sparse image representation via combined transforms - Convex ...

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Some of the figures show the basis functions for various transforms. The reason for
illustrating these basis functions is the following: the ith coefficient of a linear transform is
the inner product of the ith basis function and the image:
coefficient i = 〈basis function i , image〉,
where 〈·, ·〉 is the inner product of two functions. Hence the pattern of basis functions
determines the characteristics of the linear transform.
In functional analysis, a basis function is called a Riesz representer of the linear transform.
Notations
We follow some conventions in signal processing and statistics. A function X(t) is a continuous
function with a continuous time variable t. A function X[k] is a continuous function
with variable k that only takes integral values. Function ̂X is the Fourier transform of X.
We use ω to denote a continuous frequency variable.
More specifics of notation can be found in the main body of this chapter.
Organization
The rest of this chapter is organized as follows. The first three sections describe the three
most dominant transforms that are used in this thesis: Section 3.1 is about the discrete
cosine transform (DCT); Section 3.2 is about the wavelet transform; and Section 3.3 is
about the edgelet transform. Section 3.4 is an attempt at a survey of other activities. 1
Section 3.5 contains some discussion. Section 3.6 gives conclusions. Section 3.7 contains
some proofs.
1 It is interesting to note that there are many other activities in this field. Some examples are the Gabor
transform, wavelet packets, cosine packets, brushlets, ridgelets, wedgelets, chirplets, and so on. Unfortunately,
owing to space constraints, we can hardly provide many details. We try to maintain most of the
description at an introductory level, unless we believe our detailed description either gives a new and useful
perspective or is essential for some later discussions.