Sparsity Based Methods for Overparameterized Variational Problems
Abstract:
Two complementary approaches have been extensively used in signal and image processing leading to novel results, the sparse representation methodology and the variational strategy. Recently, a new sparsity based model has been proposed, the cosparse analysis framework, that has established a very interesting connection between the two, highlighting how the traditional total variation minimization problem can be viewed as a sparse approximation problem. Based on this work we introduce a sparsity based framework for solving overparameterized variational problems. The latter has been used to improve the estimation of optical flow and also for general denoising of signals and images. However, the recovery of the space varying parameters involved was not adequately solved by traditional variational methods. We first demonstrate the efficiency of the new framework for one dimensional signals in recovering a piecewise linear and polynomial function. Then, we illustrate how the new technique can be used for denoising, segmentation and inpainting of images.
The paper is coauthored by Raja Giryes, Michael Elad and Alfred Bruckstein, and will soon appear in SIAM J. on Imaging
The paper is coauthored by Raja Giryes, Michael Elad and Alfred Bruckstein, and will soon appear in SIAM J. on Imaging
Short Bio:
Alfred M. Bruckstein, born in Transylvania, Romania, in 1954, received his BSc and MSc degrees at the Technion, Haifa, in 1976 and 1980, respectively and then earned a Ph.D. degree in Electrical Engineering in Stanford University, California in 1984, his advisor being Professor Thomas Kailath.
From October 1984 he has been with the Technion, where he now holds of the Ollendorff Chair in Science, in the Computer Science Department. His research interests are in Ants and Swarm Robotics, Signal and Image Processing, Image Analysis and Synthesis, Pattern Recognition, and various aspects of Applied Geometry. Professor Bruckstein authored and co-authored over one hundred and fifty journal papers in the fields of interest mentioned.
Professor Bruckstein held visiting positions at MIT, Groningen University in Holland, Stanford University, and TsingHua University in Beijing, China, Evry University and at CEREMADE, Dauphine University in Paris, France, and was a visiting Member of Technical Staff at Bell Laboratories at Murray Hill, from 1987 to 2000, working with Dr. Arun Netravali and several colleagues there on Image Processing and Computer Vision topics. Since 2009 he is also a Visiting Professor at the Nanyang Technological University in Singapore, at the School of Mathematical and Physical Sciences.
From 2002 till 2005 he served as the Dean of Technion’s Graduate School, and from 2006-2011 as the Head of Technion’s Excellence Program for Undergraduate Studies.
Professor Bruckstein is a member of the AMS, and MAA, and a SIAM Fellow for contributions to Signal Processing, Image Analysis, and Ant Robotics, and received SIAM’s 2014 SIAG-Imaging Science Prize (with David Donoho and Michael Elad, for the paper “From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images”)
Professor Bruckstein is happily married to Rita and they have one son, Ariel, with whom they wrote and illustrated a bestiary of imaginary animals of Ariel’s invention called “The Knocktopus and His Friends”, published by Panopticum Press in 2013. He also illustrated several books published by his late father Ludovic Bruckstein, in Romanian, Hebrew and French, and a collection of comical verse in Hebrew, by Professor Irad Yavne, entitled”Comical Relief”, describing Academic Life in general, and at the Technion, in particular.
From October 1984 he has been with the Technion, where he now holds of the Ollendorff Chair in Science, in the Computer Science Department. His research interests are in Ants and Swarm Robotics, Signal and Image Processing, Image Analysis and Synthesis, Pattern Recognition, and various aspects of Applied Geometry. Professor Bruckstein authored and co-authored over one hundred and fifty journal papers in the fields of interest mentioned.
Professor Bruckstein held visiting positions at MIT, Groningen University in Holland, Stanford University, and TsingHua University in Beijing, China, Evry University and at CEREMADE, Dauphine University in Paris, France, and was a visiting Member of Technical Staff at Bell Laboratories at Murray Hill, from 1987 to 2000, working with Dr. Arun Netravali and several colleagues there on Image Processing and Computer Vision topics. Since 2009 he is also a Visiting Professor at the Nanyang Technological University in Singapore, at the School of Mathematical and Physical Sciences.
From 2002 till 2005 he served as the Dean of Technion’s Graduate School, and from 2006-2011 as the Head of Technion’s Excellence Program for Undergraduate Studies.
Professor Bruckstein is a member of the AMS, and MAA, and a SIAM Fellow for contributions to Signal Processing, Image Analysis, and Ant Robotics, and received SIAM’s 2014 SIAG-Imaging Science Prize (with David Donoho and Michael Elad, for the paper “From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images”)
Professor Bruckstein is happily married to Rita and they have one son, Ariel, with whom they wrote and illustrated a bestiary of imaginary animals of Ariel’s invention called “The Knocktopus and His Friends”, published by Panopticum Press in 2013. He also illustrated several books published by his late father Ludovic Bruckstein, in Romanian, Hebrew and French, and a collection of comical verse in Hebrew, by Professor Irad Yavne, entitled”Comical Relief”, describing Academic Life in general, and at the Technion, in particular.
