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Inverse Problems & Imaging

2011 , Volume 5 , Issue 3

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Direct electrical impedance tomography for nonsmooth conductivities
Kari Astala, Jennifer L. Mueller, Lassi Päivärinta, Allan Perämäki and Samuli Siltanen
2011, 5(3): 531-549 doi: 10.3934/ipi.2011.5.531 +[Abstract](207) +[PDF](768.1KB)
A new reconstruction algorithm is presented for eit in dimension two, based on the constructive uniqueness proof given by Astala and Päivärinta in [Ann. of Math. 163 (2006)]. The method is non-iterative, provides a noise-robust solution of the full nonlinear eit problem, and applies to more general conductivities than previous approaches. In particular, the new algorithm applies to piecewise smooth conductivities. Reconstructions from noisy and non-noisy simulated data from conductivity distributions representing a cross-sections of a chest and a layered medium such as stratified flow in a pipeline are presented. The results suggest that the new method can recover useful and reasonably accurate eit images from data corrupted by realistic amounts of measurement noise. In particular, the dynamic range in medium-contrast conductivities is reconstructed remarkably well.
Alpha divergences based mass transport models for image matching problems
Pengwen Chen and Changfeng Gui
2011, 5(3): 551-590 doi: 10.3934/ipi.2011.5.551 +[Abstract](76) +[PDF](938.2KB)
Registration methods could be roughly divided into two groups: area-based methods and feature-based methods. In the literature, the Monge-Kantorovich (MK) mass transport problem has been applied to image registration as an area-based method. In this paper, we propose to use Monge-Kantorovich (MK) mass transport model as a feature-based method. This novel image matching model is a coupling of the MK problem with the well-known alpha divergence from the probability theory. The optimal matching scheme is the one which minimizes the weighted alpha divergence between two images. A primal-dual approach is employed to analyze the existence and uniqueness/non-uniqueness of the optimal matching scheme. A block coordinate method, analogous to the Sinkhorn matrix balancing method, can be used to compute the optimal matching scheme. We also derive a distance function for image morphing. Similar to elastic distances proposed by Younes, the geodesic under this distance function has an explicit expression.
Anisotropic total variation regularized $L^1$ approximation and denoising/deblurring of 2D bar codes
Rustum Choksi, Yves van Gennip and Adam Oberman
2011, 5(3): 591-617 doi: 10.3934/ipi.2011.5.591 +[Abstract](107) +[PDF](649.2KB)
We consider variations of the Rudin-Osher-Fatemi functional which are particularly well-suited to denoising and deblurring of 2D bar codes. These functionals consist of an anisotropic total variation favoring rectangles and a fidelity term which measure the $L^1$ distance to the signal, both with and without the presence of a deconvolution operator. Based upon the existence of a certain associated vector field, we find necessary and sufficient conditions for a function to be a minimizer. We apply these results to 2D bar codes to find explicit regimes -- in terms of the fidelity parameter and smallest length scale of the bar codes -- for which the perfect bar code is attained via minimization of the functionals. Via a discretization reformulated as a linear program, we perform numerical experiments for all functionals demonstrating their denoising and deblurring capabilities.
Errors of regularisation under range inclusions using variable Hilbert scales
Markus Hegland and Bernd Hofmann
2011, 5(3): 619-643 doi: 10.3934/ipi.2011.5.619 +[Abstract](69) +[PDF](499.9KB)
Based on the variable Hilbert scale interpolation inequality, bounds for the error of regularisation methods are derived under range inclusions. In this context, new formulae for the modulus of continuity of the inverse of bounded operators with non-closed range are given. Even if one can show the equivalence of this approach to the version used previously in the literature, the new formulae and corresponding conditions are simpler than the former ones. Several examples from image processing and spectral enhancement illustrate how the new error bounds can be applied.
A fast algorithm for global minimization of maximum likelihood based on ultrasound image segmentation
Jie Huang, Xiaoping Yang and Yunmei Chen
2011, 5(3): 645-657 doi: 10.3934/ipi.2011.5.645 +[Abstract](117) +[PDF](655.1KB)
