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Inverse Problems & Imaging

April 2019 , Volume 13 , Issue 2

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Microlocal analysis of a spindle transform
James W. Webber and Sean Holman
2019, 13(2): 231-261 doi: 10.3934/ipi.2019013 +[Abstract](1563) +[HTML](216) +[PDF](1333.16KB)
Abstract:

An analysis of the stability of the spindle transform, introduced in [16], is presented. We do this via a microlocal approach and show that the normal operator for the spindle transform is a type of paired Lagrangian operator with "blowdown–blowdown" singularities analogous to that of a limited data synthetic aperture radar (SAR) problem studied by Felea et. al. [4]. We find that the normal operator for the spindle transform belongs to a class of distibutions \begin{document}$ I^{p, l}(\Delta, \Lambda)+I^{p, l}(\widetilde{\Delta}, \Lambda) $\end{document} studied by Felea and Marhuenda in [4,10], where \begin{document}$ \widetilde{\Delta} $\end{document} is reflection through the origin, and \begin{document}$ \Lambda $\end{document} is associated to a rotation artefact. Later, we derive a filter to reduce the strength of the image artefact and show that it is of convolution type. We also provide simulated reconstructions to show the artefacts produced by \begin{document}$ \Lambda $\end{document} and show how the filter we derived can be applied to reduce the strength of the artefact.

On finding the surface admittance of an obstacle via the time domain enclosure method
Masaru Ikehata
2019, 13(2): 263-284 doi: 10.3934/ipi.2019014 +[Abstract](1355) +[HTML](383) +[PDF](497.96KB)
Abstract:

An inverse obstacle scattering problem for the electromagnetic wave governed by the Maxwell system over a finite time interval is considered. It is assumed that the wave satisfies the Leontovich boundary condition on the surface of an unknown obstacle. The condition is described by using an unknown positive function on the surface of the obstacle which is called the surface admittance. The wave is generated at the initial time by a volumetric current source supported on a very small ball placed outside the obstacle and only the electric component of the wave is observed on the same ball over a finite time interval. It is shown that from the observed data one can extract information about the value of the surface admittance and the curvatures at the points on the surface nearest to the center of the ball. This shows that a single shot contains a meaningful information about the quantitative state of the surface of the obstacle.

Incorporating structural prior information and sparsity into EIT using parallel level sets
Ville Kolehmainen, Matthias J. Ehrhardt and Simon R. Arridge
2019, 13(2): 285-307 doi: 10.3934/ipi.2019015 +[Abstract](1683) +[HTML](336) +[PDF](1075.82KB)
Abstract:

EIT is a non-linear ill-posed inverse problem which requires sophisticated regularisation techniques to achieve good results. In this paper we consider the use of structural information in the form of edge directions coming from an auxiliary image of the same object being reconstructed. In order to allow for cases where the auxiliary image does not provide complete information we consider in addition a sparsity regularization for the edges appearing in the EIT image. The combination of these approaches is conveniently described through the parallel level sets approach. We present an overview of previous methods for structural regularisation and then provide a variational setting for our approach and explain the numerical implementation. We present results on simulations and experimental data for different cases with accurate and inaccurate prior information. The results demonstrate that the structural prior information improves the reconstruction accuracy, even in cases when there is reasonable uncertainty in the prior about the location of the edges or only partial edge information is available.

An augmented lagrangian method for solving a new variational model based on gradients similarity measures and high order regulariation for multimodality registration
Anis Theljani and Ke Chen
2019, 13(2): 309-335 doi: 10.3934/ipi.2019016 +[Abstract](1598) +[HTML](248) +[PDF](1470.91KB)
Abstract:

In this work we propose a variational model for multi-modal image registration. It minimizes a new functional based on using reformulated normalized gradients of the images as the fidelity term and higher-order derivatives as the regularizer. We first present a theoretical analysis of the proposed model. Then, to solve the model numerically, we use an augmented Lagrangian method (ALM) to reformulate it to a few more amenable subproblems (each giving rise to an Euler-Lagrange equation that is discretized by finite difference methods) and solve iteratively the main linear systems by the fast Fourier transform; a multilevel technique is employed to speed up the initialisation and avoid likely local minima of the underlying functional. Finally we show the convergence of the ALM solver and give numerical results of the new approach. Comparisons with some existing methods are presented to illustrate its effectiveness and advantages.

