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Inverse Problems & Imaging

August 2018 , Volume 12 , Issue 4

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Geometric mode decomposition
Siwei Yu, Jianwei Ma and Stanley Osher
2018, 12(4): 831-852 doi: 10.3934/ipi.2018035 +[Abstract](727) +[HTML](230) +[PDF](3108.34KB)
Abstract:

We propose a new decomposition algorithm for seismic data based on a band-limited a priori knowledge on the Fourier or Radon spectrum. This decomposition is called geometric mode decomposition (GMD), as it decomposes a 2D signal into components consisting of linear or parabolic features. Rather than using a predefined frame, GMD adaptively obtains the geometric parameters in the data, such as the dominant slope or curvature. GMD is solved by alternatively pursuing the geometric parameters and the corresponding modes in the Fourier or Radon domain. The geometric parameters are obtained from the weighted center of the corresponding mode's energy spectrum. The mode is obtained by applying a Wiener filter, the design of which is based on a certain band-limited property. We apply GMD to seismic events splitting, noise attenuation, interpolation, and demultiple. The results show that our method is a promising adaptive tool for seismic signal processing, in comparisons with the Fourier and curvelet transforms, empirical mode decomposition (EMD) and variational mode decomposition (VMD) methods.

Convergence theorems for the Non-Local Means filter
Qiyu Jin, Ion Grama and Quansheng Liu
2018, 12(4): 853-881 doi: 10.3934/ipi.2018036 +[Abstract](700) +[HTML](81) +[PDF](4312.9KB)
Abstract:

We introduce an oracle filter for removing the Gaussian noise with weights depending on a similarity function. The usual Non-Local Means filter is obtained from this oracle filter by substituting the similarity function by an estimator based on similarity patches. When the sizes of the search window are chosen appropriately, it is shown that the oracle filter converges with the optimal rate. The same optimal convergence rate is preserved when the similarity function has suitable errors-in measurements. We also provide a statistical estimator of the similarity which converges at a convenient rate. Based on our convergence theorems, we propose some simple formulas for the choice of the parameters. Simulation results show that our choice of parameters improves the restoration quality of the filter compared with the usual choice of parameters in the original algorithm.

Use of an optimized spatial prior in D-bar reconstructions of EIT tank data
Melody Alsaker and Jennifer L. Mueller
2018, 12(4): 883-901 doi: 10.3934/ipi.2018037 +[Abstract](324) +[HTML](88) +[PDF](2369.9KB)
Abstract:

The aim of this paper is to demonstrate the feasibility of using spatial a priori information in the 2-D D-bar method to improve the spatial resolution of EIT reconstructions of experimentally collected data. The prior consists of imperfectly known information about the spatial locations of inclusions and the assumption that the conductivity is a mollified piecewise constant function. The conductivity values for the prior are constructed using a novel method in which a nonlinear constrained optimization routine is used to select the values for the piecewise constant function that give the best fit to the scattering transform computed from the measured data in a disk. The prior is then included in the high-frequency components of the scattering transform and in the computation of the solution of the D-bar equation, with weights to control the influence of the prior. In addition, a new technique is described for selecting regularization parameters to truncate the measured scattering data, in which complex scattering frequencies for which the values of the scattering transform differ greatly from those in the scattering prior are omitted. The effectiveness of the method is demonstrated on EIT data collected on saline-filled tanks with agar heart and lungs with various added inhomogeneities.

Recursive reconstruction of piecewise constant signals by minimization of an energy function
Anass Belcaid, Mohammed Douimi and Abdelkader Fassi Fihri
2018, 12(4): 903-920 doi: 10.3934/ipi.2018038 +[Abstract](475) +[HTML](67) +[PDF](1053.96KB)
Abstract:

The problem of denoising piecewise constant signals while preserving their jumps is a challenging problem that arises in many scientific areas. Several denoising algorithms exist such as total variation, convex relaxation, Markov random fields models, etc. The DPS algorithm is a combinatorial algorithm that excels the classical GNC in term of speed and SNR resistance. However, its running time slows down considerably for large signals. The main reason for this bottleneck is the size and the number of linear systems that need to be solved. We develop a recursive implementation of the DPS algorithm that uses the conditional independence, created by a confirmed discontinuity between two parts, to separate the reconstruction process of each part. Additionally, we propose an accelerated Cholesky solver which reduces the computational cost and memory usage. We evaluate the new implementation on a set of synthetic and real world examples to compare the quality of our solver. The results show a significant speed up, especially with a higher number of discontinuities.

