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20
An Analysis of Electrical Impedance Tomography with Applications to Tikhonov Regularization
, 2010
"... The consecutive numbering of the publications is determined by their chronological order. The aim of this preprint series is to make new research rapidly available for scientific discussion. Therefore, the responsibility for the contents is solely due to the authors. The publications will be distrib ..."
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Cited by 74 (0 self)
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The consecutive numbering of the publications is determined by their chronological order. The aim of this preprint series is to make new research rapidly available for scientific discussion. Therefore, the responsibility for the contents is solely due to the authors. The publications will be distributed by the authors.
Image deblurring in the presence of saltandpepper noise
 Lecture Notes in Computer Science
, 2005
"... 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 ..."
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Cited by 38 (4 self)
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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 mediantype filtering is however difficult, due to the amplification (in the deconvolution stage) of medianinduced 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 piecewisesmooth image model is formulated. A modified L 1 data fidelity term integrates deblurring with robustness to outliers. Elements from the MumfordShah functional, that favor piecewise smooth images with simple edgesets, are used for regularization. Promising experimental results are shown for several blur models. 1
Electrical Impedance Tomography Using Level Set Representation and Total Variational Regularization
, 2003
"... 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#er ..."
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Cited by 29 (2 self)
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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.
Image deblurring in the presence of impulsive noise
 Int. J. Comput. Vision
, 2006
"... 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 impu ..."
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Cited by 28 (2 self)
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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. Medianbased 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 gradientbased edgepreserving 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 MumfordShah functional are used for regularization. We show that the MumfordShah 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
A SURVEY ON MULTIPLE LEVEL SET METHODS WITH APPLICATIONS FOR IDENTIFYING PIECEWISE CONSTANT FUNCTIONS
, 2004
"... 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 g ..."
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Cited by 26 (8 self)
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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.
Probing for electrical inclusions with complex spherical waves
 Comm. Pure Appl. Math
"... Abstract. Let a physical body Ω in R2 or R3 be given. Assume that the electric conductivity distribution inside Ω consists of conductive inclusions in a known smooth background. Further, assume that a subset Γ ⊂ ∂Ω is available for boundary measurements. It is proved using hyperbolic geometry that ..."
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Cited by 16 (9 self)
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Abstract. Let a physical body Ω in R2 or R3 be given. Assume that the electric conductivity distribution inside Ω consists of conductive inclusions in a known smooth background. Further, assume that a subset Γ ⊂ ∂Ω is available for boundary measurements. It is proved using hyperbolic geometry that certain information about the location of the inclusions can be exactly recovered from static electric measurements on Γ. More precisely: given a ball B with center outside the convex hull of Ω and satisfying (B∩∂Ω) ⊂ Γ, boundary measurements on Γ with explicitly given Dirichlet data are enough to determine whether or not B intersects the inclusion. An approximate detection algorithm is introduced based on the theory. Numerical experiments in dimension two with simulated noisy data suggest that the algorithm finds the inclusionfree domain near Γ and is robust against measurement noise.
Blind Deconvolution of Bar Code Signals
, 2004
"... Bar code reconstruction involves recovering a clean signal from an observed one that is corrupted by convolution with a kernel and additive noise. The precise form of the convolution kernel is also unknown, making reconstruction harder than in the case of standard deblurring. On the other hand, b ..."
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Cited by 9 (0 self)
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Bar code reconstruction involves recovering a clean signal from an observed one that is corrupted by convolution with a kernel and additive noise. The precise form of the convolution kernel is also unknown, making reconstruction harder than in the case of standard deblurring. On the other hand, bar codes are functions that have a very special form  this makes reconstruction feasible. We develop and analyze a total variation based variational model for the solution of this problem. This new technique models systematically the interaction of neighboring bars in the bar code under convolution with a kernel, as well as the estimation of the unknown parameters of the kernel from global information contained in the observed signal.
HIERARCHICAL MODELS IN STATISTICAL INVERSE PROBLEMS AND THE MUMFORD–SHAH FUNCTIONAL ∗
, 908
"... 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 ..."
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Cited by 5 (0 self)
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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, edgepreserving reconstruction. AMS subject classifications. 60F17, 35A15, 65C20 1. Introduction. We
EXISTENCE OF MINIMIZERS OF THE MUMFORDSHAH FUNCTIONAL WITH SINGULAR OPERATORS IN TWO SPACE DIMENSIONS
"... 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 MumfordShah functional term. We are partially inspired by classical linea ..."
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Cited by 4 (1 self)
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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 MumfordShah 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. MumfordShah functional, inverse problems, firstorder differential equations, inpainting, deblurring, image restoration AMS subject classifications. 49J99, 35J70, 49N60, 65J22, 68U10 1. Introduction. Freediscontinuity 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