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Inverse Problems and Imaging

October 2021 , Volume 15 , Issue 5

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Large region inpainting by re-weighted regularized methods
Yiting Chen, Jia Li and Qingyun Yu
2021, 15(5): 827-842 doi: 10.3934/ipi.2021015 +[Abstract](1247) +[HTML](362) +[PDF](2698.84KB)

In the development of imaging science and image processing request in our daily life, inpainting large regions always plays an important role. However, the existing local regularized models and some patch manifold based non-local models are often not effective in restoring the features and patterns in the large missing regions. In this paper, we will apply a strategy of inpainting from outside to inside and propose a re-weighted matching algorithm by closest patch (RWCP), contributing to further enhancing the features in the missing large regions. Additionally, we propose another re-weighted matching algorithm by distance-based weighted average (RWWA), leading to a result with higher PSNR value in some cases. Numerical simulations will demonstrate that for large region inpainting, the proposed method is more applicable than most canonical methods. Moreover, combined with image denoising methods, the proposed model is also applicable for noisy image restoration with large missing regions.

Inbetweening auto-animation via Fokker-Planck dynamics and thresholding
Yuan Gao, Guangzhen Jin and Jian-Guo Liu
2021, 15(5): 843-864 doi: 10.3934/ipi.2021016 +[Abstract](2364) +[HTML](336) +[PDF](7296.3KB)

We propose an equilibrium-driven deformation algorithm (EDDA) to simulate the inbetweening transformations starting from an initial image to an equilibrium image, which covers images varying from a greyscale type to a colorful type on planes or manifolds. The algorithm is based on the Fokker-Planck dynamics on manifold, which automatically incorporates the manifold structure suggested by dataset and satisfies positivity, unconditional stability, mass conservation law and exponentially convergence. The thresholding scheme is adapted for the sharp interface dynamics and is used to achieve the finite time convergence. Using EDDA, three challenging examples, (I) facial aging process, (II) coronavirus disease 2019 (COVID-19) pneumonia invading/fading process, and (III) continental evolution process are computed efficiently.

On the identification of the nonlinearity parameter in the Westervelt equation from boundary measurements
Barbara Kaltenbacher and William Rundell
2021, 15(5): 865-891 doi: 10.3934/ipi.2021020 +[Abstract](1325) +[HTML](336) +[PDF](506.72KB)

We consider an undetermined coefficient inverse problem for a nonlinear partial differential equation occurring in high intensity ultrasound propagation as used in acoustic tomography. In particular, we investigate the recovery of the nonlinearity coefficient commonly labeled as \begin{document}$ B/A $\end{document} in the literature which is part of a space dependent coefficient \begin{document}$ \kappa $\end{document} in the Westervelt equation governing nonlinear acoustics. Corresponding to the typical measurement setup, the overposed data consists of time trace measurements on some zero or one dimensional set \begin{document}$ \Sigma $\end{document} representing the receiving transducer array. After an analysis of the map from \begin{document}$ \kappa $\end{document} to the overposed data, we show injectivity of its linearisation and use this as motivation for several iterative schemes to recover \begin{document}$ \kappa $\end{document}. Numerical simulations will also be shown to illustrate the efficiency of the methods.

Quantum tomography and the quantum Radon transform
Alberto Ibort and Alberto López-Yela
2021, 15(5): 893-928 doi: 10.3934/ipi.2021021 +[Abstract](1711) +[HTML](348) +[PDF](647.38KB)

A general framework for the tomographical description of states, that includes, among other tomographical schemes, the classical Radon transform, quantum state tomography and group quantum tomography, in the setting of \begin{document}$ C^* $\end{document}-algebras is presented. Given a \begin{document}$ C^* $\end{document}-algebra, the main ingredients for a tomographical description of its states are identified: A generalized sampling theory and a positive transform. A generalization of the notion of dual tomographic pair provides the background for a sampling theory on \begin{document}$ C^* $\end{document}-algebras and, an extension of Bochner's theorem for functions of positive type, the positive transform.

