# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

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## Inverse Problems & Imaging

February 2010 , Volume 4 , Issue 1

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2010, 4(1): 1-9 doi: 10.3934/ipi.2010.4.1 +[Abstract](1682) +[PDF](158.0KB)
Abstract:
We extend the classical spectral estimation problem to the infinite-dimensional case and propose a new approach to this problem using the Boundary Control (BC) method. Several applications to inverse problems for partial differential equations are provided.
2010, 4(1): 11-17 doi: 10.3934/ipi.2010.4.11 +[Abstract](1376) +[PDF](157.1KB)
Abstract:
Let $z=Au+\gamma$ be an ill-posed, linear operator equation. Such a model arises, for example, in both astronomical and medical imaging, in which case $\gamma$ corresponds to background, $u$ the unknown true image, $A$ the forward operator, and $z$ the data. Regularized solutions of this equation can be obtained by solving

$R_\alpha(A,z)= arg\min_{u\geq 0} \{T_0(Au;z)+\alpha J(u)\},$

where $T_0(Au;z)$ is the negative-log of the Poisson likelihood functional, and $\alpha>0$ and $J$ are the regularization parameter and functional, respectively. Our goal in this paper is to determine general conditions which guarantee that $R_\alpha$ defines a regularization scheme for $z=Au+\gamma$. Determining the appropriate definition for regularization scheme in this context is important: not only will it serve to unify previous theoretical arguments in this direction, it will provide a framework for future theoretical analyses. To illustrate the latter, we end the paper with an application of the general framework to a case in which an analysis has not been done.

2010, 4(1): 19-38 doi: 10.3934/ipi.2010.4.19 +[Abstract](1636) +[PDF](249.3KB)
Abstract:
In the context of scattering problems in the harmonic regime, we consider the problem of identification of some Generalized Impedance Boundary Conditions (GIBC) at the boundary of an object (which is supposed to be known) from far field measurements associated with a single incident plane wave at a fixed frequency. The GIBCs can be seen as approximate models for thin coatings, corrugated surfaces or highly absorbing media. After pointing out that uniqueness does not hold in the general case, we propose some additional assumptions for which uniqueness can be restored. We also consider the question of stability when uniqueness holds. We prove in particular Lipschitz stability when the impedance parameters belong to a compact subset of a finite dimensional space.
2010, 4(1): 39-48 doi: 10.3934/ipi.2010.4.39 +[Abstract](1297) +[PDF](162.1KB)
Abstract:
We consider the interior transmission eigenvalue problem corresponding to the inverse scattering problem for an isotropic inhomogeneous medium. We first prove that transmission eigenvalues exist for media with index of refraction greater or less than one without assuming that the contrast is sufficiently large. Then we show that for an arbitrary Lipshitz domain with constant index of refraction there exists an infinite discrete set of transmission eigenvalues that accumulate at infinity. Finally, for the general case of non constant index of refraction we provide a lower and an upper bound for the first transmission eigenvalue in terms of the first transmission eigenvalue for appropriate balls with constant index of refraction.
2010, 4(1): 49-91 doi: 10.3934/ipi.2010.4.49 +[Abstract](1439) +[PDF](464.7KB)
Abstract:
It is proved that, in two dimensions, the Calderón inverse conductivity problem in Lipschitz domains is stable in the $L^p$ sense when the conductivities are uniformly bounded in any fractional Sobolev space $W^{\alpha,p}$ $\alpha>0, 1 < p < \infty$.
2010, 4(1): 93-110 doi: 10.3934/ipi.2010.4.93 +[Abstract](1226) +[PDF](234.7KB)
Abstract:
For finite dimensional CMV matrices the classical inverse spectral problems are considered. We solve the inverse problem of reconstructing a CMV matrix by its Weyl's function, the problem of reconstructing the matrix by two spectra of CMV operators with different "boundary condition'', and the problem of reconstructing a CMV matrix by its spectrum and the spectrum of the CMV matrix obtained from it by unitary truncation.
2010, 4(1): 111-130 doi: 10.3934/ipi.2010.4.111 +[Abstract](1294) +[PDF](285.0KB)
Abstract:
We consider the tomography problem of recovering a covector field on a simple Riemannian manifold based on its weighted Doppler transformation over a family of curves $\Gamma$. This is a generalization of the attenuated Doppler transform. Uniqueness is proven for a generic set of weights and families of curves under a condition on the weight function. This condition is satisfied in particular if the weight function is never zero, and its derivatives along the curves in $\Gamma$ are never zero.
2010, 4(1): 131-149 doi: 10.3934/ipi.2010.4.131 +[Abstract](1618) +[PDF](929.9KB)
Abstract:
The inverse problem for time-harmonic acoustic wave scattering to recover a sound-soft obstacle from a given incident field and the far field pattern of the scattered field is considered. We split this problem into two subproblems; first to reconstruct the shape from the modulus of the data and this is followed by employing the full far field pattern in a few measurement points to find the location of the obstacle. We extend a nonlinear integral equation approach for shape reconstruction from the modulus of the far field data [6] to the three-dimensional case. It is known, see [13], that the location of the obstacle cannot be reconstructed from only the modulus of the far field pattern since it is invariant under translations. However, employing the underlying invariance relation and using only few far field measurements in the backscattering direction we propose a novel approach for the localization of the obstacle. The efficient implementation of the method is described and the feasibility of the approach is illustrated by numerical examples.
2010, 4(1): 151-167 doi: 10.3934/ipi.2010.4.151 +[Abstract](1358) +[PDF](242.4KB)
Abstract:
Three dimensional anisotropic attenuating and scattering media sharing the same albedo operator have been shown to be related via a gauge transformation. Such transformations define an equivalence relation. We show that the gauge equivalence is also valid in media with non-constant index of refraction, modeled by a Riemannian metric. The two dimensional model is also investigated.
2010, 4(1): 169-190 doi: 10.3934/ipi.2010.4.169 +[Abstract](1462) +[PDF](1083.8KB)
Abstract:
We present a comparison of three methods for the solution of the magnetoencephalography inverse problem. The methods are: a linearly constrained minimum variance beamformer, an algorithm implementing multiple signal classification with recursively applied projection and a particle filter for Bayesian tracking. Synthetic data with neurophysiological significance are analyzed by the three methods to recover position, orientation and amplitude of the active sources. Finally, a real data set evoked by a simple auditory stimulus is considered.
2010, 4(1): 191-210 doi: 10.3934/ipi.2010.4.191 +[Abstract](2226) +[PDF](538.1KB)
Abstract:
Wavelet inpainting problem consists of filling in missed data in the wavelet domain. In [17], Chan, Shen, and Zhou proposed an efficient method to recover piecewise constant or smooth images by combining total variation regularization and wavelet representations. In this paper, we extend it to nonlocal total variation regularization in order to recover textures and local geometry structures simultaneously. Moreover, we apply an efficient algorithm framework for both local and nonlocal regularizers. Extensive experimental results on a variety of loss scenarios and natural images validate the performance of this approach.

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