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Inverse Problems & Imaging

June 2021 , Volume 15 , Issue 3

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Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods
Wenbin Li and Jianliang Qian
2021, 15(3): 387-413 doi: 10.3934/ipi.2020073 +[Abstract](921) +[HTML](261) +[PDF](1282.76KB)
Abstract:

We develop new efficient algorithms for a class of inverse problems of gravimetry to recover an anomalous volume mass distribution (measure) in the sense that we design fast local level-set methods to simultaneously reconstruct both unknown domain and varying density of the anomalous measure from modulus of gravity force rather than from gravity force itself. The equivalent-source principle of gravitational potential forces us to consider only measures of the form \begin{document}$ \mu = f\,\chi_{D} $\end{document}, where \begin{document}$ f $\end{document} is a density function and \begin{document}$ D $\end{document} is a domain inside a closed set in \begin{document}$ \bf{R}^n $\end{document}. Accordingly, various constraints are imposed upon both the density function and the domain so that well-posedness theories can be developed for the corresponding inverse problems, such as the domain inverse problem, the density inverse problem, and the domain-density inverse problem. Starting from uniqueness theorems for the domain-density inverse problem, we derive a new gradient from the misfit functional to enforce the directional-independence constraint of the density function and we further introduce a new labeling function into the level-set method to enforce the geometrical constraint of the corresponding domain; consequently, we are able to recover simultaneously both unknown domain and varying density from given modulus of gravity force. Our fast level-set method is built upon localizing the level-set evolution around a narrow band near the zero level-set and upon accelerating numerical modeling by novel low-rank matrix multiplication. Numerical results demonstrate that uniqueness theorems are crucial for solving the inverse problem of gravimetry and will be impactful on gravity prospecting. To the best of our knowledge, our inversion algorithm is the first of such for the domain-density inverse problem since it is based upon the conditional well-posedness theory of the inverse problem.

Some application examples of minimization based formulations of inverse problems and their regularization
Kha Van Huynh and Barbara Kaltenbacher
2021, 15(3): 415-443 doi: 10.3934/ipi.2020074 +[Abstract](793) +[HTML](321) +[PDF](649.69KB)
Abstract:

In this paper we extend a recent idea of formulating and regularizing inverse problems as minimization problems, so without using a forward operator, thus avoiding explicit evaluation of a parameter-to-state map. We do so by rephrasing three application examples in this minimization form, namely (a) electrical impedance tomography with the complete electrode model (b) identification of a nonlinear magnetic permeability from magnetic flux measurements (c) localization of sound sources from microphone array measurements. To establish convergence of the proposed regularization approach for these problems, we first of all extend the existing theory. In particular, we take advantage of the fact that observations are finite dimensional here, so that inversion of the noisy data can to some extent be done separately, using a right inverse of the observation operator. This new approach is actually applicable to a wide range of real world problems.

The interior transmission eigenvalue problem for elastic waves in media with obstacles
Fioralba Cakoni, Pu-Zhao Kow and Jenn-Nan Wang
2021, 15(3): 445-474 doi: 10.3934/ipi.2020075 +[Abstract](736) +[HTML](253) +[PDF](507.14KB)
Abstract:

In this paper, we investigate the interior transmission eigenvalue problem for elastic waves propagating outside a sound-soft or a sound-hard obstacle surrounded by an anisotropic layer. This study is motivated by the inverse problem of identifying an object embedded in an inhomogeneous media in the presence of elastic waves. Our analysis of this non-selfadjoint eigenvalue problem relies on the weak formulation of involved boundary value problems and some fundamental tools in functional analysis.

Tensor train rank minimization with nonlocal self-similarity for tensor completion
Meng Ding, Ting-Zhu Huang, Xi-Le Zhao, Michael K. Ng and Tian-Hui Ma
2021, 15(3): 475-498 doi: 10.3934/ipi.2021001 +[Abstract](766) +[HTML](216) +[PDF](13711.18KB)
Abstract:

The tensor train (TT) rank has received increasing attention in tensor completion due to its ability to capture the global correlation of high-order tensors (\begin{document}$ \rm{order} >3 $\end{document}). For third order visual data, direct TT rank minimization has not exploited the potential of TT rank for high-order tensors. The TT rank minimization accompany with ket augmentation, which transforms a lower-order tensor (e.g., visual data) into a higher-order tensor, suffers from serious block-artifacts. To tackle this issue, we suggest the TT rank minimization with nonlocal self-similarity for tensor completion by simultaneously exploring the spatial, temporal/spectral, and nonlocal redundancy in visual data. More precisely, the TT rank minimization is performed on a formed higher-order tensor called group by stacking similar cubes, which naturally and fully takes advantage of the ability of TT rank for high-order tensors. Moreover, the perturbation analysis for the TT low-rankness of each group is established. We develop the alternating direction method of multipliers tailored for the specific structure to solve the proposed model. Extensive experiments demonstrate that the proposed method is superior to several existing state-of-the-art methods in terms of both qualitative and quantitative measures.

