An iterative thresholding algorithm for linear inverse problems with a sparsity constraint

We consider linear inverse problems where the solution is assumed to have a sparse expansion on an arbitrary pre-assigned orthonormal basis. We prove that replacing the usual quadratic regularizing penalties by weighted l^p-penalties on the coefficients of such expansions, with 1 < or = p < or =2, still regularizes the problem. If p < 2, regularized solutions of such l^p-penalized problems will have sparser expansions, with respect to the basis under consideration. To compute the corresponding regularized solutions we propose an iterative algorithm that amounts to a Landweber iteration with thresholding (or nonlinear shrinkage) applied at each iteration step. We prove that this algorithm converges in norm. We also review some potential applications of this method.Comment: 30 pages, 3 figures; this is version 2 - changes with respect to v1: small correction in proof (but not statement of) lemma 3.15; description of Besov spaces in intro and app A clarified (and corrected); smaller pointsize (making 30 instead of 38 pages

Wavelet Electrodynamics I

A new representation for solutions of Maxwell's equations is derived. Instead of being expanded in plane waves, the solutions are given as linear superpositions of spherical wavelets dynamically adapted to the Maxwell field and well-localized in space at the initial time. The wavelet representation of a solution is analogous to its Fourier representation, but has the advantage of being local. It is closely related to the relativistic coherent-state representations for the Klein-Gordon and Dirac fields developed in earlier work.Comment: 8 Pages in Plain Te

Deconvolution under Poisson noise using exact data fidelity and synthesis or analysis sparsity priors

In this paper, we propose a Bayesian MAP estimator for solving the deconvolution problems when the observations are corrupted by Poisson noise. Towards this goal, a proper data fidelity term (log-likelihood) is introduced to reflect the Poisson statistics of the noise. On the other hand, as a prior, the images to restore are assumed to be positive and sparsely represented in a dictionary of waveforms such as wavelets or curvelets. Both analysis and synthesis-type sparsity priors are considered. Piecing together the data fidelity and the prior terms, the deconvolution problem boils down to the minimization of non-smooth convex functionals (for each prior). We establish the well-posedness of each optimization problem, characterize the corresponding minimizers, and solve them by means of proximal splitting algorithms originating from the realm of non-smooth convex optimization theory. Experimental results are conducted to demonstrate the potential applicability of the proposed algorithms to astronomical imaging datasets

Symmetry, Hamiltonian Problems and Wavelets in Accelerator Physics

In this paper we consider applications of methods from wavelet analysis to nonlinear dynamical problems related to accelerator physics. In our approach we take into account underlying algebraical, geometrical and topological structures of corresponding problems.Comment: LaTeX2e, aipproc.sty, 25 pages, typos correcte