We introduce a new iterative regularization procedure for inverse problems based on the use of Bregman distances, with particular focus on problems arising in image processing. We are motivated by the problem of restoring noisy and blurry images via variational methods by using total variation regularization. We obtain rigorous convergence results and effective stopping criteria for the general procedure. The numerical results for denoising appear to give significant improvement over standard models, and preliminary results for deblurring/denoising are very encouraging.

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An Iterative Regularization Method for Total Variation-Based Image Restoration

This paper focuses on the problem of multiplicative noise removal. We draw our inspiration from the modeling of speckle noise. By using a MAP estimator, we can derive a functional whose minimizer corresponds to the denoised image we want to recover. Although the functional is not convex, we prove the existence of a minimizer and we show the capability of our model on some numerical examples. We study the associated evolution problem, for which we derive existence and uniqueness results for the solution. We prove the convergence of an implicit scheme to compute the solution.

A Variational Approach to Removing Multiplicative Noise

Speckle is a granular disturbance, usually modeled as a multiplicative noise, that affects synthetic aperture radar (SAR) images, as well as all coherent images. Over the last three decades, several methods have been proposed for the reduction of speckle, or despeckling, in SAR images. Goal of this paper is making a comprehensive review of despeckling methods since their birth, over thirty years ago, highlighting trends and changing approaches over years. The concept of fully developed speckle is explained. Drawbacks of homomorphic filtering are pointed out. Assets of multiresolution despeckling, as opposite to spatial-domain despeckling, are highlighted. Also advantages of undecimated, or stationary, wavelet transforms over decimated ones are discussed. Bayesian estimators and probability density function (pdf) models in both spatial and multiresolution domains are reviewed. Scale-space varying pdf models, as opposite to scale varying models, are promoted. Promising methods following non-Bayesian approaches, like nonlocal (NL) filtering and total variation (TV) regularization, are reviewed and compared to spatial- and wavelet-domain Bayesian filters. Both established and new trends for assessment of despeckling are presented. A few experiments on simulated data and real COSMO-SkyMed SAR images highlight, on one side the costperformance tradeoff of the different methods, on the other side the effectiveness of solutions purposely designed for SAR heterogeneity and not fully developed speckle. Eventually, upcoming methods based on new concepts of signal processing, like compressive sensing, are foreseen as a new generation of despeckling, after spatial-domain and multiresolution-domain methods.

A Tutorial on Speckle Reduction in Synthetic Aperture Radar Images

This paper investigates the use of regularization functionals with nonlocal correlation terms for the problem of image denoising and image deblurring. These functionals are expressed as integrals o...

Deblurring and Denoising of Images by Nonlocal Functionals

Ultrasound imaging systems provide the clinician with noninvasive, low-cost, and real-time images that can help them in diagnosis, planning, and therapy. However, although the human eye is able to derive the meaningful information from these images, automatic processing is very difficult due to noise and artifacts present in the image. The speckle reducing anisotropic diffusion filter was recently proposed to adapt the anisotropic diffusion filter to the characteristics of the speckle noise present in the ultrasound images and to facilitate automatic processing of images. We analyze the properties of the numerical scheme associated with this filter, using a semi-explicit scheme. We then extend the filter to a matrix anisotropic diffusion, allowing different levels of filtering across the image contours and in the principal curvature directions. We also show a relation between the local directional variance of the image intensity and the local geometry of the image, which can justify the choice of the gradient and the principal curvature directions as a basis for the diffusion matrix. Finally, different filtering techniques are compared on a 2-D synthetic image with two different levels of multiplicative noise and on a 3-D synthetic image of a Y-junction, and the new filter is applied on a 3-D real ultrasound image of the liver

Oriented Speckle Reducing Anisotropic Diffusion

In [447, 449, 450], a constrained optimization type of numerical algorithm for restoring blurry, noisy images was developed and successfully tested. In this paper we present both theoretical and experimental justification for the method. Our main theoretical results involve constrained nonlinear partial differential equations. Our main experimental results involve blurry images which have been further corrupted with multiplicative noise. As in additive noise case of [447, 450] our numerical algorithm is simple to implement and is nonoscillatory (minimal ringing) and noninvasive (recovers sharp edges).

Pierre-Luis Lions

Papers

Multiplicative Denoising and Deblurring: Theory and Algorithms