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Inverse Problems & Imaging

February 2022 , Volume 16 , Issue 1

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A stable non-iterative reconstruction algorithm for the acoustic inverse boundary value problem
Tianyu Yang and Yang Yang
2022, 16(1): 1-18 doi: 10.3934/ipi.2021038 +[Abstract](772) +[HTML](331) +[PDF](4816.67KB)

We present a non-iterative algorithm to reconstruct the isotropic acoustic wave speed from the measurement of the Neumann-to-Dirichlet map. The algorithm is designed based on the boundary control method and involves only computations that are stable. We prove the convergence of the algorithm and present its numerical implementation. The effectiveness of the algorithm is validated on both constant speed and variable speed, with full and partial boundary measurement as well as different levels of noise.

Overcomplete representation in a hierarchical Bayesian framework
Monica Pragliola, Daniela Calvetti and Erkki Somersalo
2022, 16(1): 19-38 doi: 10.3934/ipi.2021039 +[Abstract](647) +[HTML](293) +[PDF](5687.93KB)

A common task in inverse problems and imaging is finding a solution that is sparse, in the sense that most of its components vanish. In the framework of compressed sensing, general results guaranteeing exact recovery have been proven. In practice, sparse solutions are often computed combining \begin{document}$ \ell_1 $\end{document}-penalized least squares optimization with an appropriate numerical scheme to accomplish the task. A computationally efficient alternative for finding sparse solutions to linear inverse problems is provided by Bayesian hierarchical models, in which the sparsity is encoded by defining a conditionally Gaussian prior model with the prior parameter obeying a generalized gamma distribution. An iterative alternating sequential (IAS) algorithm has been demonstrated to lead to a computationally efficient scheme, and combined with Krylov subspace iterations with an early termination condition, the approach is particularly well suited for large scale problems. Here the Bayesian approach to sparsity is extended to problems whose solution allows a sparse coding in an overcomplete system such as composite frames. It is shown that among the multiple possible representations of the unknown, the IAS algorithm, and in particular, a hybrid version of it, is effectively identifying the most sparse solution. Computed examples show that the method is particularly well suited not only for traditional imaging applications but also for dictionary learning problems in the framework of machine learning.

Inverse problems for a half-order time-fractional diffusion equation in arbitrary dimension by Carleman estimates
Xinchi Huang and Atsushi Kawamoto
2022, 16(1): 39-67 doi: 10.3934/ipi.2021040 +[Abstract](610) +[HTML](311) +[PDF](375.81KB)

We consider a half-order time-fractional diffusion equation in arbitrary dimension and investigate inverse problems of determining the source term or the diffusion coefficient from spatial data at an arbitrarily fixed time under some additional assumptions. We establish the stability estimate of Lipschitz type in the inverse problems and the proofs are based on the Bukhgeim-Klibanov method by using Carleman estimates.

A mathematical approach towards THz tomography for non-destructive imaging
Simon Hubmer, Alexander Ploier, Ronny Ramlau, Peter Fosodeder and Sandrine van Frank
2022, 16(1): 68-88 doi: 10.3934/ipi.2021041 +[Abstract](725) +[HTML](301) +[PDF](14473.15KB)

In this paper, we consider the imaging problem of terahertz (THz) tomography, in particular as it appears in non-destructive testing. We derive a nonlinear mathematical model describing a full THz tomography experiment, and consider linear approximations connecting THz tomography with standard computerized tomography and the Radon transform. Based on the derived models we propose different reconstruction approaches for solving the THz tomography problem, which we then compare on experimental data obtained from THz measurements of a plastic sample.

