Multiplicative Noise Removal Using L1 Fidelity on Frame Coefficients

We address the denoising of images contaminated with multiplicative noise, e.g. speckle noise. Classical ways to solve such problems are filtering, statistical (Bayesian) methods, variational methods, and methods that convert the multiplicative noise into additive noise (using a logarithmic function), shrinkage of the coefficients of the log-image data in a wavelet basis or in a frame, and transform back the result using an exponential function. We propose a method composed of several stages: we use the log-image data and apply a reasonable under-optimal hard-thresholding on its curvelet transform; then we apply a variational method where we minimize a specialized criterion composed of an $\ell^1$ data-fitting to the thresholded coefficients and a Total Variation regularization (TV) term in the image domain; the restored image is an exponential of the obtained minimizer, weighted in a way that the mean of the original image is preserved. Our restored images combine the advantages of shrinkage and variational methods and avoid their main drawbacks. For the minimization stage, we propose a properly adapted fast minimization scheme based on Douglas-Rachford splitting. The existence of a minimizer of our specialized criterion being proven, we demonstrate the convergence of the minimization scheme. The obtained numerical results outperform the main alternative methods

Average performance of the sparsest approximation in a dictionary

International audienceGiven data d ∈ RN, we consider its representation u* involving the least number of non-zero elements (denoted by ℓ0(u*)) using a dictionary A (represented by a matrix) under the constraint kAu − dk ≤ τ, for τ > 0 and a norm k.k. This (nonconvex) optimization problem leads to the sparsest approximation of d. We assume that data d are uniformly distributed in θBfd (1) where θ>0 and Bfd (1) is the unit ball for a norm fd. Our main result is to estimate the probability that the data d give rise to a K−sparse solution u*: we prove that P (ℓ0(u*) ≤ K) = CK( τ θ )(N−K) + o(( τ θ )(N−K)), where u* is the sparsest approximation of the data d and CK > 0. The constants CK are an explicit function of k.k, A, fd and K which allows us to analyze the role of these parameters for the obtention of a sparsest K−sparse approximation. Consequently, given fd and θ, we have a tool to build A and k.k in such a way that CK (and hence P (ℓ0(u*) ≤ K)) are as large as possible for K small. In order to obtain the above estimate, we give a precise characterization of the set [\zigma τK] of all data leading to a K−sparse result. The main difficulty is to estimate accurately the Lebesgue measure of the sets {[\zigma τ K] ∩ Bfd (θ)}. We sketch a comparative analysis between our Average Performance in Approximation (APA) methodology and the well known Nonlinear Approximation (NA) which also assess the performance in approximation

A Convex Model for Edge-Histogram Specification with Applications to Edge-preserving Smoothing

The goal of edge-histogram specification is to find an image whose edge image has a histogram that matches a given edge-histogram as much as possible. Mignotte has proposed a non-convex model for the problem [M. Mignotte. An energy-based model for the image edge-histogram specification problem. IEEE Transactions on Image Processing, 21(1):379--386, 2012]. In his work, edge magnitudes of an input image are first modified by histogram specification to match the given edge-histogram. Then, a non-convex model is minimized to find an output image whose edge-histogram matches the modified edge-histogram. The non-convexity of the model hinders the computations and the inclusion of useful constraints such as the dynamic range constraint. In this paper, instead of considering edge magnitudes, we directly consider the image gradients and propose a convex model based on them. Furthermore, we include additional constraints in our model based on different applications. The convexity of our model allows us to compute the output image efficiently using either Alternating Direction Method of Multipliers or Fast Iterative Shrinkage-Thresholding Algorithm. We consider several applications in edge-preserving smoothing including image abstraction, edge extraction, details exaggeration, and documents scan-through removal. Numerical results are given to illustrate that our method successfully produces decent results efficiently

Non-convex optimization for 3D point source localization using a rotating point spread function

We consider the high-resolution imaging problem of 3D point source image recovery from 2D data using a method based on point spread function (PSF) engineering. The method involves a new technique, recently proposed by S.~Prasad, based on the use of a rotating PSF with a single lobe to obtain depth from defocus. The amount of rotation of the PSF encodes the depth position of the point source. Applications include high-resolution single molecule localization microscopy as well as the problem addressed in this paper on localization of space debris using a space-based telescope. The localization problem is discretized on a cubical lattice where the coordinates of nonzero entries represent the 3D locations and the values of these entries the fluxes of the point sources. Finding the locations and fluxes of the point sources is a large-scale sparse 3D inverse problem. A new nonconvex regularization method with a data-fitting term based on Kullback-Leibler (KL) divergence is proposed for 3D localization for the Poisson noise model. In addition, we propose a new scheme of estimation of the source fluxes from the KL data-fitting term. Numerical experiments illustrate the efficiency and stability of the algorithms that are trained on a random subset of image data before being applied to other images. Our 3D localization algorithms can be readily applied to other kinds of depth-encoding PSFs as well.Comment: 28 page