This paper presents a novel variational model for ultrasound image segmentation that uses a maximum likelihood estimator based on Fisher-Tippett distribution of the intensities of ultrasound images. A convex relaxation method is applied to get a convex model of the subproblem with fixed distribution parameters. The relaxed subproblem, which is convex, can be fast solved by using a primal-dual hybrid gradient algorithm. The experimental results on simulated and real ultrasound images indicate the effectiveness of the method presented.
Microlocal aspects of common offset synthetic aperture radar imaging
Venkateswaran P. Krishnan and Eric Todd Quinto
2011, 5(3): 659-674 doi: 10.3934/ipi.2011.5.659 +[Abstract](93) +[PDF](398.4KB)
In this article, we analyze the microlocal properties of the linearized forward scattering operator $F$ and the reconstruction operator $F^{*}F$ appearing in bistatic synthetic aperture radar imaging. In our model, the radar source and detector travel along a line a fixed distance apart. We show that $F$ is a Fourier integral operator, and we give the mapping properties of the projections from the canonical relation of $F$, showing that the right projection is a blow-down and the left projection is a fold. We then show that $F^{*}F$ is a singular FIO belonging to the class $I^{3,0}$.
Explicit characterization of the support of non-linear inclusions
Armin Lechleiter
2011, 5(3): 675-694 doi: 10.3934/ipi.2011.5.675 +[Abstract](87) +[PDF](459.9KB)
We study inverse problems for non-linear penetrable media in the context of scattering theory and impedance tomography. Using a general description of the range of the non-linear far-field operator we show an explicit characterization of the support of a weakly non-linear inhomogeneous scattering object. Application of the same technique to the impedance tomography problem for a monotonic non-linear inclusion yields a characterization of the inclusion's support from the non-linear Neumann-to-Dirichlet operator.
Identification of a real constant in linear evolution equations in Hilbert spaces
Alfredo Lorenzi and Gianluca Mola
2011, 5(3): 695-714 doi: 10.3934/ipi.2011.5.695 +[Abstract](72) +[PDF](427.5KB)
Let $H$ be a real separable Hilbert space and $A:\mathcal{D}(A) \to H$ be a positive and self-adjoint (unbounded) operator, and denote by $A^\sigma$ its power of exponent $\sigma \in [-1,1)$. We consider the identification problem consisting in searching for a function $u:[0,T] \to H$ and a real constant $\mu$ that fulfill the initial-value problem $$ u' + Au = \mu \, A^\sigma u, \quad t \in (0,T), \quad u(0) = u_0, $$ and the additional condition $$ \alpha \|u(T)\|^{2} + \beta \int_{0}^{T}\|A^{1/2}u(\tau)\|^{2}d\tau = \rho, $$ where $u_{0} \in H$, $u_{0} \neq 0$ and $\alpha, \beta \geq 0$, $\alpha+\beta > 0$ and $\rho >0$ are given. By means of a finite-dimensional approximation scheme, we construct a unique solution $(u,\mu)$ of suitable regularity on the whole interval $[0,T]$, and exhibit an explicit continuous dependence estimate of Lipschitz-type with respect to the data $u_{0}$ and $\rho $. Also, we provide specific applications to second and fourth-order parabolic initial-boundary value problems.
Inverse boundary value problems for discrete Schrödinger operators on the multi-dimensional square lattice
Hisashi Morioka
2011, 5(3): 715-730 doi: 10.3934/ipi.2011.5.715 +[Abstract](82) +[PDF](379.3KB)
We consider an inverse boundary value problem for a discrete Schrödinger operator $-\Delta + \hat{q} $ on a bounded domain in the square lattice. We define an analogue of the Dirichlet-to-Neumann map, and give a reconstruction procedure of the potential $\hat{q} $ from the D-to-N map for all energies.
Solving an inverse problem for the wave equation by using a minimization algorithm and time-reversed measurements
Lauri Oksanen
2011, 5(3): 731-744 doi: 10.3934/ipi.2011.5.731 +[Abstract](90) +[PDF](415.4KB)
We consider the inverse problem for the wave equation on a compact Riemannian manifold or on a bounded domain of $\mathbb{R}^n$, and generalize the concept of domain of influence. We present an efficient minimization algorithm to compute the volume of a domain of influence using boundary measurements and time-reversed boundary measurements. Moreover, we show that if the manifold is simple, then the volumes of the domains of influence determine the manifold. For a continuous real valued function $\tau$ on the boundary of the manifold, the domain of influence is the set of those points on the manifold from which the travel time to some boundary point $y$ is less than $\tau(y)$.

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