Inverse acoustic scattering using high-order small-inclusion expansion of misfit function
Marc Bonnet
2018, 12(4): 921-953 doi: 10.3934/ipi.2018039 +[Abstract](392) +[HTML](103) +[PDF](1935.99KB)
Abstract:

This article concerns an extension of the topological derivative concept for 3D inverse acoustic scattering problems involving the identification of penetrable obstacles, whereby the featured data-misfit cost function \begin{document} $\mathbb{J}$ \end{document} is expanded in powers of the characteristic radius \begin{document} $a$ \end{document} of a single small inhomogeneity. The \begin{document} $O(a^6)$ \end{document} approximation \begin{document} $\mathbb{J}_6$ \end{document} of \begin{document} $\mathbb{J}$ \end{document} is derived and justified for a single obstacle of given location, shape and material properties embedded in a 3D acoustic medium of arbitrary shape. The generalization of \begin{document} $\mathbb{J}_6$ \end{document} to multiple small obstacles is outlined. Simpler and more explicit expressions of \begin{document} $\mathbb{J}_6$ \end{document} are obtained when the scatterer is centrally-symmetric or spherical. An approximate and computationally light global search procedure, where the location and size of the unknown object are estimated by minimizing \begin{document} $\mathbb{J}_6$ \end{document} over a search grid, is proposed and demonstrated on numerical experiments, where the identification from known acoustic pressure on the surface of a penetrable scatterer embedded in a acoustic semi-infinite medium, and whose shape may differ from that of the trial obstacle assumed in the expansion of \begin{document} $\mathbb{J}$ \end{document}, is considered.

Inverse source problems without (pseudo) convexity assumptions
Victor Isakov and Shuai Lu
2018, 12(4): 955-970 doi: 10.3934/ipi.2018040 +[Abstract](393) +[HTML](67) +[PDF](3176.0KB)
Abstract:

We study the inverse source problem for the Helmholtz equation from boundary Cauchy data with multiple wave numbers. The main goal of this paper is to study the uniqueness and increasing stability when the (pseudo)convexity or non-trapping conditions for the related hyperbolic problem are not satisfied. We consider general elliptic equations of the second order and arbitrary observation sites. To show the uniqueness we use the analytic continuation, the Fourier transform with respect to the wave numbers and uniqueness in the lateral Cauchy problem for hyperbolic equations. Numerical examples in 2 spatial dimension support the analysis and indicate the increasing stability for large intervals of the wave numbers, while analytic proofs of the increasing stability are not available.

Asymptotic expansions of transmission eigenvalues for small perturbations of media with generally signed contrast
Fioralba Cakoni, Shari Moskow and Scott Rome
2018, 12(4): 971-992 doi: 10.3934/ipi.2018041 +[Abstract](375) +[HTML](88) +[PDF](412.72KB)
Abstract:

In this paper we revisit the transmission eigenvalue problem for an inhomogeneous media of compact support perturbed by small penetrable homogeneous inclusions. Assuming that the inhomogeneous background media is known and smooth, we investigate how these small volume inclusions affect the transmission eigenvalues. Our perturbation analysis makes use of the formulation of the transmission eigenvalue problem introduced Kirsch in [8], which requires that the contrast of the inhomogeneity is of one-sign only near the boundary. Thus, our approach can handle small perturbations with positive, negative or zero (voids) contrasts. In addition to proving the convergence rate for the eigenvalues corresponding to the perturbed media as inclusions' volume goes to zero, we also provide the explicit first correction term in the asymptotic expansion for simple eigenvalues. The correction term involves computable information about the known inhomogeneity as well as the location, size and refractive index of small perturbations. Our asymptotic formula has the potential to be used to recover information about small inclusions from knowledge of the real transmission eigenvalues, which can be determined from scattering data.

Reconstruction of a compact manifold from the scattering data of internal sources
Matti Lassas, Teemu Saksala and Hanming Zhou
2018, 12(4): 993-1031 doi: 10.3934/ipi.2018042 +[Abstract](334) +[HTML](59) +[PDF](583.09KB)
Abstract:

Given a smooth non-trapping compact manifold with strictly convex boundary, we consider an inverse problem of reconstructing the manifold from the scattering data initiated from internal sources. These data consist of the exit directions of geodesics that are emaneted from interior points of the manifold. We show that under certain generic assumption of the metric, the scattering data measured on the boundary determine the Riemannian manifold up to isometry.

On the transmission eigenvalue problem for the acoustic equation with a negative index of refraction and a practical numerical reconstruction method
Tiexiang Li, Tsung-Ming Huang, Wen-Wei Lin and Jenn-Nan Wang
2018, 12(4): 1033-1054 doi: 10.3934/ipi.2018043 +[Abstract](418) +[HTML](85) +[PDF](6938.26KB)
Abstract:

In this paper, we consider the two-dimensional Maxwell's equations with the TM mode in pseudo-chiral media. The system can be reduced to the acoustic equation with a negative index of refraction. We first study the transmission eigenvalue problem (TEP) for this equation. By the continuous finite element method, we discretize the reduced equation and transform the study of TEP to a quadratic eigenvalue problem by deflating all nonphysical zeros. We then estimate half of the eigenvalues are negative with order of \begin{document}$O(1)$\end{document} and the other half of eigenvalues are positive with order of \begin{document}$O(10^2)$\end{document}. In the second part of the paper, we present a practical numerical method to reconstruct the support of the inhomogeneity by the near-field measurements, i.e., Cauchy data. Based on the linear sampling method, we propose the truncated singular value decomposition to solve the ill-posed near-field integral equation, at one wave number which is not a transmission eigenvalue. By carefully chosen an indicator function, this method produce different jumps for the sampling points inside and outside the support. Numerical results show that our method is able to reconstruct the support reliably.

2017  Impact Factor: 1.465

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