Wavelets, Fractals, and Radial Basis Functions

Wavelets and radial basis functions (RBFs) lead to two distinct ways of representing signals in terms of shifted basis functions. RBFs, unlike wavelets, are nonlocal and do not involve any scaling, which makes them applicable to nonuniform grids. Despite these fundamental differences, we show that the two types of representation are closely linked together …through fractals. First, we identify and characterize the whole class of self-similar radial basis functions that can be localized to yield conventional multiresolution wavelet bases. Conversely, we prove that for any compactly supported scaling function φ(x), there exists a one-sided central basis function $\rho _{ + }(x)$ that spans the same multiresolution subspaces. The central property is that the multiresolution bases are generated by simple translation of $\rho _{ + }$ without any dilation. We also present an explicit time-domain representation of a scaling function as a sum of harmonic splines. The leading term in the decomposition corresponds to the fractional splines: a recent, continuous-order generalization of the polynomial splines.

Wavelet Theory Demystified

Thierry Blu, Michaël Unser

In this paper, we revisit wavelet theory starting from the representation of a scaling function as the convolution of a B-spline (the regular part of it) and a distribution (the irregular or residual part). This formulation leads to some new insights on wavelets and makes it possible to rederive the main results of the classical theory—including some new extensions for fractional orders—in a self-contained, accessible fashion. In particular, we prove that the B-spline component is entirely responsible for five key wavelet properties: order of approximation, reproduction of polynomials, vanishing moments, multiscale differentiation property, and smoothness (regularity) of the basis functions. We also investigate the interaction of wavelets with differential operators giving explicit time domain formulas for the fractional derivatives of the basis functions. This allows us to specify a corresponding dual wavelet basis and helps us understand why the wavelet transform provides a stable characterization of the derivatives of a signal. Additional results include a new peeling theory of smoothness, leading to the extended notion of wavelet differentiability in the

L _{ p }

-sense and a sharper theorem stating that smoothness implies order.

IEEE2003

A Time-Domain Wavelet-Based Approach for Fluorescence Diffuse Optical Tomography

Chandra Sekhar Seelamantula, Michaël Unser

Purpose: In the context of fluorescence diffuse optical tomography, determining the optimal way to exploit the time-resolved information has been receiving much attention and different features of the time-resolved signals have been introduced. In this article, the authors revisit and generalize the notion of feature, considering the projection of the measurements onto some basis functions. This leads the authors to propose a novel approach based on the wavelet transform of the measurements. Methods: A comparative study between the reconstructions obtained from the proposed wavelet-based approach and the reconstructions obtained from the reference temporal moments is provided. An inhomogeneous cubic medium is considered. Reconstructions are performed from synthetic measurements assuming Poisson noise statistics. In order to provide fairly comparable reconstructions, the reconstruction scheme is associated with a particular procedure for selecting the regularization parameter. Results: In the noise-free case, the reconstruction quality is shown to be mainly driven by the number of selected features. In the presence of noise, however, the reconstruction quality depends on the type of the features. In this case, the wavelet approach is shown to outperform the moment approach. While the optimal time-resolved reconstruction quality, which is obtained considering the whole set of time samples, is recovered using only eight wavelet functions, it cannot be attained using moments. It is finally observed that the time-resolved information is of limited utility, in terms of reconstruction, when the maximum number of detected photons is lower than 105. Conclusions: The wavelet approach allows for better exploiting the time-resolved information, especially when the number of detected photons is low. However, when the number of detected photons decreases below a certain threshold, the time-resolved information itself is shown to be of limited utility.

AAPM2010