Inverse $N$-body scattering with the time-dependent hartree-fock approximation
Michiyuki Watanabe
2021, 15(3): 499-517 doi: 10.3934/ipi.2021002 +[Abstract](558) +[HTML](197) +[PDF](472.74KB)
Abstract:

We consider an inverse \begin{document}$N$\end{document}-body scattering problem of determining two potentials—an external potential acting on all particles and a pair interaction potential—from the scattering particles. This paper finds that the time-dependent Hartree-Fock approximation for a three-dimensional inverse \begin{document}$N$\end{document}-body scattering in quantum mechanics enables us to recover the two potentials from the scattering states with high-velocity initial states. The main ingredient of mathematical analysis in this paper is based on the asymptotic analysis of the scattering operator defined in terms of a scattering solution to the Hartree-Fock equation at high energies. We show that the leading part of the asymptotic expansion of the scattering operator uniquely reconstructs the Fourier transform of the pair interaction, and the second term of the expansion uniquely reconstructs the \begin{document}$X$\end{document}-ray transform of the external potential.

A nonlocal low rank model for poisson noise removal
Mingchao Zhao, You-Wei Wen, Michael Ng and Hongwei Li
2021, 15(3): 519-537 doi: 10.3934/ipi.2021003 +[Abstract](836) +[HTML](310) +[PDF](11942.48KB)
Abstract:

Patch-based methods, which take the advantage of the redundancy and similarity among image patches, have attracted much attention in recent years. However, these methods are mainly limited to Gaussian noise removal. In this paper, the Poisson noise removal problem is considered. Unlike Gaussian noise which has an identical and independent distribution, Poisson noise is signal dependent, which makes the problem more challenging. By incorporating the prior that a group of similar patches should possess a low-rank structure, and applying the maximum a posterior (MAP) estimation, the Poisson noise removal problem is formulated as an optimization one. Then, an alternating minimization algorithm is developed to find the minimizer of the objective function efficiently. Convergence of the minimizing sequence will be established, and the efficiency and effectiveness of the proposed algorithm will be demonstrated by numerical experiments.

The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation
Jianli Xiang and Guozheng Yan
2021, 15(3): 539-554 doi: 10.3934/ipi.2021004 +[Abstract](727) +[HTML](199) +[PDF](418.99KB)
Abstract:

This paper considers the inverse elastic wave scattering by a bounded penetrable or impenetrable scatterer. We propose a novel technique to show that the elastic obstacle can be uniquely determined by its far-field pattern associated with all incident plane waves at a fixed frequency. In the first part of this paper, we establish the mixed reciprocity relation between the far-field pattern corresponding to special point sources and the scattered field corresponding to plane waves, and the mixed reciprocity relation is the key point to show the uniqueness results. In the second part, besides the mixed reciprocity relation, a priori estimates of solution to the transmission problem with boundary data in \begin{document}$ [L^{\mathrm{q}}(\partial\Omega)]^{3} $\end{document} (\begin{document}$ 1<\mathrm{q}<2 $\end{document}) is deeply investigated by the integral equation method and also we have constructed a well-posed modified static interior transmission problem on a small domain to obtain the uniqueness result.

Mathematical analysis of the acoustic imaging modality using bubbles as contrast agents at nearly resonating frequencies
Alexander Dabrowski, Ahcene Ghandriche and Mourad Sini
2021, 15(3): 555-597 doi: 10.3934/ipi.2021005 +[Abstract](582) +[HTML](200) +[PDF](624.17KB)
Abstract:

We analyze mathematically the acoustic imaging modality using bubbles as contrast agents. These bubbles are modeled by mass densities and bulk moduli enjoying contrasting scales. These contrasting scales allow them to resonate at certain incident frequencies. We consider two types of such contrasts. In the first one, the bubbles are light with small bulk modulus, as compared to the ones of the background, so that they generate the Minnaert resonance (corresponding to a local surface wave). In the second one, the bubbles have moderate mass density but still with small bulk modulus so that they generate a sequence of resonances (corresponding to local body waves).

We propose to use as measurements the far-fields collected before and after injecting a bubble, set at a given location point in the target domain, generated at a band of incident frequencies and at a fixed single backscattering direction. Then, we scan the target domain with such bubbles and collect the corresponding far-fields. The goal is to reconstruct both the, variable, mass density and bulk modulus of the background in the target region.

1.We show that, for each fixed used bubble, the contrasted far-fields reach their maximum value at, incident, frequencies close to the Minnaert resonance (or the body-wave resonances depending on the types of bubbles we use). Hence, we can reconstruct this resonance from our data. The explicit dependence of these resonances in terms of the background mass density of the background allows us to recover it, i.e. the mass density, in a straightforward way.

2.In addition, this measured contrasted far-fields allow us to recover the total field at the location points of the bubbles (i.e. the total field in the absence of the bubbles). A numerical differentiation argument, for instance, allows us to recover the bulk modulus of the targeted region as well.

2020 Impact Factor: 1.639
5 Year Impact Factor: 1.720
2020 CiteScore: 2.6

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