sparse image representation via combined transforms - Convex ...

Chapter 4
Combined Image Representation
This chapter is about combined image representation and sparse decomposition. First, in
Section 4.1, we discuss the motivation for using combined image representation, the key being
that we can obtain benefits from different image transforms. Because we have combined
representations, we have an overcomplete system, or an overcomplete dictionary. Section
4.2 surveys research developments in finding sparse decompositions in an overcomplete dictionary.
Section 4.3 explains the optimality of using the minimum l 1 norm decomposition
and explains the formulation we used in this thesis. Section 4.4 explains how we use Lagrange
multipliers to transform a constrained optimization problem into an unconstrained
optimization problem. Section 4.5 is about how to choose the parameters in our method.
Section 4.6 points out that a homotopic method converges to the minimum l 1 norm decomposition.
Section 4.7 describes the Newton method. Section 4.8 surveys existing methods
and softwares and explains some advantages of our approach. Section 4.9 gives a preview
about the importance of iterative methods and why we use them. Section 4.10 gives more
detail to the numerical solution of the problem. Finally, in Section 4.11, we make some
general remarks. Section 4.12 contains all relevant proofs in this chapter.
4.1 Why Combined Image Representation?
Recently, many new methods for signal/image representation have been proposed, including
wavelets, wavelet packets, cosine packets, brushlets, edgelets, and ridgelets. Typically, each
of these is good for a specific class of features, but not for others. For example, for 1-D
signal representation, wavelets are effective at representing signals made by impulses—where
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