sparse image representation via combined transforms - Convex ...
sparse image representation via combined transforms - Convex ... 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.
Chapter 2 Sparsity in Image Coding This chapter explains the connection between sparse image decomposition and efficient image coding. The technique we have developed promises to improve the efficiency of transform coding, which is ubiquitous in the world of digital signal and image processing. We start with an overview of image coding and emphasize the importance of transform coding. Then we review Donoho’s work that answers in a deterministic way the question: why does sparsity lead to good compression? Finally we review some important principles in image coding. 2.1 Image Coding This section consists of three parts. The first begins with an overview of Shannon’s mathematical formulation of communication systems, and then describes the difference between source coding and channel coding. The second part describes a way of measuring the sparsity of a sequence and the connection between sparsity and coding, concluding with an asymptotic result. Finally, we discuss some of the requirements in designing a realistic transform coding scheme. Since all the work described here can easily be found in the literature, we focus on developing a historic perspective and overview. 2.1.1 Shannon’s Mathematical Communication System Nowadays, it is almost impossible to review the field of theoretical signal processing without mentioning the contributions of Shannon. In his 1948 paper [127], Shannon writes, “The 3
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Chapter 2<br />
Sparsity in Image Coding<br />
This chapter explains the connection between <strong>sparse</strong> <strong>image</strong> decomposition and efficient<br />
<strong>image</strong> coding. The technique we have developed promises to improve the efficiency of<br />
transform coding, which is ubiquitous in the world of digital signal and <strong>image</strong> processing.<br />
We start with an overview of <strong>image</strong> coding and emphasize the importance of transform<br />
coding. Then we review Donoho’s work that answers in a deterministic way the question:<br />
why does sparsity lead to good compression? Finally we review some important principles<br />
in <strong>image</strong> coding.<br />
2.1 Image Coding<br />
This section consists of three parts. The first begins with an overview of Shannon’s mathematical<br />
formulation of communication systems, and then describes the difference between<br />
source coding and channel coding. The second part describes a way of measuring the sparsity<br />
of a sequence and the connection between sparsity and coding, concluding with an<br />
asymptotic result. Finally, we discuss some of the requirements in designing a realistic<br />
transform coding scheme.<br />
Since all the work described here can easily be found in the literature, we focus on<br />
developing a historic perspective and overview.<br />
2.1.1 Shannon’s Mathematical Communication System<br />
Nowadays, it is almost impossible to review the field of theoretical signal processing without<br />
mentioning the contributions of Shannon. In his 1948 paper [127], Shannon writes, “The<br />
3
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