# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

All Issues

## Inverse Problems and Imaging

August 2022 , Volume 16 , Issue 4

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2022, 16(4): 673-690 doi: 10.3934/ipi.2021069 +[Abstract](1083) +[HTML](332) +[PDF](884.84KB)
Abstract:

In this paper, the Bayesian method is proposed for the interior inverse scattering problem to reconstruct the interface of a two-layered cavity. The scattered field is measured by the point sources located on a closed curve inside the interior interface. The well-posedness of the posterior distribution in the Bayesian framework is proved. The Markov Chain Monte Carlo algorithm is employed to explore the posterior density. Some numerical experiments are presented to demonstrate the effectiveness of the proposed method.

2022, 16(4): 691-702 doi: 10.3934/ipi.2021071 +[Abstract](810) +[HTML](189) +[PDF](699.9KB)
Abstract:

We consider the recovering of the shape of a cavity from the Cauchy datum on an accessible boundary in case of semilinear boundary value problems. Existence and a characterization of the domain derivative of solutions of semilinear elliptic equations are proven. Furthermore, the result is applied to solve an inverse obstacle problem with an iterative regularization scheme. By some numerical examples its performance in case of a Kerr type nonlinearity is illustrated.

2022, 16(4): 703-737 doi: 10.3934/ipi.2021072 +[Abstract](729) +[HTML](188) +[PDF](622.14KB)
Abstract:

We compute the Euler equations of a functional useful for simultaneous video inpainting and motion estimation, which was obtained in [17] as the relaxation of a modified version of the functional proposed in [16]. The functional is defined on vectorial functions of bounded variations, therefore we also get the Euler equations holding on the singular sets of minimizers, highlighting in particular the conditions on the jump sets. Such conditions are expressed by means of traces of geometrically meaningful vector fields and characterized as pointwise limits of averages on cylinders with axes parallel to the unit normals to the jump sets.

2022, 16(4): 739-751 doi: 10.3934/ipi.2021073 +[Abstract](750) +[HTML](195) +[PDF](367.09KB)
Abstract:

A discrete analog is considered for the inverse transmission eigenvalue problem, having applications in acoustics. We provide a well-posed inverse problem statement, develop a constructive procedure for solving this problem, prove uniqueness of solution, global solvability, local solvability, and stability. Our approach is based on the reduction of the discrete transmission eigenvalue problem to a linear system with polynomials of the spectral parameter in the boundary condition.

2022, 16(4): 753-769 doi: 10.3934/ipi.2021074 +[Abstract](848) +[HTML](286) +[PDF](3103.62KB)
Abstract:

Electrical impedance tomography (EIT) is a sensing technique with which conductivity distribution can be reconstructed. It should be mentioned that the reconstruction is a highly ill-posed inverse problem. Currently, the regularization method has been an effective approach to deal with this problem. Especially, total variation regularization method is advantageous over Tikhonov method as the edge information can be well preserved. Nevertheless, the reconstructed image shows severe staircase effect. In this work, to enhance the quality of reconstruction, a novel hybrid regularization model which combines a total generalized variation method with a wavelet frame approach (TGV-WF) is proposed. An efficient mean doubly augmented Lagrangian algorithm has been developed to solve the TGV-WF model. To demonstrate the effectiveness of the proposed method, numerical simulation and experimental validation are conducted for imaging conductivity distribution. Furthermore, some comparisons are made with typical regularization methods. From the results, it can be found that the proposed method shows better performance in the reconstruction since the edge of the inclusion can be well preserved and the staircase effect is effectively relieved.

2022, 16(4): 771-786 doi: 10.3934/ipi.2021075 +[Abstract](748) +[HTML](172) +[PDF](2565.11KB)
Abstract:

In this paper, we address an alternative formulation for the exact inverse formula of the Radon transform on circle arcs arising in a modality of Compton Scattering Tomography in translational geometry proposed by Webber and Miller (Inverse Problems (36)2, 025007, 2020). The original study proposes a first method of reconstruction, using the theory of Volterra integral equations. The numerical realization of such a type of inverse formula may exhibit some difficulties, mainly due to stability issues. Here, we provide a suitable formulation for exact inversion that can be straightforwardly implemented in the Fourier domain. Simulations are carried out to illustrate the efficiency of the proposed reconstruction algorithm.

