sparse image representation via combined transforms - Convex ...

Chapter 1
Introduction
1.1 Overview
Recently, many new methods of image representation have been proposed, including wavelets,
cosine packets, brushlets, edgelets, ridgelets, and so on. Typically each of these is good for a
specific class of features, but not good for others. We propose a method of combining image
representations, more particularly, a method based on the 2-D wavelet transform and the
edgelet-like transform. The 2-D wavelet transform is good at capturing point singularities,
while the newly proposed edgelet-like transform is good at capturing linear singularities
(edges). Both transforms have fast algorithms for digital images.
Wavelets and edgelet-like features (together) form an overcomplete dictionary. To find
a sparse representation, we have had success by minimizing the objective function: ‖y −
Φx‖ 2 2 + λρ(x), where y is the image, Φ is the matrix whose columns are all the basis vectors
in all the image representations, x is the vector of coefficients and ρ(x) is a penalty function
that is convex, l 1 -like, and separable. Sparsity of representation is achieved by adding the
penalty term: λρ(x) to balance the goodness of fit measure ‖y − Φx‖.
To develop an efficient algorithm to solve this problem, we utilize insights from convex
optimization and the LSQR solver from computational linear algebra. These methods
combine to give a fast algorithm for our problem. Numerical experiments provide promising
results.
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