professor Martin M. Fejer for agreeing to serve as the chair of my oral defense committee. I would like to thank Professor Haldun M. Ozaktas of Bilkent University for all his contributions to my academic development. I started research in an undergraduate senior year project under his supervision. With his passion for research and his vast knowledge, he was a role model for me and greatly affected my decision to pursue a Ph.D. I also need to thank Professor Peter Peumans for his support, especially during the early hard times. My Master’s Program Advisor, Professor Stephen Boyd should also be acknowledged for his help during my early years of studies. Dr. Yuzuru Takashima and Ludwig Galambos, senior researchers in our department, have also contributed to my development at Stanford. I would like to especially acknowl- edge Dr. Yuzuru Takashima for his support, guidance and help in the projects that I was involved in. I would like to thank for all the help, companionship and discussions provided by my past and present research group members: Paul, Yao-Te, Yin, Xiaobo, Brian, Yuxin, Eu- gene and Toan. Particularly, I cannot forget the friendships of Paul and Yao-Te. Thank you guys, for listening my absurd business ideas during lunches we enjoyed. Our research group’s administrative assistants Ms. Lilyan Sequeira and Ms. Ann Guerra deserve to be acknowledged for the administrative issues that have superbly been taken care of for our group. I also send my thanks and gratitude to Ms. Natasha Newson, the Student Services Officer of Electrical Engineering Department, for her helpful attitude that have made lots of official issues easy. I couldn’t complete this long process without the support and enjoyment provided by great companions and friends. I had the chance to befriend great people during my years at Stanford. I have countless good moments and memoirs with them. I am grateful for their support and friendship. Although I cannot list every friend of mine, I still want to x name at least some of them: Berk Atiko˘glu, Duygu Ozuysal,¨ Yusuf Ozuysal,¨ Ays¸e T¨urker C¸ınar, Murat C¸ınar, Bihter Padak, Erdem S¸as¸maz, G¨urer Kıratlı, T¨urev Dara Acar, Hakan Baba, Emel Tas¸y¨urek, Ozge¨ Is¸legen,˙ Onur Kılıc¸, Uygar S¨umb¨ul, Ali Ozer¨ Ercan, Murat Aksoy, Emine Ulk¨uSarıtas¸,¨ Tolga C¸ukur, Ipek˙ Kasımo˘glu, Ismail˙ Kasımo˘glu, Vinay Ma- jjigi, Maryam Etezadi-Amoli, Michelle Hewlett, Thomas Sushil John, and Zuley Rivera- Alvidrez. I owe my deepest gratitude to my mother for raising me and being there always at all costs. I am indebted everything to her, who gave me the first key steps of my education. Her dedication to education was, in my opinion, key to my future educational success. I would not be here without her unconditional love and support. I am also grateful to my family, especially to my grandparents who helped raise me while I was a little kid and who gave me their love. Finally, I want to acknowledge my greatest support since the last three years, Ozlem.¨ Without her love, I would not finish, even dare to continue my studies. She is always happy, cheerful, optimistic and every such good thing that one can imagine. Most importantly, she has offered these to me generously and I have found the necessary encouragement even when I am feeling ‘down’. Thank you very much my love... March 2011 Stanford, CA, USA. xi Don't be proud of your knowledge, Consult the ignorant and the wise; The limits of art are not reached, No artist's skills are perfect. † †the Vizier Ptahhotep, from the Papyrus Prisse, dating back the Middle Kingdom of Ancient Egypt xii Contents Abstract v Acknowledgments ix 1 Introduction 1 1.1 MotivationandPreviousWork . 9 1.2 OrganizationoftheThesis . 15 2 Fundamentals 17 2.1 Preliminaries ................................. 17 2.1.1 Quadratic-PhaseSystemsandMatrixOptics. .... 17 2.1.2 LinearCanonicalTransforms-1D . 18 2.1.3 LinearCanonicalTransforms-2D . 19 2.1.4 LinearCanonicalTransforms-Complex. .. 23 2.1.5 RelationofLCTstotheWignerdistribution . .... 24 2.2 SpecialLinearCanonicalTransforms. ..... 26 2.2.1 Scaling ................................ 26 2.2.2 FourierTransformation. 26 2.2.3 ChirpMultiplication . 27 2.2.4 ChirpConvolution . .. .. .. .. .. .. .. 28 xiii 2.2.5 FractionalFouriertransformation . ... 30 3 The Algorithm for 1D Quadratic-Phase Systems 33 3.1 Introduction.................................