2022, 16(4): 787-826 doi: 10.3934/ipi.2021076 +[Abstract](807) +[HTML](198) +[PDF](646.11KB)
Abstract:

For an integer \begin{document}$r\ge0$\end{document}, we prove the \begin{document}$r^{\mathrm{th}}$\end{document} order Reshetnyak formula for the ray transform of rank \begin{document}$m$\end{document} symmetric tensor fields on \begin{document}${{\mathbb R}}^n$\end{document}. Roughly speaking, for a tensor field \begin{document}$f$\end{document}, the order \begin{document}$r$\end{document} refers to \begin{document}$L^2$\end{document}-integrability of higher order derivatives of the Fourier transform \begin{document}$\widehat f$\end{document} over spheres centered at the origin. Certain differential operators \begin{document}$A^{(m,r,l)}\ (0\le l\le r)$\end{document} on the sphere \begin{document}${{\mathbb S}}^{n-1}$\end{document} are main ingredients of the formula. The operators are defined by an algorithm that can be applied for any \begin{document}$r$\end{document} although the volume of calculations grows fast with \begin{document}$r$\end{document}. The algorithm is realized for small values of \begin{document}$r$\end{document} and Reshetnyak formulas of orders \begin{document}$0,1,2$\end{document} are presented in an explicit form.

2022, 16(4): 827-870 doi: 10.3934/ipi.2022001 +[Abstract](1444) +[HTML](318) +[PDF](12518.4KB)
Abstract:

We propose a new variational framework to remove a mixture of Gaussian and impulse noise from images. This framework is based on a non-convex PDE-constrained with a fractional-order operator. The non-convex norm is applied to the impulse component controlled by a weighted parameter \begin{document}$\gamma$\end{document}, which depends on the level of the impulse noise and image feature. Furthermore, the fractional operator is used to preserve image texture and edges. In a first part, we study the theoretical properties of the proposed PDE-constrained, and we show some well-posdnees results. In a second part, after having demonstrated how to numerically find a minimizer, a proximal linearized algorithm combined with a Primal-Dual approach is introduced. Moreover, a bi-level optimization framework with a projected gradient algorithm is proposed in order to automatically select the parameter \begin{document}$\gamma$\end{document}. Denoising tests confirm that the non-convex term and learned parameter \begin{document}$\gamma$\end{document} lead in general to an improved reconstruction when compared to results of convex norm and other competitive denoising methods. Finally, we show extensive denoising experiments on various images and noise intensities and we report conventional numerical results which confirm the validity of the non-convex PDE-constrained, its analysis and also the proposed bi-level optimization with learning data.

2022, 16(4): 871-894 doi: 10.3934/ipi.2022002 +[Abstract](690) +[HTML](196) +[PDF](1235.68KB)
Abstract:

Electrical Impedance Tomography gives rise to the severely ill-posed Calderón problem of determining the electrical conductivity distribution in a bounded domain from knowledge of the associated Dirichlet-to-Neumann map for the governing equation. The uniqueness and stability questions for the three-dimensional problem were largely answered in the affirmative in the 1980's using complex geometrical optics solutions, and this led further to a direct reconstruction method relying on a non-physical scattering transform. In this paper, the reconstruction problem is taken one step further towards practical applications by considering data contaminated by noise. Indeed, a regularization strategy for the three-dimensional Calderón problem is presented based on a suitable and explicit truncation of the scattering transform. This gives a certified, stable and direct reconstruction method that is robust to small perturbations of the data. Numerical tests on simulated noisy data illustrate the feasibility and regularizing effect of the method, and suggest that the numerical implementation performs better than predicted by theory.

2022, 16(4): 895-924 doi: 10.3934/ipi.2022004 +[Abstract](784) +[HTML](144) +[PDF](7369.39KB)
Abstract:

Image denoising has always been a challenging task. For performing this task, one of the most effective methods is based on variational PDE. Inspired by the LLT model, we first propose a new adaptive LLT model by adding a weighted function, and then we propose a class of fourth-order diffusion equations based on the new functional. Owing to the adaptive function, the new functional is better than the LLT model and other fourth-order models in terms of edge preservation. While generalizing the Euler-Lagrange equation of the new functional, we discuss a new fourth-order diffusion framework for image denoising. Different from those of other fourth-order diffusion models, the new diffusion coefficients depend on the first-order and second-order derivatives, which can preserve edges and smooth images, respectively. Regarding numerical implementations, we first design an explicit scheme for the proposed model. However, fourth-order diffusion equations require strict stability conditions, and the number of iterations needed is considerable. Consequently, we apply the fast explicit diffusion algorithm (FED) to the explicit scheme to reduce the time consumption of the proposed approach. Furthermore, the additive operator splitting (AOS) scheme is applied for the numerical implementation, and it is the most efficient among all of our algorithms. Finally, compared with other models, the new model exhibits superior effectiveness and efficiency.