The abstract theory is realized by using dynamical systems, that is, groups represented on \begin{document}$ C^* $\end{document}-algebra. Using a fiducial state and the corresponding GNS construction, explicit expressions for tomograms associated with states defined by density operators on the corresponding Hilbert spade are obtained. In particular a general quantum version of the classical definition of the Radon transform is presented. The theory is completed by proving that if the representation of the group is square integrable, the representation itself defines a dual tomographic map and explicit reconstruction formulas are obtained by making a judiciously use of the theory of frames. A few significant examples are discussed that illustrates the use and scope of the theory.

Stable recovery of a non-compactly supported coefficient of a Schrödinger equation on an infinite waveguide
Yosra Soussi
2021, 15(5): 929-950 doi: 10.3934/ipi.2021022 +[Abstract](1099) +[HTML](269) +[PDF](352.4KB)

We study the stability issue for the inverse problem of determining a coefficient appearing in a Schrödinger equation defined on an infinite cylindrical waveguide. More precisely, we prove the stable recovery of some general class of non-compactly and non periodic coefficients appearing in an unbounded cylindrical domain. We consider both results of stability from full and partial boundary measurements associated with the so called Dirichlet-to-Neumann map.

Convergence rates of Tikhonov regularization for recovering growth rates in a Lotka-Volterra competition model with diffusion
De-han Chen and Daijun jiang
2021, 15(5): 951-974 doi: 10.3934/ipi.2021023 +[Abstract](1224) +[HTML](292) +[PDF](390.08KB)

In this paper, we shall study the convergence rates of Tikhonov regularizations for the recovery of the growth rates in a Lotka-Volterra competition model with diffusion. The ill-posed inverse problem is transformed into a nonlinear minimization system by an appropriately selected version of Tikhonov regularization. The existence of the minimizers to the minimization system is demonstrated. We shall propose a new variational source condition, which will be rigorously verified under a Hölder type stability estimate. We will also derive the reasonable convergence rates under the new variational source condition.

Near-field imaging for an obstacle above rough surfaces with limited aperture data
Lei Zhang and Luming Jia
2021, 15(5): 975-997 doi: 10.3934/ipi.2021024 +[Abstract](994) +[HTML](304) +[PDF](1511.98KB)

This paper is concerned with the scattering and inverse scattering problems for a point source incident wave by an obstacle embedded in a two-layered background medium. It is a nontrivial extension of the previous theoretical work on the inverse obstacle scattering in an unbounded structure [Commun. Comput. Phys., 26 (2019), 1274-1306]. By the potential theory of boundary integral equations, we derive a novel integral equation formula for the scattering problem, then the well-posedness of the system is proved. Based on the singularity analysis of integral kernels, we presented a numerical method for the integral equations. Furthermore, we developed a reverse time migration method for the corresponding composite inverse scattering problem with the limited aperture data. Numerical experiments show that the proposed method is effective to recover the support of an unknown obstacle and the shape, location of the surfaces.

A note on transmission eigenvalues in electromagnetic scattering theory
Fioralba Cakoni, Shixu Meng and Jingni Xiao
2021, 15(5): 999-1014 doi: 10.3934/ipi.2021025 +[Abstract](1208) +[HTML](339) +[PDF](331.62KB)

This short note was motivated by our efforts to investigate whether there exists a half plane free of transmission eigenvalues for Maxwell's equations. This question is related to solvability of the time domain interior transmission problem which plays a fundamental role in the justification of linear sampling and factorization methods with time dependent data. Our original goal was to adapt semiclassical analysis techniques developed in [21,23] to prove that for some combination of electromagnetic parameters, the transmission eigenvalues lie in a strip around the real axis. Unfortunately we failed. To try to understand why, we looked at the particular example of spherically symmetric media, which provided us with some insight on why we couldn't prove the above result. Hence this paper reports our findings on the location of all transmission eigenvalues and the existence of complex transmission eigenvalues for Maxwell's equations for spherically stratified media. We hope that these results can provide reasonable conjectures for general electromagnetic media.