On numerical aspects of parameter identification for the Landau-Lifshitz-Gilbert equation in Magnetic Particle Imaging
Tram Thi Ngoc Nguyen and Anne Wald
2022, 16(1): 89-117 doi: 10.3934/ipi.2021042 +[Abstract](682) +[HTML](348) +[PDF](3243.75KB)

The Landau-Lifshitz-Gilbert equation yields a mathematical model to describe the evolution of the magnetization of a magnetic material, particularly in response to an external applied magnetic field. It allows one to take into account various physical effects, such as the exchange within the magnetic material itself. In particular, the Landau-Lifshitz-Gilbert equation encodes relaxation effects, i.e., it describes the time-delayed alignment of the magnetization field with an external magnetic field. These relaxation effects are an important aspect in magnetic particle imaging, particularly in the calibration process. In this article, we address the data-driven modeling of the system function in magnetic particle imaging, where the Landau-Lifshitz-Gilbert equation serves as the basic tool to include relaxation effects in the model. We formulate the respective parameter identification problem both in the all-at-once and the reduced setting, present reconstruction algorithms that yield a regularized solution and discuss numerical experiments. Apart from that, we propose a practical numerical solver to the nonlinear Landau-Lifshitz-Gilbert equation, not via the classical finite element method, but through solving only linear PDEs in an inverse problem framework.

A mathematical perspective on radar interferometry
Mikhail Gilman and Semyon Tsynkov
2022, 16(1): 119-152 doi: 10.3934/ipi.2021043 +[Abstract](555) +[HTML](291) +[PDF](2124.16KB)

Radar interferometry is an advanced remote sensing technology that utilizes complex phases of two or more radar images of the same target taken at slightly different imaging conditions and/or different times. Its goal is to derive additional information about the target, such as elevation. While this kind of task requires centimeter-level accuracy, the interaction of radar signals with the target, as well as the lack of precision in antenna position and other disturbances, generate ambiguities in the image phase that are orders of magnitude larger than the effect of interest.

Yet the common exposition of radar interferometry in the literature often skips such topics. This may lead to unrealistic requirements for the accuracy of determining the parameters of imaging geometry, unachievable precision of image co-registration, etc. To address these deficiencies, in the current work we analyze the problem of interferometric height reconstruction and provide a careful and detailed account of all the assumptions and requirements to the imaging geometry and data processing needed for a successful extraction of height information from the radar data. We employ two most popular scattering models for radar targets: an isolated point scatterer and delta-correlated extended scatterer, and highlight the similarities and differences between them.

Smoothing Newton method for $ \ell^0 $-$ \ell^2 $ regularized linear inverse problem
Peili Li, Xiliang Lu and Yunhai Xiao
2022, 16(1): 153-177 doi: 10.3934/ipi.2021044 +[Abstract](543) +[HTML](393) +[PDF](1625.17KB)

Sparse regression plays a very important role in statistics, machine learning, image and signal processing. In this paper, we consider a high-dimensional linear inverse problem with \begin{document}$ \ell^0 $\end{document}-\begin{document}$ \ell^2 $\end{document} penalty to stably reconstruct the sparse signals. Based on the sufficient and necessary condition of the coordinate-wise minimizers, we design a smoothing Newton method with continuation strategy on the regularization parameter. We prove the global convergence of the proposed algorithm. Several numerical examples are provided, and the comparisons with the state-of-the-art algorithms verify the effectiveness and superiority of the proposed method.

Learning to scan: A deep reinforcement learning approach for personalized scanning in CT imaging
Ziju Shen, Yufei Wang, Dufan Wu, Xu Yang and Bin Dong
2022, 16(1): 179-195 doi: 10.3934/ipi.2021045 +[Abstract](677) +[HTML](270) +[PDF](3446.9KB)

