# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

All Issues

## Inverse Problems and Imaging

December 2020 , Volume 14 , Issue 6

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2020, 14(6): 967-983 doi: 10.3934/ipi.2020044 +[Abstract](1782) +[HTML](239) +[PDF](603.83KB)
Abstract:

We study an inverse problem where an unknown radiating source is observed with collimated detectors along a single line and the medium has a known attenuation. The research is motivated by applications in SPECT and beam hardening. If measurements are carried out with frequencies ranging in an open set, we show that the source density is uniquely determined by these measurements up to averaging over levelsets of the integrated attenuation. This leads to a generalized Laplace transform. We also discuss some numerical approaches and demonstrate the results with several examples.

2020, 14(6): 985-1000 doi: 10.3934/ipi.2020052 +[Abstract](2395) +[HTML](251) +[PDF](5409.15KB)
Abstract:

Seismic data are often undersampled owing to physical or financial limitations. However, complete and regularly sampled data are becoming increasingly critical in seismic processing. In this paper, we present an efficient two-dimensional (2D) seismic data reconstruction method that works on texture-based patches. It performs completion on a patch tensor, which folds texture-based patches into a tensor. Reconstruction is performed by reducing the rank using tensor completion algorithms. This approach differs from past methods, which proceed by unfolding matrices into columns and then applying common matrix completion approaches to deal with 2D seismic data reconstruction. Here, we first re-arrange the seismic data matrix into a third-order patch tensor, by stacking texture-based patches that are divided from seismic data. Then, the seismic data reconstruction problem is formulated into a low-rank tensor completion problem. This formulation avoids destroying the spatial structure, and better extracts the underlying useful information. The proposed method is efficient and gives an improved performance compared with traditional approaches. The effectiveness of our patch tensor-based framework is validated using two classical tensor completion algorithms, low-rank tensor completion (LRTC), and the parallel matrix factorization algorithm (TMac), on both synthetic and field data experiments.

2020, 14(6): 1001-1024 doi: 10.3934/ipi.2020053 +[Abstract](1969) +[HTML](205) +[PDF](1521.29KB)
Abstract:

In this work, an inverse problem in the fractional diffusion equation with random source is considered. The measurements we use are the statistical moments of the realizations of single point observation \begin{document}$u(x_0,t,\omega).$\end{document} We build a representation of the solution \begin{document}$u$\end{document} in the integral sense, then prove some theoretical results like uniqueness and stability. After that, we establish a numerical algorithm to solve the unknowns, where a mollification method is used.

2020, 14(6): 1025-1056 doi: 10.3934/ipi.2020054 +[Abstract](1234) +[HTML](184) +[PDF](1318.16KB)
Abstract:

This paper is concerned with the time-harmonic direct and inverse elastic scattering by an extended rigid elastic body surrounded by a finite number of point-like obstacles. We first justify the point-interaction model for the Lamé operator within the singular perturbation approach. For a general family of pointwise-supported singular perturbations, including anisotropic and non-local interactions, we derive an explicit representation of the scattered field.

In the case of isotropic and local point-interactions, our result is consistent with the ones previously obtained by Foldy's formal method as well as by the renormalization technique. In the case of multiple scattering with pointwise and extended obstacles, we show that the scattered field consists of two parts: one is due to the diffusion by the extended scatterer and the other one is a linear combination of the interactions between the point-like obstacles and the interaction between the point-like obstacles with the extended one.

As to the inverse problem, the factorization method by Kirsch is adapted to recover simultaneously the shape of an extended elastic body and the location of point-like scatterers in the case of isotropic and local interactions. The inverse problems using only one type of elastic waves (i.e. pressure or shear waves) are also investigated and numerical examples are presented to confirm the inversion schemes.

2020, 14(6): 1057-1105 doi: 10.3934/ipi.2020055 +[Abstract](1088) +[HTML](178) +[PDF](611.16KB)
Abstract:

We analyze the behavior of the singularities of solutions of semilinear wave equations after the interaction of three transversal conormal waves. Our results hold for space dimensions two and higher, and for arbitrary \begin{document}${{C}^{\infty }}$\end{document} nonlinearity. The case of two space dimensions in which the nonlinearity is a polynomial was studied by the author and Yiran Wang. We also indicate possible applications to inverse problems.

2020, 14(6): 1107-1133 doi: 10.3934/ipi.2020056 +[Abstract](1847) +[HTML](214) +[PDF](3092.83KB)
Abstract:

This paper concerns the numerical resolution of a data completion problem for the time-harmonic Maxwell equations in the electric field. The aim is to recover the missing data on the inaccessible part of the boundary of a bounded domain from measured data on the accessible part. The non-iterative quasi-reversibility method is studied and different mixed variational formulations are proposed. Well-posedness, convergence and regularity results are proved. Discretization is performed by means of edge finite elements. Various two- and three-dimensional numerical simulations attest the efficiency of the method, in particular for noisy data.

2020, 14(6): 1135-1156 doi: 10.3934/ipi.2020058 +[Abstract](1272) +[HTML](145) +[PDF](3581.38KB)
Abstract:

An ideal image is desirable to faithfully represent the real-world scene. However, the observed images from imaging system are typically involved in the illumination of light. As the human visual system (HVS) is capable of perceiving identical color with spatially varying illumination, retinex theory is established to probe the rationale of HVS on perceiving color. In the realm of image processing, the retinex models are devoted to diminishing illumination effects from observed images. In this paper, following the recent work by Ng and Wang (SIAM J. Imaging Sci. 4:345-356, 2011), we develop a parallel operator splitting algorithm tailored for the constrained total-variation retinex model, in which all the resulting subproblems admit closed form solutions or can be tractably solved by some subroutines without any internally nested iterations. The global convergence of the novel algorithm is analysed on the perspective of variational inequality in optimization community. Preliminary numerical simulations demonstrate the promising performance of the proposed algorithm.

2020, 14(6): 1157-1184 doi: 10.3934/ipi.2020059 +[Abstract](1147) +[HTML](129) +[PDF](8415.43KB)
Abstract:

Following the pioneering works of Rudin, Osher and Fatemi on total variation (TV) and of Buades, Coll and Morel on non-local means (NL-means), the last decade has seen a large number of denoising methods mixing these two approaches, starting with the nonlocal total variation (NLTV) model. The present article proposes an analysis of the NLTV model for image denoising as well as a number of improvements, the most important of which being to apply the denoising both in the space domain and in the Fourier domain, in order to exploit the complementarity of the representation of image data. A local version obtained by a regionwise implementation followed by an aggregation process, called Local Spatial-Frequency NLTV (L-SFNLTV) model, is finally proposed as a new reference algorithm for image denoising among the family of approaches mixing TV and NL operators. The experiments show the great performance of L-SFNLTV in terms of image quality and of computational speed, comparing with other recently proposed NLTV-related methods.

2021 Impact Factor: 1.483
5 Year Impact Factor: 1.462
2021 CiteScore: 2.6