Propagation of boundary-induced discontinuity in stationary radiative transfer and its application to the optical tomography
I-Kun Chen and Daisuke Kawagoe
2019, 13(2): 337-351 doi: 10.3934/ipi.2019017 +[Abstract](1070) +[HTML](166) +[PDF](345.98KB)
Abstract:

We consider a boundary value problem of the stationary transport equation with the incoming boundary condition in two or three dimensional bounded convex domains. We discuss discontinuity of the solution to the boundary value problem arising from discontinuous incoming boundary data, which we call the boundary-induced discontinuity. In particular, we give two kinds of sufficient conditions on the incoming boundary data for the boundary-induced discontinuity. We propose a method to reconstruct the attenuation coefficient from jumps in boundary measurements.

Electrical networks with prescribed current and applications to random walks on graphs
Christina Knox and Amir Moradifam
2019, 13(2): 353-375 doi: 10.3934/ipi.2019018 +[Abstract](1363) +[HTML](167) +[PDF](444.17KB)
Abstract:

In this paper we study Current Density Impedance Imaging (CDII) on Electrical Networks. The inverse problem is to determine the conductivity matrix of an electrical network from the prescribed knowledge of the magnitude of the induced current along the edges coupled with the imposed voltage or injected current on the boundary nodes. This problem leads to a weighted \begin{document}$ l^1 $\end{document} minimization problem for the corresponding voltage potential. We also investigate the problem of determining the transition probabilities of random walks on graphs from the prescribed expected net number of times the walker passes along the edges of the graph. Convergent numerical algorithms for solving such problems are also presented. Our results can be utilized in the design of electrical networks when certain current flow on the network is desired as well as the design of random walk models on graphs when the expected net number of the times the walker passes along the edges is prescribed. We also show that a mass preserving flow \begin{document}$ J = (J_{ij}) $\end{document} on a network can be uniquely recovered from the knowledge of \begin{document}$ |J| = (|J_{ij}|) $\end{document} and the flux of the flow on the boundary nodes, where \begin{document}$ J_{ij} $\end{document} is the flow from node \begin{document}$ i $\end{document} to node \begin{document}$ j $\end{document} and \begin{document}$ J_{ij} = -J_{ji} $\end{document}, and discuss its potential application in cryptography.

An inverse obstacle problem for the wave equation in a finite time domain
Laurent Bourgeois, Dmitry Ponomarev and Jérémi Dardé
2019, 13(2): 377-400 doi: 10.3934/ipi.2019019 +[Abstract](1595) +[HTML](182) +[PDF](1903.18KB)
Abstract:

We consider an inverse obstacle problem for the acoustic transient wave equation. More precisely, we wish to reconstruct an obstacle characterized by a Dirichlet boundary condition from lateral Cauchy data given on a subpart of the boundary of the domain and over a finite interval of time. We first give a proof of uniqueness for that problem and then propose an "exterior approach" based on a mixed formulation of quasi-reversibility and a level set method in order to actually solve the problem. Some 2D numerical experiments are provided to show that our approach is effective.

Regularization of a backwards parabolic equation by fractional operators
Barbara Kaltenbacher and William Rundell
2019, 13(2): 401-430 doi: 10.3934/ipi.2019020 +[Abstract](1679) +[HTML](311) +[PDF](566.78KB)
Abstract:

The backwards diffusion equation is one of the classical ill-posed inverse problems, related to a wide range of applications, and has been extensively studied over the last 50 years. One of the first methods was that of quasireversibility whereby the parabolic operator is replaced by a differential operator for which the backwards problem in time is well posed. This is in fact the direction we will take but will do so with a nonlocal operator; an equation of fractional order in time for which the backwards problem is known to be "almost well posed."

We shall look at various possible options and strategies but our conclusion for the best of these will exploit the linearity of the problem to break the inversion into distinct frequency bands and to use a different fractional order for each. The fractional exponents will be chosen using the discrepancy principle under the assumption we have an estimate of the noise level in the data. An analysis of the method is provided as are some illustrative numerical examples.

2018  Impact Factor: 1.469

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