2022, 16(4): 925-942 doi: 10.3934/ipi.2022005 +[Abstract](523) +[HTML](121) +[PDF](1660.0KB)
Abstract:

In this paper, we consider the inverse problem of determining the location and the shape of a sound-soft or sound-hard obstacle from the modulus of the total-field collected on a measured curve for an incident point source. We propose a system of nonlinear integral equations based iterative scheme to reconstruct both the location and the shape of the obstacle. Several validating numerical examples are provided to illustrate the effectiveness and robustness of the proposed inversion algorithm.

2022, 16(4): 943-966 doi: 10.3934/ipi.2022006 +[Abstract](661) +[HTML](119) +[PDF](524.93KB)
Abstract:

We consider inverse boundary value problems for polyharmonic operators and in particular, the problem of recovering the coefficients of terms up to order one. The main interest of our result is that it further relaxes the regularity required to establish uniqueness. The proof relies on an averaging technique introduced by Haberman and Tataru for the study of an inverse boundary value problem for a second order operator.

2022, 16(4): 967-995 doi: 10.3934/ipi.2022007 +[Abstract](720) +[HTML](144) +[PDF](3246.13KB)
Abstract:

Abel inversion tomography plays an important role in dynamic experiments, while most known studies are started with a single Gaussian assumption. This paper proposes a mixed Poisson-Laplace-Gaussian distribution to characterize the noise in charge-coupled-device (CCD) sensed radiographic data, and develops a multi-convex optimization model to address the reconstruction problem. The proposed model is derived by incorporating varying amplitude Gaussian approximation and expectation maximization algorithm into an infimal convolution process. To solve it numerically, variable splitting and augmented Lagrangian method are integrated into a block coordinate descent framework, in which anisotropic diffusion and additive operator splitting are employed to gain edge preserving and computation efficiency. Supplementarily, a space of functions of adaptive bounded Hessian is introduced to prove the existence and uniqueness of solution to a higher-order regularized, quadratic subproblem. Moreover, a simplified algorithm with higher order regularizer is derived for Poisson noise removal. To illustrate the performance of the proposed algorithms, numerical tests on synthesized and real digital data are performed.

2022, 16(4): 997-1017 doi: 10.3934/ipi.2022009 +[Abstract](459) +[HTML](102) +[PDF](1138.98KB)
Abstract:

Consider the two-dimensional inverse elastic wave scattering by an infinite rough surface with a Dirichlet boundary condition. A non-iterative sampling method is proposed for detecting the rough surface by taking elastic field measurements on a bounded line segment above the surface, based on reconstructing a modified near-field equation associated with a special surface, which generalized our previous work for the Helmholtz equation (SIAM J. Imag. Sci. 10(3) (2017), 1579-1602) to the Navier equation. Several numerical examples are carried out to illustrate the effectiveness of the inversion algorithm.

2022, 16(4): 1019-1046 doi: 10.3934/ipi.2022010 +[Abstract](639) +[HTML](109) +[PDF](3459.17KB)
Abstract:

Image registration has been widely studied over the past several decades, with numerous applications in science, engineering and medicine. Most of the conventional mathematical models for large deformation image registration rely on prescribed landmarks, which usually require tedious manual labeling. In recent years, there has been a surge of interest in the use of machine learning for image registration. In this paper, we develop a novel method for large deformation image registration by a fusion of quasiconformal theory and convolutional neural network (CNN). More specifically, we propose a quasiconformal energy model with a novel fidelity term that incorporates the features extracted using a pre-trained CNN, thereby allowing us to obtain meaningful registration results without any guidance of prescribed landmarks. Moreover, unlike many prior image registration methods, the bijectivity of our method is guaranteed by quasiconformal theory. Experimental results are presented to demonstrate the effectiveness of the proposed method. More broadly, our work sheds light on how rigorous mathematical theories and practical machine learning approaches can be integrated for developing computational methods with improved performance.

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