Image Denoising Via Sparse and Redundant Representations Over Learned Dictionaries Michael Elad and Michal Aharon
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Five Lectures on Sparse and Redundant Representations Modelling of Images
Five Lectures on Sparse and Redundant Representations Modelling of Images Michael Elad IAS/Park City Mathematics Series Volume 19, 2010 Five Lectures on Sparse and Redundant Representations Modelling of Images Michael Elad Preface The field of sparse and redundant representations has evolved tremendously over the last decade or so. In this short series of lectures, we intend to cover this progress through its various aspects: theoretical claims, numerical problems (and solutions), and lastly, applications, focusing on tasks in image processing. Our story begins with a search for a proper model, and of ways to use it. We are surrounded by huge masses of data, coming from various sources, such as voice signals, still images, radar images, CT images, traffic data and heart signals just to name a few. Despite their diversity, these sources of data all have something in common: each and everyone of these signals has some kind of internal structure, which we wish to exploit for processing it better. Consider the simplest problem of removing noise from data. Given a set of noisy samples (e.g., value as a function of time), we wish to recover the true (noise- free) samples with as high accuracy as possible. Suppose that we know the exact statistical characteristics of the noise. Could we perform denoising of the given data? The answer is negative: Without knowing anything about the properties of the signal in mind, this is an impossible task. However, if we could assume, for example, that the signal is piecewise-linear, the noise removal task becomes feasible. The same can be said for almost any data processing problem, such as de- blurring, super-resolution, inpainting, compression, anomaly detection, detection, recognition, and more - relying on a good model is in the base of any useful solution. -
From Sparse Solutions of Systems of Equations to Sparse Modeling Of
SIAM REVIEW c 2009 Society for Industrial and Applied Mathematics Vol. 51,No. 1,pp. 34–81 From Sparse Solutions of Systems of Equations to Sparse ∗ Modeling of Signals and Images Alfred M. Bruckstein† David L. Donoho‡ Michael Elad§ Abstract. A full-rank matrix A ∈ Rn×m with n<mgenerates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries. Can it ever be unique? If so, when? As opti- mization of sparsity is combinatorial in nature, are there efficient methods for finding the sparsest solution? These questions have been answered positively and constructively in recent years, exposing a wide variety of surprising phenomena, in particular the existence of easily verifiable conditions under which optimally sparse solutions can be found by con- crete, effective computational methods. Such theoretical results inspire a bold perspective on some important practical problems in signal and image processing. Several well-known signal and image processing problems can be cast as demanding solutions of undetermined systems of equations. Such problems have previously seemed, to many, intractable, but there is considerable evidence that these problems often have sparse solutions. Hence, ad- vances in finding sparse solutions to underdetermined systems have energized research on such signal and image processing problems—to striking effect. In this paper we review the theoretical results on sparse solutions of linear systems, empirical results on sparse mod- eling of signals and images, and recent applications in inverse problems and compression in image processing. This work lies at the intersection of signal processing and applied mathematics, and arose initially from the wavelets and harmonic analysis research commu- nities. -
Learning Low-Dimensional Models for Heterogeneous Data by David
Learning Low-Dimensional Models for Heterogeneous Data by David Hong A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Electrical Engineering: Systems) in The University of Michigan 2019 Doctoral Committee: Professor Laura Balzano, Co-Chair Professor Jeffrey A. Fessler, Co-Chair Professor Rina Foygel Barber, The University of Chicago Professor Anna Gilbert Professor Raj Rao Nadakuditi David Hong [email protected] ORCID iD: 0000-0003-4698-4175 © David Hong 2019 To mom and dad. ii ACKNOWLEDGEMENTS As always, no list of dissertation acknowledgements can hope to be complete, and I have many people to thank for their wonderful influence over the past six years of graduate study. To start, I thank my advisors, Professor Laura Balzano and Professor Jeffrey Fessler, who introduced me to the exciting challenges of learning low-dimensional image models from heterogeneous data and taught me the funda- mental ideas and techniques needed to tackle it; this dissertation is the direct result. Laura and Jeff, I am always amazed by your deep and broad technical insights, your sincere care for students and your great patience (with me!). Working with you has changed the way I think, write and teach, and I cannot say how thankful I am to have had the opportunity to learn from you. I hope to one day be as wonderful an advisor to students of my own. I am also thankful for my excellent committee. Professor Rina Barber has many times provided deep statistical insight into my work, and her wonderful suggestions can be found throughout this dissertation.