Sparse Signal Representation: Image Compression using Sparse Bayesian Learning

with Φ ∈ RN×M , M ≥ N , and some noise . The challenge is to determine the sparsest representation of reconstruction coefficients w = [w1, . . . , wM ] . Finding a sparse representation of a signal in an overcomplete dictionary is equivalent to solving a regularized linear inverse. For a given dictionary Φ, finding the maximally sparse w is an NP-hard problem [1]. A great deal of recent research has focused on computationally feasible methods for determining highly sparse representations and is fueled by applications in signal processing, compression and feature extraction [2]. In section II of this paper we formulate the problem of finding a sparse inverse solution. In section III we give an overview of several popular techniques: Method of Frames (MOF), Matching Pursuits (MP), Basis Pursuit (BP), Focal Underdetermined System Solution (FOCUSS), and Sparse Bayesian Learning (SBL). We give a general comparison of problems solved by each method and the strengths and weaknesses of each approach. In sections IV and V we apply these techniques to two applications: image compression and medical image reconstruction. Each application highlights one of the two goals of sparse signal representation: sparsity and hyper-resolution.