. Computed Tomography (CT) takes X-ray measurements on the subjects to reconstruct tomographic images. As X-ray is radioactive, it is desirable to control the total amount of dose of X-ray for safety concerns. Therefore, we can only select a limited number of measurement angles and assign each of them limited amount of dose. Traditional methods such as compressed sensing usually randomly select the angles and equally distribute the allowed dose on them. In most CT reconstruction models, the emphasize is on designing effective image representations, while much less emphasize is on improving the scanning strategy. The simple scanning strategy of random angle selection and equal dose distribution performs well in general, but they may not be ideal for each individual subject. It is more desirable to design a personalized scanning strategy for each subject to obtain better reconstruction result. In this paper, we propose to use Reinforcement Learning (RL) to learn a personalized scanning policy to select the angles and the dose at each chosen angle for each individual subject. We first formulate the CT scanning process as an Markov Decision Process (MDP), and then use modern deep RL methods to solve it. The learned personalized scanning strategy not only leads to better reconstruction results, but also shows strong generalization to be combined with different reconstruction algorithms.

Identification and stability of small-sized dislocations using a direct algorithm
Batoul Abdelaziz, Abdellatif El Badia and Ahmad El Hajj
2022, 16(1): 197-214 doi: 10.3934/ipi.2021046 +[Abstract](472) +[HTML](239) +[PDF](492.65KB)

This paper considers the problem of identifying dislocation lines of curvilinear form in three-dimensional materials from boundary measurements, when the areas surrounded by the dislocation lines are assumed to be small-sized. The objective of this inverse problem is to reconstruct the number, the initial position and certain characteristics of these dislocations and establish, using certain test functions, a Hölder stability of the centers. This paper can be considered as a generalization of [9], where instead of reconstructing point-wise dislocations, as done in the latter paper, our aim is to recover the parameters of line dislocations by employing a direct algebraic algorithm.

Partial inversion of the 2D attenuated $ X $-ray transform with data on an arc
Hiroshi Fujiwara, Kamran Sadiq and Alexandru Tamasan
2022, 16(1): 215-228 doi: 10.3934/ipi.2021047 +[Abstract](619) +[HTML](276) +[PDF](895.16KB)

In two dimensions, we consider the problem of inversion of the attenuated \begin{document}$ X $\end{document}-ray transform of a compactly supported function from data restricted to lines leaning on a given arc. We provide a method to reconstruct the function on the convex hull of this arc. The attenuation is assumed known. The method of proof uses the Hilbert transform associated with \begin{document}$ A $\end{document}-analytic functions in the sense of Bukhgeim.

Refined stability estimates in electrical impedance tomography with multi-layer structure
Haigang Li, Jenn-Nan Wang and Ling Wang
2022, 16(1): 229-249 doi: 10.3934/ipi.2021048 +[Abstract](565) +[HTML](281) +[PDF](388.27KB)

In this paper we study the inverse problem of determining an electrical inclusion in a multi-layer composite from boundary measurements in 2D. We assume the conductivities in different layers are different and derive a stability estimate for the linearized map with explicit formulae on the conductivity and the thickness of each layer. Intuitively, if an inclusion is surrounded by a highly conductive layer, then, in view of "the principle of the least work", the current will take a path in the highly conductive layer and disregard the existence of the inclusion. Consequently, a worse stability of identifying the hidden inclusion is expected in this case. Our estimates indeed show that the ill-posedness of the problem increases as long as the conductivity of some layer becomes large. This work is an extension of the previous result by Nagayasu-Uhlmann-Wang[15], where a depth-dependent estimate is derived when an inclusion is deeply hidden in a conductor. Estimates in this work also show the influence of the depth of the inclusion.

Runge approximation and stability improvement for a partial data Calderón problem for the acoustic Helmholtz equation
María Ángeles García-Ferrero, Angkana Rüland and Wiktoria Zatoń
2022, 16(1): 251-281 doi: 10.3934/ipi.2021049 +[Abstract](583) +[HTML](246) +[PDF](490.1KB)

In this article, we discuss quantitative Runge approximation properties for the acoustic Helmholtz equation and prove stability improvement results in the high frequency limit for an associated partial data inverse problem modelled on [3,35]. The results rely on quantitative unique continuation estimates in suitable function spaces with explicit frequency dependence. We contrast the frequency dependence of interior Runge approximation results from non-convex and convex sets.

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




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