Optimized Wavelet Denoising for Self-Similar alpha-Stable Processes

We investigate the performance of wavelet shrinkage methods for the denoising of symmetric-a-stable (S alpha S) self-similar stochastic processes corrupted by additive white Gaussian noise (AWGN), where a is tied to the sparsity of the process. The wavelet transform is assumed to be orthonormal and the shrinkage function minimizes the mean-square approximation error (MMSE estimator). We derive the corresponding formula for the expected value of the averaged estimation error. We show that the predicted MMSE is a monotone function of a simple criterion that depends on the wavelet and the statistical parameters of the process. Using the calculus of variations, we then optimize this criterion to find the best performing wavelet within the extended family of Meyer wavelets, which are bandlimited. These are compared with the Daubechies wavelets, which are compactly supported in time. We find that the wavelets that are shorter in time (in particular, the Haar basis) are better suited to denoise the sparser processes (say, alpha < 1.2), while the bandlimited ones (including the Held and Shannon wavelets) offer the best performance for alpha > 1.6, the limit corresponding to the Gaussian case (fBm) with alpha = 2.

On The Optimality Of Operator-Like Wavelets For Sparse Ar(1) Processes

Pedram Pad, Michaël Unser

Sinusoidal transforms such as the DCT are known to be optimal-that is, asymptotically equivalent to the Karhunen-Loeve transform (KLT)-for the representation of Gaussian stationary processes, including the classical AR(1) processes. While the KLT remains applicable for non-Gaussian signals, it loses optimality and, is outperformed by the independent-component analysis (ICA), which aims at producing the most-decoupled representation. In this paper, we consider an extension of the classical AR(1) model that is driven by symmetric-alpha-stable (S alpha S) noise which is either Gaussian (alpha = 2) or sparse (0 < alpha < 2). For the sparse (non-Gaussian) regime, we prove that an expansion in a proper wavelet basis (including the Haar transform) is much closer to the optimal orthogonal ICA solution than the classical Fourier-type representations. Our criterion for optimality, which favors independence, is the Kullback-Leibler divergence between the joint pdf of the original signal and the product of the marginals in the transformed domain. We also observe that, for very sparse AR(1) processes (alpha

Ieee2013

Wavelets on non-Euclidean manifolds

Iva Bogdanova Vandergheynst

This dissertation investigates wavelets as a multiscale tool on non-Euclidean manifolds. The growing importance of using non-Euclidean manifolds as a geometric model for data comes from the diversity of the data collected. In this work we mostly deal with the sphere and the hyperboloid. First, given the recent success of the continuous wavelet transform on the sphere a natural extension is to build discrete frames. Then, from a more theoretical perspective, having already wavelets on the sphere, which is a non-Euclidean manifold of constant positive curvature, it is interesting and even challenging to build and prove the existence of wavelets on its dual manifold-the hyperboloid as non-Euclidean manifold of constant negative curvature. This dissertation starts with detailing the construction of one- and two-dimensional Euclidean wavelets in both continuous and discrete versions. Next, it continues with details on the construction of wavelets on the sphere. In the three cases (line, plane and sphere) the group theoretical approach for constructing wavelets is used. We develop discrete wavelet frames on the sphere by discretizing the existing spherical continuous wavelet transform. First, half-continuous wavelet frames are derived. Second, we show that a controlled frame may be constructed in order to get an easy reconstruction of functions from their decomposition coefficients. Finally we completely discretize the continuous wavelet transform on the sphere and give examples of frame decomposition of spherical data. As a close parent of the wavelet transform we also implement the Laplacian Pyramid on the sphere. Another important part of this dissertation is dedicated to the hyperboloid. We build a total family of functions, in the space of square-integrable functions on the hyperboloid, by picking a probe with suitable localization properties, applying on it hyperbolic motions and supplemented by appropriate dilations. Based on a minimal set of axioms, we define appropriate dilations for the hyperbolic geometry. Then, the continuous wavelet transform on the hyperboloid is obtained by convolution of the scaled wavelets with the signal. This transform is proved to be a well-defined invertible map, provided the wavelets satisfy an admissibility condition. As a final part in this dissertation, we discuss one possible application of non-Euclidean wavelets – the processing of non-Euclidean images. This leads to implementing some other basic non-Euclidean image processing techniques, for example scale-space analysis and active contour, that we apply to catadioptric image processing.

EPFL2006

Wavelets on the 2-Hyperboloid

Iva Bogdanova Vandergheynst, Pierre Vandergheynst

We build wavelets on the 2-Hyperboloid. First, we define dilations on the hyperboloid through conic projection. Then, incorporating hyperbolic motions belonging to

SO_0(1,2)

, we define a family of hyperbolic wavelets. The continuous wavelet transform (CWT)is obtained by convolution of the scaled wavelets with the signal. This wavelet transform is proved to be invertible whenever wavelets satisfy a particular admissibility condition. Finally, the Euclidean limit of this CWT on the hyperboloid is considered.

2004