Two-dimensional inverse scattering for quasi-linear biharmonic operator
Markus Harju, Jaakko Kultima, Valery Serov and Teemu Tyni
2021, 15(5): 1015-1033 doi: 10.3934/ipi.2021026 +[Abstract](1181) +[HTML](275) +[PDF](2545.53KB)

The subject of this work concerns the classical direct and inverse scattering problems for quasi-linear perturbations of the two-dimensional biharmonic operator. The quasi-linear perturbations of the first and zero order might be complex-valued and singular. We show the existence of the scattering solutions to the direct scattering problem in the Sobolev space \begin{document}$ W^1_{\infty}( \mathbb{{R}}^2) $\end{document}. Then the inverse scattering problem can be formulated as follows: does the knowledge of the far field pattern uniquely determine the unknown coefficients for given differential operator? It turns out that the answer to this classical question is affirmative for quasi-linear perturbations of the biharmonic operator. Moreover, we present a numerical method for the reconstruction of unknown coefficients, which from the practical point of view can be thought of as recovery of the coefficients from fixed energy measurements.

On Tikhonov-type regularization with approximated penalty terms
Andreas Neubauer
2021, 15(5): 1035-1050 doi: 10.3934/ipi.2021027 +[Abstract](1244) +[HTML](300) +[PDF](421.92KB)

In this paper, we deal with (nonlinear) ill-posed problems that are regularized by minimizing Tikhonov-type functionals. If the minimization is tedious for some penalty term \begin{document}$ P_0 $\end{document}, we approximate it by a family of penalty terms \begin{document}$ ({P_\beta}) $\end{document} having nicer properties and analyze what happens as \begin{document}$ \beta\to 0 $\end{document}.

We investigate the discrepancy principle for the choice of the regularization parameter and apply all results to linear problems with sparsity constraints. Numerical results show that the proposed method yields good results.

Phase retrieval from Fourier measurements with masks
Huiping Li and Song Li
2021, 15(5): 1051-1075 doi: 10.3934/ipi.2021028 +[Abstract](951) +[HTML](348) +[PDF](449.23KB)

This paper concerns the problem of phase retrieval from Fourier measurements with random masks. Here we focus on researching two kinds of random masks. Firstly, we utilize the Fourier measurements with real masks to estimate a general signal \begin{document}$ \mathit{\boldsymbol{x}}_0\in \mathbb{R}^d $\end{document} in noiseless case when \begin{document}$ d $\end{document} is even. It is demonstrated that \begin{document}$ O(\log^2d) $\end{document} real random masks are able to ensure accurate recovery of \begin{document}$ \mathit{\boldsymbol{x}}_0 $\end{document}. Then we find that such real masks are not adaptable to reconstruct complex signals of even dimension. Subsequently, we prove that \begin{document}$ O(\log^4d) $\end{document} complex masks are enough to stably estimate a general signal \begin{document}$ \mathit{\boldsymbol{x}}_0\in \mathbb{C}^d $\end{document} under bounded noise interference, which extends E. Candès et al.'s work. Meanwhile, we establish tighter error estimations for real signals of even dimensions or complex signals of odd dimensions by using \begin{document}$ O(\log^2d) $\end{document} real masks. Finally, we intend to tackle with the noisy phase problem about an \begin{document}$ s $\end{document}-sparse signal by a robust and efficient approach, namely, two-stage algorithm. Based on the stable guarantees for general signals, we show that the \begin{document}$ s $\end{document}-sparse signal \begin{document}$ \mathit{\boldsymbol{x}}_0 $\end{document} can be stably recovered from composite measurements under near-optimal sample complexity up to a \begin{document}$ \log $\end{document} factor, namely, \begin{document}$ O(s\log(\frac{ed}{s})\log^4(s\log(\frac{ed}{s}))) $\end{document}

A Bayesian level set method for an inverse medium scattering problem in acoustics
Jiangfeng Huang, Zhiliang Deng and Liwei Xu
2021, 15(5): 1077-1097 doi: 10.3934/ipi.2021029 +[Abstract](1112) +[HTML](213) +[PDF](2057.46KB)

