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Inverse Problems & Imaging

2009 , Volume 3 , Issue 1

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Inverse problems for Einstein manifolds
Colin Guillarmou and  Antônio Sá Barreto
2009, 3(1): 1-15 doi: 10.3934/ipi.2009.3.1 +[Abstract](36) +[PDF](248.5KB)
We show that the Dirichlet-to-Neumann operator of the Laplacian on an open subset of the boundary of a connected compact Einstein manifold with boundary determines the manifold up to isometries. Similarly, for connected conformally compact Einstein manifolds of even dimension $n+1,$ we prove that the scattering matrix at energy $n$ on an open subset of its boundary determines the manifold up to isometries.
A positivity principle for parabolic integro-differential equations and inverse problems with final overdetermination
Jaan Janno and  Kairi Kasemets
2009, 3(1): 17-41 doi: 10.3934/ipi.2009.3.17 +[Abstract](44) +[PDF](330.6KB)
A positivity principle for parabolic integro-differential equations is proved. By means of this principle, uniqueness, existence and stability for an inverse source problem and two inverse coefficient problems are established.
Image recovery using functions of bounded variation and Sobolev spaces of negative differentiability
Yunho Kim and  Luminita A. Vese
2009, 3(1): 43-68 doi: 10.3934/ipi.2009.3.43 +[Abstract](37) +[PDF](794.4KB)
In this work we wish to recover an unknown image from a blurry, or noisy-blurry version. We solve this inverse problem by energy minimization and regularization. We seek a solution of the form $u + v$, where $u$ is a function of bounded variation (cartoon component), while $v$ is an oscillatory component (texture), modeled by a Sobolev function with negative degree of differentiability. We give several results of existence and characterization of minimizers of the proposed optimization problem. Experimental results show that this cartoon + texture model better recovers textured details in natural images, by comparison with the more standard models where the unknown is restricted only to the space of functions of bounded variation.
A greedy method for reconstructing polycrystals from three-dimensional X-ray diffraction data
Arun K. Kulshreshth , Andreas Alpers , Gabor T. Herman , Erik Knudsen , Lajos Rodek and  Henning F. Poulsen
2009, 3(1): 69-85 doi: 10.3934/ipi.2009.3.69 +[Abstract](44) +[PDF](446.8KB)
An iterative search method is proposed for obtaining orientation maps inside polycrystals from three-dimensional X-ray diffraction (3DXRD) data. In each step, detector pixel intensities are calculated by a forward model based on the current estimate of the orientation map. The pixel at which the experimentally measured value most exceeds the simulated one is identified. This difference can only be reduced by changing the current estimate at a location from a relatively small subset of all possible locations in the estimate and, at each such location, an increase at the identified pixel can only be achieved by changing the orientation in only a few possible ways. The method selects the location/orientation pair indicated as best by a function that measures data consistency combined with prior information on orientation maps. The superiority of the method to a previously published forward projection Monte Carlo optimization is demonstrated on simulated data.
Discretization-invariant Bayesian inversion and Besov space priors
Matti Lassas , Eero Saksman and  Samuli Siltanen
2009, 3(1): 87-122 doi: 10.3934/ipi.2009.3.87 +[Abstract](57) +[PDF](441.9KB)
Bayesian solution of an inverse problem for indirect measurement $M = AU + $ε is considered, where $U$ is a function on a domain of $\R^d$. Here $A$ is a smoothing linear operator and ε is Gaussian white noise. The data is a realization $m_k$ of the random variable $M_k = P_kA U+P_k$ε , where $P_k$ is a linear, finite dimensional operator related to measurement device. To allow computerized inversion, the unknown is discretized as $U_n=T_nU$, where $T_n$ is a finite dimensional projection, leading to the computational measurement model $M_{kn}=P_k A U_n + P_k$ε . Bayes formula gives then the posterior distribution

$\pi_{kn}(u_n\|\m_{kn})$~ Π n $(u_n)\exp(-\frac{1}{2}$||$\m_{kn} - P_kA u_n$||$\_2^2)$

in $\R^d$, and the mean $\u_{kn}$:$=\int u_n \ \pi_{kn}(u_n\|\m_k)\ du_n$ is considered as the reconstruction of $U$. We discuss a systematic way of choosing prior distributions Π n for all $n\geq n_0>0$ by achieving them as projections of a distribution in a infinite-dimensional limit case. Such choice of prior distributions is discretization-invariant in the sense that Π n represent the same a priori information for all $n$ and that the mean $\u_{kn}$ converges to a limit estimate as $k,n$→$\infty$. Gaussian smoothness priors and wavelet-based Besov space priors are shown to be discretization invariant. In particular, Bayesian inversion in dimension two with $B^1_11$ prior is related to penalizing the $\l^1$ norm of the wavelet coefficients of $U$.

The factorization method is independent of transmission eigenvalues
Armin Lechleiter
2009, 3(1): 123-138 doi: 10.3934/ipi.2009.3.123 +[Abstract](53) +[PDF](402.4KB)
As a rule of thumb, sampling methods for inverse scattering problems suffer from interior eigenvalues of the obstacle. Indeed, throughout the history of such algorithms one meets the phenomenon that if the wave number meets some resonance frequency of the scatterer, then those methods can only be shown to work under suitable modifications. Such modifications often require a-priori knowledge, corrupting thereby the main advantage of sampling methods. It was common belief that transmission eigenvalues play a role corresponding to Dirichlet or Neumann eigenvalues in this respect. We show that this is not the case for the Factorization method: when applied to inverse medium scattering problems this method is stable at transmission eigenvalues.
On the boundary control approach to inverse spectral and scattering theory for Schrödinger operators
Alexei Rybkin
2009, 3(1): 139-149 doi: 10.3934/ipi.2009.3.139 +[Abstract](47) +[PDF](193.5KB)
We link boundary control theory and inverse spectral theory for the Schrödinger operator $H=-\partial _{x}^{2}+q( x) $ on $L^{2}( 0,\infty) $ with Dirichlet boundary condition at $x=0.$ This provides a shortcut to some results on inverse spectral theory due to Simon, Gesztesy-Simon and Remling. The approach also has a clear physical interpritation in terms of boundary control theory for the wave equation.

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