sparse image representation via combined transforms - Convex ...

68 CHAPTER 3. IMAGE TRANSFORMS AND IMAGE FEATURES
In the remainder of this subsection, we introduce four transforms: Gabor, chirplet,
cosine packets and wavelet packets. They are selected because of their importance in the
literature. (Of course the selection might be biased by personal preference.)
Gabor Analysis
The Gabor transform was first introduced in 1946 [69]. A contemporary description of
the Gabor transform can be found in many books, for example, [68] and [100]. For a 1-D
continuous function f(t) and a continuous time variable t, the Gabor transform is
G(m, n) =
∫ +∞
−∞
f(t) · h(t − mT )e −iω 0nt dt, for integer m, n ∈ Z, (3.35)
where h is a function with finite support, also called a window function. The constants T
and ω 0 are sampling rates for time and frequency. A function
b m,n (t) =h(t − mT )e −iω 0nt
(3.36)
is called a basis function of Gabor analysis. (Of course we assume that the set {b m,n : m, n ∈
Z} does form a basis.) Function b m,n (t) should be both time and frequency localized.
A basis function of the Gabor transform is a shifted version of one initializing function.
The shifting is operated in both time (by mT ) and frequency (by nω 0 ) domains, so the
Gabor transform can be compared to partitioning the TFP into rectangles that have the
same width and height. Figure 3.7 (c) depicts an idealized tiling corresponding to Gabor
analysis.
Choices of T and ω 0 determine the resolution of a partitioning. The choice of the window
function h, which appears in both (3.35) and (3.36), determines how concentrated (in both
time and frequency) a basis function b m,n is. Two theoretical results are noteworthy:
• If a product T · ω 0 is greater than 2π, then a set {h(t − mT )e −iω0nt : m, n ∈ Z} can
never be a frame for L 2 (R) (Daubechies [34], Section 4.1).
• When T · ω 0 =2π, ifaset{h(t − mT )e −iω0nt : m, n ∈ Z} isaframeforL 2 (R), then
either ∫ x 2 |g(x)| 2 dx = ∞ or ∫ ξ 2 |ĝ(ξ)| 2 dξ = ∞ (Balian-Low [34]).
Gabor analysis is a powerful tool. It has become a foundation for the joint time frequency
analysis.