In this work, we are interested in the determination of the shape of the scatterer for the two dimensional time harmonic inverse medium scattering problems in acoustics. The scatterer is assumed to be a piecewise constant function with a known value inside inhomogeneities and its shape is represented by the level set functions for which we investigate the information using the Bayesian method. In the Bayesian framework, the solution of the geometric inverse problem is defined as a posterior probability distribution. The well-posedness of the posterior distribution is discussed and the Markov chain Monte Carlo (MCMC) method is applied to generate samples from the posterior distribution. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

Where Bayes tweaks Gauss: Conditionally Gaussian priors for stable multi-dipole estimation
Alessandro Viani, Gianvittorio Luria, Alberto Sorrentino and Harald Bornfleth
2021, 15(5): 1099-1119 doi: 10.3934/ipi.2021030 +[Abstract](987) +[HTML](236) +[PDF](6969.92KB)

We present a very simple yet powerful generalization of a previously described model and algorithm for estimation of multiple dipoles from magneto/electro-encephalographic data. Specifically, the generalization consists in the introduction of a log-uniform hyperprior on the standard deviation of a set of conditionally linear/Gaussian variables. We use numerical simulations and an experimental dataset to show that the approximation to the posterior distribution remains extremely stable under a wide range of values of the hyperparameter, virtually removing the dependence on the hyperparameter.

Velocity modeling based on Rayleigh wave dispersion curve and sparse optimization inversion
Yan Cui and Yanfei Wang
2021, 15(5): 1121-1134 doi: 10.3934/ipi.2021031 +[Abstract](923) +[HTML](245) +[PDF](906.21KB)

This paper studies the S wave velocity modeling based on the Rayleigh wave dispersion curve inversion. We first discuss the forward simulation, and present a fast root-finding method with cubic-order of convergence speed to obtain the Rayleigh wave dispersion curve. With the Rayleigh wave dispersion curve as the observation data, and considering the prior geological anomalies structural information, we establish a sparse constraint regularization model, and propose an iterative solution method to solve for the S wave velocity. Experimental tests are performed both on the theoretical models and on the field data. It indicates from the experimental results that our new inversion scheme possesses the characteristics of easy calculation, high computational efficiency and high precision for model characterization.

3D Electrical Impedance Tomography reconstructions from simulated electrode data using direct inversion $ \mathbf{t}^{\rm{{\textbf{exp}}}} $ and Calderón methods
Sarah J. Hamilton, David Isaacson, Ville Kolehmainen, Peter A. Muller, Jussi Toivanen and Patrick F. Bray
2021, 15(5): 1135-1169 doi: 10.3934/ipi.2021032 +[Abstract](1521) +[HTML](370) +[PDF](17485.5KB)

The first numerical implementation of a \begin{document}$ \mathbf{t}^{\rm{{\textbf{exp}}}} $\end{document} method in 3D using simulated electrode data is presented. Results are compared to Calderón's method as well as more common TV and smoothness regularization-based methods. The \begin{document}$ \mathbf{t}^{\rm{{\textbf{exp}}}} $\end{document} method for EIT is based on tailor-made non-linear Fourier transforms involving the measured current and voltage data. Low-pass filtering in the non-linear Fourier domain is used to stabilize the reconstruction process. In 2D, \begin{document}$ \mathbf{t}^{\rm{{\textbf{exp}}}} $\end{document} methods have shown great promise for providing robust real-time absolute and time-difference conductivity reconstructions but have yet to be used on practical electrode data in 3D, until now. Results are presented for simulated data for conductivity and permittivity with disjoint non-radially symmetric targets on spherical domains and noisy voltage data. The 3D \begin{document}$ \mathbf{t}^{\rm{{\textbf{exp}}}} $\end{document} and Calderón methods are demonstrated to provide comparable quality to their 2D counterparts and hold promise for real-time reconstructions due to their fast, non-optimized, computational cost.


Erratum: The name of the fifth author has been corrected from Jussi Toivainen to Jussi Toivanen. We apologize for any inconvenience this may cause.

