Sampling in the Analysis Transform Domain

Many signal and image processing applications have benefited remarkably from the fact that the underlying signals reside in a low dimensional subspace. One of the main models for such a low dimensionality is the sparsity one. Within this framework there are two main options for the sparse modeling: the synthesis and the analysis ones, where the first is considered the standard paradigm for which much more research has been dedicated. In it the signals are assumed to have a sparse representation under a given dictionary. On the other hand, in the analysis approach the sparsity is measured in the coefficients of the signal after applying a certain transformation, the analysis dictionary, on it. Though several algorithms with some theory have been developed for this framework, they are outnumbered by the ones proposed for the synthesis methodology. Given that the analysis dictionary is either a frame or the two dimensional finite difference operator, we propose a new sampling scheme for signals from the analysis model that allows to recover them from their samples using any existing algorithm from the synthesis model. The advantage of this new sampling strategy is that it makes the existing synthesis methods with their theory also available for signals from the analysis framework.Comment: 13 Pages, 2 figure

Strong Solutions for Stochastic Partial Differential Equations of Gradient Type

Unique existence of analytically strong solutions to stochastic partial differential equations (SPDE) with drift given by the subdifferential of a quasi-convex function and with general multiplicative noise is proven. The proof applies a genuinely new method of weighted Galerkin approximations based on the "distance" defined by the quasi-convex function. Spatial regularization of the initial condition analogous to the deterministic case is obtained. The results yield a unified framework which is applied to stochastic generalized porous media equations, stochastic generalized reaction diffusion equations and stochastic generalized degenerated p-Laplace equations. In particular, higher regularity for solutions of such SPDE is obtained.Comment: 30 page

Image interpolation using Shearlet based iterative refinement

This paper proposes an image interpolation algorithm exploiting sparse representation for natural images. It involves three main steps: (a) obtaining an initial estimate of the high resolution image using linear methods like FIR filtering, (b) promoting sparsity in a selected dictionary through iterative thresholding, and (c) extracting high frequency information from the approximation to refine the initial estimate. For the sparse modeling, a shearlet dictionary is chosen to yield a multiscale directional representation. The proposed algorithm is compared to several state-of-the-art methods to assess its objective as well as subjective performance. Compared to the cubic spline interpolation method, an average PSNR gain of around 0.8 dB is observed over a dataset of 200 images

Pattern fluctuations in transitional plane Couette flow

In wide enough systems, plane Couette flow, the flow established between two parallel plates translating in opposite directions, displays alternatively turbulent and laminar oblique bands in a given range of Reynolds numbers R. We show that in periodic domains that contain a few bands, for given values of R and size, the orientation and the wavelength of this pattern can fluctuate in time. A procedure is defined to detect well-oriented episodes and to determine the statistics of their lifetimes. The latter turn out to be distributed according to exponentially decreasing laws. This statistics is interpreted in terms of an activated process described by a Langevin equation whose deterministic part is a standard Landau model for two interacting complex amplitudes whereas the noise arises from the turbulent background.Comment: 13 pages, 11 figures. Accepted for publication in Journal of statistical physic

DOLPHIn - Dictionary Learning for Phase Retrieval

We propose a new algorithm to learn a dictionary for reconstructing and sparsely encoding signals from measurements without phase. Specifically, we consider the task of estimating a two-dimensional image from squared-magnitude measurements of a complex-valued linear transformation of the original image. Several recent phase retrieval algorithms exploit underlying sparsity of the unknown signal in order to improve recovery performance. In this work, we consider such a sparse signal prior in the context of phase retrieval, when the sparsifying dictionary is not known in advance. Our algorithm jointly reconstructs the unknown signal - possibly corrupted by noise - and learns a dictionary such that each patch of the estimated image can be sparsely represented. Numerical experiments demonstrate that our approach can obtain significantly better reconstructions for phase retrieval problems with noise than methods that cannot exploit such "hidden" sparsity. Moreover, on the theoretical side, we provide a convergence result for our method