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Abstract. The problem of image deblurring in the presence of salt and pepper noise is considered. Standard image deconvolution algorithms, that are designed for Gaussian noise, do not perform well in this case. Median type filtering is a common method for salt and pepper noise removal. Deblurring an image that has been preprocessed by median-type filtering is however difficult, due to the amplification (in the deconvolution stage) of median-induced distortion. A unified variational approach to salt and pepper noise removal and image deblurring is presented. An objective functional that represents the goals of deblurring, noiserobustness and compliance with the piecewise-smooth image model is formulated. A modified L 1 data fidelity term integrates deblurring with robustness to outliers. Elements from the Mumford-Shah functional, that favor piecewise smooth images with simple edge-sets, are used for regularization. Promising experimental results are shown for several blur models. 1

We try to give a brief survey about using multiple level set methods for identifying piecewise constant or piecewise smooth functions. A general framework is presented. Application using this general framework for different practical problems are shown. We try to show some details in applying the general approach for applications to: image segmentation, optimal shape design, elliptic inverse coefficient identification, electricall impedance tomography and positron emission tomography. Numerical experiments are also presented for some of the problems.

In this paper, we propose a numerical scheme for the identification of piecewise constant conductivity coe#cient for a problem arising from electrical impedance tomography. The key feature of the scheme is the use of level set method for the representation of interface between domains with di#erent values of coe#cients. Numerical tests show that our method can be able to recover a sharp interface and can tolerate higher level of noise in the observation data. Results concerning the e#ects of number of measurements, noise level in the data as well as the regularization parameters on the accuracy of the scheme are also given.

Consider the problem of image deblurring in the presence of impulsive noise. Standard image deconvolution methods rely on the Gaussian noise model and do not perform well with impulsive noise. The main challenge is to deblur the image, recover its discontinuities and at the same time remove the impulse noise. Median-based approaches are inadequate, because at high noise levels they induce nonlinear distortion that hampers the deblurring process. Distinguishing outliers from edge elements is difficult in current gradient-based edge-preserving restoration methods. The suggested approach integrates and extends the robust statistics, line process (half quadratic) and anisotropic diffusion points of view. We present a unified variational approach to image deblurring and impulse noise removal. The objective functional consists of a fidelity term and a regularizer. Data fidelity is quantified using the robust modified L 1 norm, and elements from the Mumford-Shah functional are used for regularization. We show that the Mumford-Shah regularizer can be viewed as an extended line process. It reflects spatial organization properties of the image edges, that do not appear in the common line process or anisotropic diffusion. This allows to distinguish outliers from edges and leads to superior experimental results. 1

We try to give a brief survey about using multiple level set methods for identifying piecewise constant functions. A general framework is presented. Application using this general framework for different practical problems are shown. We try to show some details in applying the general approach for different applications. Numerical experiments are also presented for some of the problems.

Reference metadata extraction using a hierarchical knowledge

Abstract. We consider the regularization of inverse problems involving certain local (differential) and nonlocal (integral) operators, by means of the minimization of a functional formed by a term of discrepancy to data and a Mumford-Shah functional term. We are partially inspired by classical linear inverse problems in image processing, namely inpainting and deblurring. We provide existence results of minimal solutions by assuming such minimizers in a reasonable class of smooth functions out of piecewise Lipschitz discontinuity sets. The compactness arguments we developed to derive the existence results stem from geometrical and regularity properties of domains, interpolation inequalities, and classical compactness arguments in Sobolev spaces. Key words. Mumford-Shah functional, inverse problems, first-order differential equations, inpainting, deblurring, image restoration AMS subject classifications. 49J99, 35J70, 49N60, 65J22, 68U10 1. Introduction. Free-discontinuity problems describe situations where the solution of interest is defined by a function and a lower dimensional set consisting of the discontinuities of the function [18]. Hence, the derivative of the solution is assumed to be a ‘small ’ function almost everywhere except on sets where it concentrates as

Abstract. The Bayesian methods for linear inverse problems is studied using hierarchical Gaussian models. The problems are considered with different discretizations, and we analyze the phenomena which appear when the discretization becomes finer. A hierarchical solution method for signal restoration problems is introduced and studied with arbitrarily fine discretization. We show that the maximum a posteriori estimate converges to a minimizer of the Mumford–Shah functional, up to a subsequence. A new result regarding the existence of a minimizer of the Mumford–Shah functional is proved. Moreover, we study the inverse problem under different assumptions on the asymptotic behavior of the noise as discretization becomes finer. We show that the maximum a posteriori and conditional mean estimates converge under different conditions. Key words. Inverse problem, Mumford–Shah functional, Bayesian inversion, hierarchical models, discretization invariance, edge-preserving reconstruction. AMS subject classifications. 60F17, 35A15, 65C20 1. Introduction. We