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Inverse Problems & Imaging

2012 , Volume 6 , Issue 3

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Transmission eigenvalues for inhomogeneous media containing obstacles
Fioralba Cakoni, Anne Cossonnière and Houssem Haddar
2012, 6(3): 373-398 doi: 10.3934/ipi.2012.6.373 +[Abstract](295) +[PDF](425.9KB)
Abstract:
We consider the interior transmission problem corresponding to the inverse scattering by an inhomogeneous (possibly anisotropic) media in which an impenetrable obstacle with Dirichlet boundary conditions is embedded. Our main focus is to understand the associated eigenvalue problem, more specifically to prove that the transmission eigenvalues form a discrete set and show that they exist. The presence of Dirichlet obstacle brings new difficulties to already complicated situation dealing with a non-selfadjoint eigenvalue problem. In this paper, we employ a variety of variational techniques under various assumptions on the index of refraction as well as the size of the Dirichlet obstacle.
Fine-tuning electrode information in electrical impedance tomography
Jérémi Dardé, Harri Hakula, Nuutti Hyvönen and Stratos Staboulis
2012, 6(3): 399-421 doi: 10.3934/ipi.2012.6.399 +[Abstract](443) +[PDF](711.2KB)
Abstract:
Electrical impedance tomography is a noninvasive imaging technique for recovering the admittivity distribution inside a body from boundary measurements of current and voltage. In practice, impedance tomography suffers from inaccurate modelling of the measurement setting: The exact electrode locations and the shape of the imaged object are not necessarily known precisely. In this work, we tackle the problem with imperfect electrode information by introducing the Fréchet derivative of the boundary measurement map of impedance tomography with respect to the electrode shapes and locations. This enables us to include the fine-tuning of the information on the electrode positions as a part of a Newton-type output least squares reconstruction algorithm; we demonstrate that this approach is feasible via a two-dimensional numerical example based on simulated data. The impedance tomography measurements are modelled by the complete electrode model, which is in good agreement with real-life electrode measurements.
Probing for inclusions in heat conductive bodies
Patricia Gaitan, Hiroshi Isozaki, Olivier Poisson, Samuli Siltanen and Janne Tamminen
2012, 6(3): 423-446 doi: 10.3934/ipi.2012.6.423 +[Abstract](305) +[PDF](994.9KB)
Abstract:
This work deals with an inverse boundary value problem arising from the equation of heat conduction. Mathematical theory and algorithm is described in dimensions 1--3 for probing the discontinuous part of the conductivity from local temperature and heat flow measurements at the boundary. The approach is based on the use of complex spherical waves, and no knowledge is needed about the initial temperature distribution. In dimension two we show how conformal transformations can be used for probing deeper than is possible with discs. Results from numerical experiments in the one-dimensional case are reported, suggesting that the method is capable of recovering locations of discontinuities approximately from noisy data.
Optimal estimation of $\ell_1$-regularization prior from a regularized empirical Bayesian risk standpoint
Hui Huang, Eldad Haber and Lior Horesh
2012, 6(3): 447-464 doi: 10.3934/ipi.2012.6.447 +[Abstract](279) +[PDF](3702.7KB)
Abstract:
We address the problem of prior matrix estimation for the solution of $\ell_1$-regularized ill-posed inverse problems. From a Bayesian viewpoint, we show that such a matrix can be regarded as an influence matrix in a multivariate $\ell_1$-Laplace density function. Assuming a training set is given, the prior matrix design problem is cast as a maximum likelihood term with an additional sparsity-inducing term. This formulation results in an unconstrained yet nonconvex optimization problem. Memory requirements as well as computation of the nonlinear, nonsmooth sub-gradient equations are prohibitive for large-scale problems. Thus, we introduce an iterative algorithm to design efficient priors for such large problems. We further demonstrate that the solutions of ill-posed inverse problems by incorporation of $\ell_1$-regularization using the learned prior matrix perform generally better than commonly used regularization techniques where the prior matrix is chosen a-priori.
