The continuous wavelet transform on conic sections

We review the known construction of the continuous wavelet transform (CWT) on the two-sphere. Next we describe the construction of a CWT on the upper sheet of a two- sheeted hyperboloid, emphasizing the similarities between the two cases. Finally we give some indications on the CWT on a paraboloid and we introduce a unified approach to the CWT on conic sections

Templates for Convex Cone Problems with Applications to Sparse Signal Recovery

This paper develops a general framework for solving a variety of convex cone problems that frequently arise in signal processing, machine learning, statistics, and other fields. The approach works as follows: first, determine a conic formulation of the problem; second, determine its dual; third, apply smoothing; and fourth, solve using an optimal first-order method. A merit of this approach is its flexibility: for example, all compressed sensing problems can be solved via this approach. These include models with objective functionals such as the total-variation norm, ||Wx||_1 where W is arbitrary, or a combination thereof. In addition, the paper also introduces a number of technical contributions such as a novel continuation scheme, a novel approach for controlling the step size, and some new results showing that the smooth and unsmoothed problems are sometimes formally equivalent. Combined with our framework, these lead to novel, stable and computationally efficient algorithms. For instance, our general implementation is competitive with state-of-the-art methods for solving intensively studied problems such as the LASSO. Further, numerical experiments show that one can solve the Dantzig selector problem, for which no efficient large-scale solvers exist, in a few hundred iterations. Finally, the paper is accompanied with a software release. This software is not a single, monolithic solver; rather, it is a suite of programs and routines designed to serve as building blocks for constructing complete algorithms.Comment: The TFOCS software is available at http://tfocs.stanford.edu This version has updated reference

Wavelets on the Two-Sphere and Other Conic Sections

We survey the construction of the continuous wavelet transform (CWT) on the twosphere. Then we discuss the discretization of the spherical CWT, obtaining various types of discrete frames. Finally, we give some indications on the construction of a CWT on other conic section

Continuous Wavelet Transform on the Hyperboloid

In this paper we build a Continuous Wavelet Transform (CWT) on the upper sheet of the 2-hyperboloid H_+^2. First, we define a class of suitable dilations on the hyperboloid through conic ptojection. Then, incorporating hyperbolic motions belonging to SO_0(1,2), we define a family of hyperbolic wavelets. The continuous wavelet transform W_f(a,x) is obtained by convolution of the scaled wavelets with the signal. The wavelet transform is proved to be invertible whenever wavelets satisfy a particular admissibility condition, which turns out to be a zero-mean condition. We then provide some basic examples and discuss the limit at null curvature

Time-scale analysis of abrupt changes corrupted by multiplicative noise

Multiplicative Abrupt Changes (ACs) have been considered in many applications. These applications include image processing (speckle) and random communication models (fading). Previous authors have shown that the Continuous Wavelet Transform (CWT) has good detection properties for ACs in additive noise. This work applies the CWT to AC detection in multiplicative noise. CWT translation invariance allows to define an AC signature. The problem then becomes signature detection in the time-scale domain. A second-order contrast criterion is defined as a measure of detection performance. This criterion depends upon the first- and second-order moments of the multiplicative process's CWT. An optimal wavelet (maximizing the contrast) is derived for an ideal step in white multiplicative noise. This wavelet is asymptotically optimal for smooth changes and can be approximated for small AC amplitudes by the Haar wavelet. Linear and quadratic suboptimal signature-based detectors are also studied. Closed-form threshold expressions are given as functions of the false alarm probability for three of the detectors. Detection performance is characterized using Receiver Operating Characteristic (ROC) curves computed from Monte-Carlo simulations

ℓ1\ell^1-Analysis Minimization and Generalized (Co-)Sparsity: When Does Recovery Succeed?

This paper investigates the problem of signal estimation from undersampled noisy sub-Gaussian measurements under the assumption of a cosparse model. Based on generalized notions of sparsity, we derive novel recovery guarantees for the $\ell^{1}$-analysis basis pursuit, enabling highly accurate predictions of its sample complexity. The corresponding bounds on the number of required measurements do explicitly depend on the Gram matrix of the analysis operator and therefore particularly account for its mutual coherence structure. Our findings defy conventional wisdom which promotes the sparsity of analysis coefficients as the crucial quantity to study. In fact, this common paradigm breaks down completely in many situations of practical interest, for instance, when applying a redundant (multilevel) frame as analysis prior. By extensive numerical experiments, we demonstrate that, in contrast, our theoretical sampling-rate bounds reliably capture the recovery capability of various examples, such as redundant Haar wavelets systems, total variation, or random frames. The proofs of our main results build upon recent achievements in the convex geometry of data mining problems. More precisely, we establish a sophisticated upper bound on the conic Gaussian mean width that is associated with the underlying $\ell^{1}$-analysis polytope. Due to a novel localization argument, it turns out that the presented framework naturally extends to stable recovery, allowing us to incorporate compressible coefficient sequences as well

Diffusive wavelets on groups and homogeneous spaces

The aim of this exposition is to explain basic ideas behind the concept of diffusive wavelets on spheres in the language of representation theory of Lie groups and within the framework of the group Fourier transform given by Peter-Weyl decomposition of $L^2(G)$ for a compact Lie group $G$. After developing a general concept for compact groups and their homogeneous spaces we give concrete examples for tori -which reflect the situation on $R^n$- and for spheres $S^2$ and $S^3$.Comment: 20 pages, 3 figure