sparse image representation via combined transforms - Convex ...

Chapter 3
Image Transforms and Image
Features
This chapter is a summary of early achievements and recent advances in the design of
transforms. It also describes features that these transforms are good at processing.
Some mathematical slogans are given, together with their quantitative explanations. As
many of these topics are too large to be covered in a single chapter, we give pointers in the
literature. The purpose of this chapter is to build a concrete foundation for the remainder
of this thesis.
One particular message that we want to deliver is: “Each transform is good for one
particular phenomenon, but not good for some others; at the same time, a typical image is
made by a variety of phenomena. It is natural to think about how to combine different transforms,
so that the combined method will have advantages from each of these transforms.”
This belief is the main motivation for this thesis.
We give emphasis to discrete algorithms and, moreover, fast linear algebra, because the
possibility of efficient implementation is one of our most important objectives.
Note that there are two important ideas in signal transform methods: decomposition and
distribution. The idea of decomposition, also called atomic decomposition, is to write the
image/signal as a superposition (which is equivalently a linear combination) of pre-selected
atoms. These atoms make a dictionary, and the dictionary has some special property;
for example, orthonormality when a dictionary is made by a single orthonormal basis, or
tightness when a dictionary is made by a single tight frame. The second idea is to map the
signal into a distribution space. An example is the Wigner-Ville transform, which maps a
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