sparse image representation via combined transforms - Convex ...

2 CHAPTER 1. INTRODUCTION
1.2 Outline
This thesis is organized as follows:
• Chapter 2 explains why sparse decomposition may lead to a more efficient image
coding and compression. It contains two parts: the first part gives a brief description
of the mathematical framework of a modern communication system; the second part
presents some quantitative results to explain (mainly in the asymptotic sense) why
sparsity in coefficients may lead to efficiency in image coding.
• Chapter 3 is a survey of existing transforms for images. It serves as a background for
this project.
• Chapter 4 explains our approach in finding a sparse decomposition in an overcomplete
dictionary. We minimize the objective function that is a sum of the residual sum
of squares and a penalty on coefficients. We apply Newton’s method to solve this
minimization problem. To find the Newton’s direction, we use an iterative method—
LSQR—to solve a system of linear equations, which happens to be equivalent to a
least squares problem. (LSQR solves a least square problem.) With carefully chosen
parameters, the solution here is close to the solution of an exact Basis Pursuit, which
is the minimum l 1 norm solution with exact equality constraints.
• Chapter 5 is a survey of iterative methods and explains why we choose LSQR. Some
alternative approaches are discussed.
• Chapter 6 presents some numerical simulations. They show that a combined approach
does provide a sparser decomposition than the existing approach that uses only one
transform.
• Chapter 7 discusses some thoughts on future research.
• Appendix A documents the details about how to implement the exact edgelet transform
in a direct way. This algorithm has high complexity. Some examples are given.
• Appendix B documents the details about how to implement an approximate edgelet
transform in a fast way. Some examples are given. This transform is the one we used
in simulations.