On the Optimality of Operator-Like Wavelets for Sparse AR(1) Processes

Sinusoidal transforms such as the DCT are known to be optimal—that is, asymptotically equivalent to the Karhunen-Loève 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 (SαS) noise which is either Gaussian (α = 2) or sparse (0 < α < 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 (α ≤ 1), the operator-like wavelet transform is indistinguishable from the ICA solution that is determined through numerical optimization.

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

Linear Compressive Networks

Michael Christoph Gastpar, Naveen Goela

A linear compressive network (LCN) is defined as a graph of sensors in which each encoding sensor compresses incoming jointly Gaussian random signals and transmits (potentially) low-dimensional linear projections to neighbors over a noisy uncoded channel. Each sensor has a maximum power to allocate over signal subspaces. The networks of focus are acyclic, directed graphs with multiple sources and multiple destinations. LCN pathways lead to decoding leaf nodes that estimate linear functions of the original high dimensional sources by minimizing a mean squared error (MSE) distortion cost function. An iterative Optimization of local compressive matrices for all graph nodes is developed using an optimal quadratically constrained quadratic program (QCQP) step. The performance of the optimization is marked by power-compression-distortion spectra, with converse bounds based on cut-set arguments. Examples include single layer and multi-layer (e.g. p-layer tree cascades, butterfly) networks. The LCN is a generalization of the Karhunen-Loeve Transform to noisy multi-layer networks, and extends previous approaches for point-to-point and distributed compression-estimation of Gaussian signals. The framework relates to network coding in the noiseless case, and uncoded transmission in the noisy case.

Ieee Service Center, 445 Hoes Lane, Po Box 1331, Piscataway, Nj 08855-1331 Usa2009

Wavelets on non-Euclidean manifolds

Iva Bogdanova Vandergheynst

EPFL2006