טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentMatalon Boaz
SubjectBayesian Methodology for Denoising Using Sparse
Representations
DepartmentDepartment of Electrical Engineering
Supervisors Dr. Michael Zibulevsky
Professor Michael Elad
Full Thesis textFull thesis text - English Version


Abstract

In this research we address the denoising problem, and specifically, retrieving signals corrupted by additive white Gaussian noise. A classic and well-known framework for handling this task is the Basis-Pursuit Denoising (BPDN). Despite being frequently deployed in many applications, the BPDN problem has proven hard to solve using classic iterative algorithms, which exhibit slow convergence rate. In a recent work, a new iterative-shrinkage method called Parallel Coordinate Descent (PCD) was devised. We provide herein a convergence analysis of the PCD algorithm, and also introduce a form of the regularization function, which permits analytical solution to the coordinate optimization. Several other recent works, which considered the deblurring problem in a Bayesian methodology, also obtained iterative-shrinkage algorithms. We show that these three methods are essentially equivalent, and the unified method is termed Separable Surrogate Functionals (SSF). We provide a convergence analysis for SSF as well, and draw a connection between this family and the PCD method. To further accelerate PCD and SSF, we merge them into a recently developed sequential subspace optimization technique (SESOP), with almost no additional complexity. A thorough numerical comparison, both with synthetic data and with real images, is presented. The advantage of our combined PCD-SESOP method is clearly demonstrated.

In this research we have also generalized the BPDN, so as to take into account the inter-dependencies between representation coefficients. Several advanced methods that model these dependencies were developed recently, and were shown to yield significant improvement over simple shrinkage. However, these methods operate on the transform domain error rather than on the image domain one, errors which are in general entirely different for redundant transforms. Our new method combines the image domain error with the transform domain dependency structure, resulting in a general objective function, applicable for any wavelet-like transform. We focus here on the Contourlet Transform (CT), a relatively new transform designed to sparsely represent images. Our method shows state-of-the-art results, thus providing a more advanced tool for image restoration.