Boundary determination of electromagnetic and Lamé parameters with corrupted data
Pedro Caro, Ru-Yu Lai, Yi-Hsuan Lin and Ting Zhou
2021, 15(5): 1171-1198 doi: 10.3934/ipi.2021033 +[Abstract](866) +[HTML](213) +[PDF](428.83KB)

We study boundary determination for an inverse problem associated to the time-harmonic Maxwell equations and another associated to the isotropic elasticity system. We identify the electromagnetic parameters and the Lamé moduli for these two systems from the corresponding boundary measurements. In a first step we reconstruct Lipschitz magnetic permeability, electric permittivity and conductivity on the surface from the ideal boundary measurements. Then, we study inverse problems for Maxwell equations and the isotropic elasticity system assuming that the data contains measurement errors. For both systems, we provide explicit formulas to reconstruct the parameters on the boundary as well as its rate of convergence formula.

An efficient multi-grid method for TV minimization problems
Zhenwei Zhang, Xue Li, Yuping Duan, Ke Yin and Xue-Cheng Tai
2021, 15(5): 1199-1221 doi: 10.3934/ipi.2021034 +[Abstract](1092) +[HTML](256) +[PDF](10678.24KB)

We propose an efficient multi-grid domain decomposition method for solving the total variation (TV) minimization problems. Our multi-grid scheme is developed based on the piecewise constant function spanned subspace correction rather than the piecewise linear one in [17], which ensures the calculation of the TV term only occurs on the boundaries of the support sets. Besides, the domain decomposition method is implemented on each layer to enable parallel computation. Comprehensive comparison results are presented to demonstrate the improvement in CPU time and image quality of the proposed method on medium and large-scale image denoising and reconstruction problems.

Edge detection with mixed noise based on maximum a posteriori approach
Yuying Shi, Zijin Liu, Xiaoying Wang and Jinping Zhang
2021, 15(5): 1223-1245 doi: 10.3934/ipi.2021035 +[Abstract](963) +[HTML](272) +[PDF](14069.12KB)

Edge detection is an important problem in image processing, especially for mixed noise. In this work, we propose a variational edge detection model with mixed noise by using Maximum A-Posteriori (MAP) approach. The novel model is formed with the regularization terms and the data fidelity terms that feature different mixed noise. Furthermore, we adopt the alternating direction method of multipliers (ADMM) to solve the proposed model. Numerical experiments on a variety of gray and color images demonstrate the efficiency of the proposed model.

A linear sampling method for inverse acoustic scattering by a locally rough interface
Jianliang Li, Jiaqing Yang and Bo Zhang
2021, 15(5): 1247-1267 doi: 10.3934/ipi.2021036 +[Abstract](1064) +[HTML](242) +[PDF](857.4KB)

This paper is concerned with the inverse problem of time-harmonic acoustic scattering by an unbounded, locally rough interface which is assumed to be a local perturbation of a plane. The purpose of this paper is to recover the local perturbation of the interface from the near-field measurement given on a straight line segment with a finite distance above the interface and generated by point sources. Precisely, we propose a novel version of the linear sampling method to recover the location and shape of the local perturbation of the interface numerically. Our method is based on a modified near-field operator equation associated with a special rough surface, constructed by reformulating the forward scattering problem into an equivalent integral equation formulation in a bounded domain, leading to a fast imaging algorithm. Numerical experiments are presented to illustrate the effectiveness of the imaging method.

Inverse obstacle scattering for acoustic waves in the time domain
Lu Zhao, Heping Dong and Fuming Ma
2021, 15(5): 1269-1286 doi: 10.3934/ipi.2021037 +[Abstract](1235) +[HTML](381) +[PDF](460.67KB)

This paper concerns an inverse acoustic scattering problem which is to determine the location and shape of a rigid obstacle from time domain scattered field data. An efficient convolution quadrature method combined with nonlinear integral equation method is proposed to solve the inverse problem. In particular, replacing the classic Fourier transform with the convolution quadrature method for time discretization, the boundary integral equations for the Helmholtz equation with complex wave numbers can be obtained to guarantee the numerically approximate causality property of the scattered field under some condition. Numerical experiments are presented to demonstrate the effectiveness and robustness of the proposed method.

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




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