On the inverse doping profile problem
Victor Isakov and Joseph Myers
2012, 6(3): 465-486 doi: 10.3934/ipi.2012.6.465 +[Abstract](399) +[PDF](622.6KB)
Abstract:
We obtain new analytic results for the problem of the recovery of a doped region $D$ in semiconductor devices from the total flux of electrons/holes through a part of the boundary for various applied potentials on some complementary part of the boundary. We consider the stationary two-dimensional case and we use the index of the gradient of solutions of the linear elliptic equation modeling a unipolar device. Under mild assumptions we prove local uniqueness of smooth $D$ and global uniqueness of polygonal $D$ satisfying some geometrical (star-shapednedness or convexity in some direction) assumptions. We design a nonlinear minimization algorithm for numerical solution and we demonstrate its effectiveness on some basic examples. An essential ingredient of this algorithm is a numerical solution of the direct problem by using single layer potentials.
The Green function of the interior transmission problem and its applications
Kyoungsun Kim, Gen Nakamura and Mourad Sini
2012, 6(3): 487-521 doi: 10.3934/ipi.2012.6.487 +[Abstract](382) +[PDF](524.0KB)
Abstract:
The interior transmission problem appears naturally in the scattering theory. In this paper, we construct the Green function associated to this problem. In addition, we provide point-wise estimates of this Green function similar to those known for the Green function related to the classical transmission problems. These estimates are, in particular, useful to the study of various inverse scattering problems. Here, we apply them to justify some asymptotic formulas already used for detecting partially coated dielectric mediums from far field measurements.
Nowhere conformally homogeneous manifolds and limiting Carleman weights
Tony Liimatainen and Mikko Salo
2012, 6(3): 523-530 doi: 10.3934/ipi.2012.6.523 +[Abstract](309) +[PDF](310.0KB)
Abstract:
In this note we prove that a generic Riemannian manifold of dimension $\geq 3$ does not admit any nontrivial local conformal diffeomorphisms. This is a conformal analogue of a result of Sunada concerning local isometries, and makes precise the principle that generic manifolds in high dimensions do not have conformal symmetries. Consequently, generic manifolds of dimension $\geq 3$ do not admit nontrivial conformal Killing vector fields near any point. As an application to the inverse problem of Calderón on manifolds, this implies that generic manifolds of dimension $\geq 3$ do not admit limiting Carleman weights near any point.
Non rigid geometric distortions correction - Application to atmospheric turbulence stabilization
Yu Mao and Jérôme Gilles
2012, 6(3): 531-546 doi: 10.3934/ipi.2012.6.531 +[Abstract](284) +[PDF](5533.0KB)
Abstract:
A novel approach is presented to recover an image degraded by atmospheric turbulence. Given a sequence of frames affected by turbulence, we construct a variational model to characterize the static image. The optimization problem is solved by Bregman Iteration and the operator splitting method. Our algorithm is simple, efficient, and can be easily generalized for different scenarios.
Alternating algorithms for total variation image reconstruction from random projections
Yunhai Xiao, Junfeng Yang and Xiaoming Yuan
2012, 6(3): 547-563 doi: 10.3934/ipi.2012.6.547 +[Abstract](409) +[PDF](515.4KB)
Abstract:
Total variation (TV) regularization is popular in image reconstruction due to its edge-preserving property. In this paper, we extend the alternating minimization algorithm recently proposed in [37] to the case of recovering images from random projections. Specifically, we propose to solve the TV regularized least squares problem by alternating minimization algorithms based on the classical quadratic penalty technique and alternating minimization of the augmented Lagrangian function. The per-iteration cost of the proposed algorithms is dominated by two matrix-vector multiplications and two fast Fourier transforms. Convergence results, including finite convergence of certain variables and $q$-linear convergence rate, are established for the quadratic penalty method. Furthermore, we compare numerically the new algorithms with some state-of-the-art algorithms. Our experimental results indicate that the new algorithms are stable, efficient and competitive with the compared ones.

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