Inverse Problems & Imaging
August 2008 , Volume 2 , Issue 3
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We consider a Cauchy problem for the Laplace equation in a two-dimensional semi-infinite region with a bounded inclusion, i.e. the region is the intersection between a half-plane and the exterior of a bounded closed curve contained in the half-plane. The Cauchy data are given on the unbounded part of the boundary of the region and the aim is to construct the solution on the boundary of the inclusion. In 1989, Kozlov and Maz'ya  proposed an alternating iterative method for solving Cauchy problems for general strongly elliptic and formally self-adjoint systems in bounded domains. We extend their approach to our setting and in each iteration step mixed boundary value problems for the Laplace equation in the semi-infinite region are solved. Well-posedness of these mixed problems are investigated and convergence of the alternating procedure is examined. For the numerical implementation an efficient boundary integral equation method is proposed, based on the indirect variant of the boundary integral equation approach. The mixed problems are reduced to integral equations over the (bounded) boundary of the inclusion. Numerical examples are included showing the feasibility of the proposed method.
We consider scattering by a smooth obstacle in $R^d$, $d\geq 3 $ odd. We show that for the Neumann Laplacian if an obstacle has the same resonances as the ball of radius $\rho$ does, then the obstacle is a ball of radius $\rho$. We give related results for obstacles which are disjoint unions of several balls of the same radius.
Integral invariants have been proven to be useful for shape matching and recognition, but fundamental mathematical questions have not been addressed in the computer vision literature. In this article we are concerned with the identifiability and numerical algorithms for the reconstruction of a star-shaped object from its integral invariants. In particular we analyse two integral invariants and prove injectivity for one of them. Additionally, numerical experiments are performed.
In various imaging problems the task is to use the Cauchy data of the solutions to an elliptic boundary value problem to reconstruct the coefficients of the corresponding partial differential equation. Often the examined object has known background properties but is contaminated by inhomogeneities that cause perturbations of the coefficient functions. The factorization method of Kirsch provides a tool for locating such inclusions. In this paper, the factorization technique is studied in the framework of coercive elliptic partial differential equations of the divergence type: Earlier it has been demonstrated that the factorization algorithm can reconstruct the support of a strictly positive (or negative) definite perturbation of the leading order coefficient, or if that remains unperturbed, the support of a strictly positive (or negative) perturbation of the zeroth order coefficient. In this work we show that these two types of inhomogeneities can, in fact, be located simultaneously. Unlike in the earlier articles on the factorization method, our inclusions may have disconnected complements and we also weaken some other a priori assumptions of the method. Our theoretical findings are complemented by two-dimensional numerical experiments that are presented in the framework of the diffusion approximation of optical tomography.
We reconsider the Linear Sampling Method by Colton and Kirsch, and provide an analysis which may serve as a justification of the method for problems where the Factorization Method is known to work. As a by-product, however, we obtain convincing arguments that one popular implementation of the Linear Sampling Method may not be as robust as is commonly believed. Our approach stems from the theory of regularization methods for linear ill-posed operator equations. More precisely, we derive a novel asymptotic analysis of the Tikhonov method if the exact right-hand side is inconsistent, i.e., does not belong to the (dense) range of the corresponding operator. It appears possible that our results can be a starting point to derive a calibration of standard implementations of the Linear Sampling Method, in order to obtain reconstructions of the scattering obstacles that go beyond an approximate localization of their respective positions.
We consider the regularization of the inverse conductivity problem with discontinuous conductivities, like for example the so-called inclusion problem. We theoretically validate the use of some of the most widely adopted regularization operators, like for instance total variation and the Mumford-Shah functional, by proving a convergence result for the solutions to the regularized minimum problems.
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