Other Multiscale Representations

Wavelet transforms are part of a general framework of multiscale analysis. Various multiscale representations have been derived from the spatial-frequency framework offered by wavelet expansion, many of which were introduced to provide more flexibility for the spatial-frequency selectivity or better adaptation to real-world applications.

In this section, we briefly review several multiscale representations derived from wavelet transforms. Readers with an intention to investigate more theoretical and technical details are referred to the textbooks on Gabor analysis [20], wavelet packets [21], and the original paper on brushlet [22].

6.2.3.1 Gabor Transform and Gabor Wavelets

In his early work, Gabor [23] suggested an expansion of a signal s(t) in terms of time-frequency atoms gmn(t) defined as:

m, n where gmn(t), m,n e Z, are constructed with a window function g(x), combined to a complex exponential:

Gabor also suggested that an appropriate choice for the window function g(x) is the Gaussian function due to the fact that a Gaussian function has the theoretically best joint spatial-frequency resolution (uncertainty principle). It is important to note here that the Gabor elementary functions gmn(t) are not orthogonal and therefore require a biorthogonal dual function y (x) for reconstruction [24]. This dual window function is used for the computation of the expansion coefficients cmn as:

while the Gaussian window is used for the reconstruction.

The biorthogonality of the two window functions y (x) and g(x) is expressed as:

From Eq. (6.21), it is easy to see that all spatial-frequency atom gmn(t) share the same spatial-frequency resolution defined by the Gaussian function g(x). As pointed out in the discussion on short-time Fourier transforms, such design is suboptimal for the analysis of signals with different frequency components.

A wavelet-type generalization of Gabor expansion can be constructed such that different window functions are used instead of a single one [25] according to their spatial-frequency location. Following the design of wavelets, a Gabor wavelet ^ (x) = g(t)eiT]t is then obtained with a Gaussian function

Extension of Gabor wavelet to 2D is expressed as:

Different translation and scaling parameters of ^k(x, y) constitute the wavelet basis for expansion. An extra parameter ak provides selectivity for the orientation of the function. We observe here that the 2D Gabor wavelet has a non-separable structure that provides more flexibility on orientation selection than separable wavelet functions.

It is well known that optical sensitive cells in animal's visual cortex respond selectively to stimuli with particular frequency and orientation [26]. Equation (6.24) described a wavelet representation that naturally reflects this neurophysi-ological phenomenon. Gabor expansion and Gabor wavelets have therefore been widely used for visual discrimination tasks and especially texture recognition [27, 28].

6.2.3.2 Wavelet Packets

Unlike dyadic wavelet transform, wavelet packets decompose the low-frequency component as well as the high-frequency component in every subbands [29]. Such adaptive expansion can be represented with binary trees where each subband high- or low-frequency component is a node with two children corresponding to the pair of high- and low-frequency expansion at the next scale. An admissible tree for an adaptive expansion is therefore defined as a binary tree where each node has either 0 or 2 children, as illustrated in Fig. 6.6(c). The number of all different wavelet packet orthogonal basis (also called a wavelet packets dictionary) is equal to the number of different admissible binary trees, which is of the order of 22J, where J is the depth of decomposition [14].

Obviously, wavelet packets provide more flexibility on partitioning the spatial-frequency domain, and therefore improve the separation of noise and signal into different subbands in an approximated sense (this is referred to the near-diagonalization of signal and noise). This property can greatly facilitate

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Figure 6.6: (a) Dyadic wavelet decomposition tree. (b) Wavelet packets decomposition tree. (c) An example of an orthogonal basis tree with wavelet packets decomposition.

the enhancement and denoising task of a noisy signal if the wavelet packets basis are selected properly [30]. In practical applications for various medical imaging modalities and applications, features of interest and noise properties have significantly different characteristics that can be efficiently characterized separately with this framework.

A fast algorithm for wavelet-packets best basis selection was introduced by Coifman and Wickerhauser in [30]. This algorithm identifies the "best" basis for a specific problem inside the wavelet packets dictionary according to a criterion (referred to as a cost function) that is minimized. This cost function typically reflects the entropy of the coefficients or the energy of the coefficients inside each subband and the optimal choice minimizes the cost function comparing values at a node and its children. The complexity of the algorithm is O (N log N) for a signal of N samples.

6.2.3.3 Brushlets

Brushlet functions were introduced to build an orthogonal basis of transient functions with good time-frequency localization. For this purpose, lapped orthogonal transforms with windowed complex exponential functions, such as Gabor functions, have been used for many years in the context of sine-cosine transforms [31].

Brushlet functions are defined with true complex exponential functions on subintervals of the real axis as:

Uj,n(x) = bn(x - Cn)ejn(x) + v(x - an)ej^n(2an - x)-v(x - an+i)ej^n(2an+i -x),

where ln = an+1 — an and cn = ln/2. The two window functions bn and v are derived from the ramp function r:

The bump function v is defined as:

The bell function bn is defined by:

An illustration of the windowing functions is provided in Fig. 6.7. Finally, the complex-valued exponentials ejn are defined as:

In order to decompose a given signal f along directional texture components, the Fourier transform f of the signal and not the signal itself is projected on the

Figure 6.7: Windowing functions bn and bump functions v defined on the interval [an — e, an+1 + e].

brushlet basis functions:

n j with Un^j being the brushlet basis functions and fj being the brushlet coefficients.

The original signal f can then be reconstructed by:

nj where j is the inverse Fourier transform of un j, which is expressed as:

wn, j (x) = VTne2™ anXeinlnX^y (-1)jbn(lnX - j) - 2i sin(n lnX)v (InX + j)^ ,

with bn and v being the Fourier transforms of the window functions bn and v. Since the Fourier operator is a unitary operator, the family of functions j is also an orthogonal basis of the real axis. We observe here the wavelet-like structure of the j functions with scaling factor ln and translation factor j. An illustration of the brushlet analysis and synthesis functions is provided in Fig. 6.8.

Projection on the analysis functions un j can be implemented efficiently by a folding operator and Fourier transform. The folding technique was introduced by Malvar [31] and is described for multidimensional implementation by Wick-erhauser in [21]. These brushlet functions share many common properties with Gabor wavelets and wavelet packets regarding the orientation and frequency selection of the analysis but only brushlet can offer an orthogonal framework

Figure 6.8: (a) Real part of analysis brushlet function j. (b) Real part of synthesis brushlet function wy j.

with a single expansion coefficient for a particular pair of frequency and orientation.

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