sparse image representation via combined transforms - Convex ...
sparse image representation via combined transforms - Convex ... 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.
3.4. OTHER TRANSFORMS 69 Chirplets A chirp is a signal whose instantaneous frequency is a linear function of the time. A detailed and concise definition of instantaneous frequency is in Flandrin [68]. A chirplet can be viewed as a piece of a chirp, or equivalently a windowized version of a chirp. The following is an example of a chirplet function: h(t − nT )e iq(t) , (3.37) where h is still a window function, T is the time sampling rate, and q(t) is a quadratic polynomial of t. An example of h is the Gaussian function h(t) =e −t2 . Note h is a real function. Suppose the general form for the quadratic polynomial is q(t) =a 2 t 2 + a 1 t + a 0 . The instantaneous frequency is (in this case) q ′ (t) =2a 2 t + a 1 , which is linear in t. On the TFP, a support of a chirplet is a needle-like atom. Figure 3.7(e) shows a chirplet. An early reference on this subject is [109]. A more general but complicated way to formulate chirplets is described in [103]. An idealistic description of the chirplet formulation in [103] is that we consider the chirplet as a quadrilateral on the time frequency plane. Since each corner of the quadrilateral has two degrees of freedom, the degrees of freedom under this formulation can go up to 8. A signal associated with this quadrilateral is actually a transform of a signal associated with a rectangle in TFP; for example, the original signal could be a Gabor basis function. There are eight allowed transforms: (1) translation in time, (2) dilation in time, (3) translation in frequency, also called modulation, (4) dilation in frequency, (5) shear in time (Fourier transformation, followed by a multiplication by a chirp, then an inverse Fourier transformation), (6) shear in frequency (multiplication by a chirp), (7) perspective projection in the time domain, and finally (8) perspective projection in the frequency domain. Generally, a chirplet system is overcomplete. The following is about how to build a chirplet-like orthonormal basis. It is instructive to see that a new orthonormal basis can be built out of an existing orthonormal basis. The following idea is described in [7]. They introduced an operator called axis warping: iff(x) is a function in L 2 (R), the axis warping of function f(x) is (W c f)(x) =|c| 1/2 |x| (c−1)/2 f(|x| c sign(x)),
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3.4. OTHER TRANSFORMS 69<br />
Chirplets<br />
A chirp is a signal whose instantaneous frequency is a linear function of the time. A<br />
detailed and concise definition of instantaneous frequency is in Flandrin [68]. A chirplet<br />
can be viewed as a piece of a chirp, or equivalently a windowized version of a chirp. The<br />
following is an example of a chirplet function:<br />
h(t − nT )e iq(t) , (3.37)<br />
where h is still a window function, T is the time sampling rate, and q(t) is a quadratic<br />
polynomial of t. An example of h is the Gaussian function h(t) =e −t2 . Note h is a real<br />
function. Suppose the general form for the quadratic polynomial is q(t) =a 2 t 2 + a 1 t + a 0 .<br />
The instantaneous frequency is (in this case) q ′ (t) =2a 2 t + a 1 , which is linear in t.<br />
On the TFP, a support of a chirplet is a needle-like atom. Figure 3.7(e) shows a chirplet.<br />
An early reference on this subject is [109]. A more general but complicated way to formulate<br />
chirplets is described in [103]. An idealistic description of the chirplet formulation in [103]<br />
is that we consider the chirplet as a quadrilateral on the time frequency plane. Since<br />
each corner of the quadrilateral has two degrees of freedom, the degrees of freedom under<br />
this formulation can go up to 8. A signal associated with this quadrilateral is actually a<br />
transform of a signal associated with a rectangle in TFP; for example, the original signal<br />
could be a Gabor basis function. There are eight allowed <strong>transforms</strong>: (1) translation in<br />
time, (2) dilation in time, (3) translation in frequency, also called modulation, (4) dilation<br />
in frequency, (5) shear in time (Fourier transformation, followed by a multiplication by a<br />
chirp, then an inverse Fourier transformation), (6) shear in frequency (multiplication by a<br />
chirp), (7) perspective projection in the time domain, and finally (8) perspective projection<br />
in the frequency domain.<br />
Generally, a chirplet system is overcomplete. The following is about how to build a<br />
chirplet-like orthonormal basis. It is instructive to see that a new orthonormal basis can<br />
be built out of an existing orthonormal basis. The following idea is described in [7]. They<br />
introduced an operator called axis warping: iff(x) is a function in L 2 (R), the axis warping<br />
of function f(x) is<br />
(W c f)(x) =|c| 1/2 |x| (c−1)/2 f(|x| c sign(x)),
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