# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

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## Inverse Problems & Imaging

June 2019 , Volume 13 , Issue 3

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2019, 13(3): 431-447 doi: 10.3934/ipi.2019021 +[Abstract](1017) +[HTML](196) +[PDF](439.8KB)
Abstract:

We consider the inverse dynamic problem for a dynamical system with discrete time associated with a semi-infinite Jacobi matrix. We derive discrete analogs of Krein equations and answer a question on the characterization of dynamic inverse data. As a consequence we obtain a necessary and sufficient condition for a measure on a real line to be a spectral measure of a semi-infinite discrete Schrödinger operator.

2019, 13(3): 449-460 doi: 10.3934/ipi.2019022 +[Abstract](1159) +[HTML](230) +[PDF](2343.31KB)
Abstract:

The patch manifold of a natural image has a low dimensional structure and accommodates rich structural information. Inspired by the recent work of the low-dimensional manifold model (LDMM), we apply the LDMM for regularizing X-ray computed tomography (CT) image reconstruction. This proposed method recovers detailed structural information of images, significantly enhancing spatial and contrast resolution of CT images. Both numerically simulated data and clinically experimental data are used to evaluate the proposed method. The comparative studies are also performed over the simultaneous algebraic reconstruction technique (SART) incorporated the total variation (TV) regularization to demonstrate the merits of the proposed method. Results indicate that the LDMM-based method enables a more accurate image reconstruction with high fidelity and contrast resolution.

2019, 13(3): 461-478 doi: 10.3934/ipi.2019023 +[Abstract](1040) +[HTML](307) +[PDF](16613.36KB)
Abstract:

Image contrast enhancement plays an important role in computer vision and pattern recognition by improving image quality. The main aim of this paper is to propose and develop a variational model for contrast enhancement of color images based on local gamma correction. The proposed variational model contains an energy functional to determine a local gamma function such that the gamma values can be set according to the local information of the input image. A spatial regularization of the gamma function is incorporated into the functional so that the contrast in an image can be modified by using the information of each pixel and its neighboring pixels. Another regularization term is also employed to preserve the ordering of pixel values. Theoretically, the existence and uniqueness of the minimizer of the proposed model are established. A fast algorithm can be developed to solve the resulting minimization model. Experimental results on benchmark images are presented to show that the performance of the proposed model are better than that of the other testing methods.

2019, 13(3): 479-511 doi: 10.3934/ipi.2019024 +[Abstract](927) +[HTML](213) +[PDF](2090.23KB)
Abstract:

We introduce in this study an algorithm for the imaging of faults and of slip fields on those faults. The physics of this problem are modeled using the equations of linear elasticity. We define a regularized functional to be minimized for building the image. We first prove that the minimum of that functional converges to the unique solution of the related fault inverse problem. Due to inherent uncertainties in measurements, rather than seeking a deterministic solution to the fault inverse problem, we then consider a Bayesian approach. The randomness involved in the unknown slip is teased out by assuming independence of the priors, and we show how the regularized error functional introduced earlier can be used to recover the probability density of the geometry parameter. The advantage of this Bayesian approach is that we obtain a way of quantifying uncertainties as part of our final answer. On the downside, this approach leads to a very large computation which we implemented on a parallel platform. After showing how this algorithm performs on simulated data, we apply it to measured data. The data was recorded during a slow slip event in Guerrero, Mexico.

2019, 13(3): 513-543 doi: 10.3934/ipi.2019025 +[Abstract](1054) +[HTML](272) +[PDF](2119.19KB)
Abstract:

Periodic parameters are common and important in stochastic differential equations (SDEs) arising in many contemporary scientific and engineering fields involving dynamical processes. These parameters include the damping coefficient, the volatility or diffusion coefficient and possibly an external force. Identification of these periodic parameters allows a better understanding of the dynamical processes and their hidden intermittent instability. Conventional approaches usually assume that one of the parameters is known and focus on the recovery of rest parameters. By introducing the decorrelation time and calculating the standard Gaussian statistics (mean, variance) explicitly for the scalar Langevin equations with periodic parameters, we propose a parameter identification approach to simultaneously recovering all these parameters by observing a single trajectory of SDEs. Such an approach is summarized in form of regularization schemes with noisy operators and noisy right-hand sides and is further extended to parameter identification of SDEs which are indirectly observed by other random processes. Numerical examples show that our approach performs well in stable and weakly unstable regimes but may fail in strongly unstable regime which is induced by the strong intermittent instability itself.

2019, 13(3): 545-573 doi: 10.3934/ipi.2019026 +[Abstract](1099) +[HTML](220) +[PDF](1030.42KB)
Abstract:

Consider an exterior problem of the three-dimensional elastic wave equation, which models the scattering of a time-harmonic plane wave by a rigid obstacle. The scattering problem is reformulated into a boundary value problem by introducing a transparent boundary condition. Given the incident field, the direct problem is to determine the displacement of the wave field from the known obstacle; the inverse problem is to determine the obstacle's surface from the measurement of the displacement on an artificial boundary enclosing the obstacle. In this paper, we consider both the direct and inverse problems. The direct problem is shown to have a unique weak solution by examining its variational formulation. The domain derivative is studied and a frequency continuation method is developed for the inverse problem. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

2019, 13(3): 575-596 doi: 10.3934/ipi.2019027 +[Abstract](906) +[HTML](217) +[PDF](1355.72KB)
Abstract:

An inverse boundary value problem for the 1+1 dimensional wave equation \begin{document}$(\partial_t^2 - c(x)^2 \partial_x^2)u(x,t) = 0,\quad x\in\mathbb{R}_+$\end{document} is considered. We give a discrete regularization strategy to recover wave speed \begin{document}$c(x)$\end{document} when we are given the boundary value of the wave, \begin{document}$u(0,t)$\end{document}, that is produced by a single pulse-like source. The regularization strategy gives an approximative wave speed \begin{document}$\widetilde c$\end{document}, satisfying a Hölder type estimate \begin{document}$\| \widetilde c-c\|\leq C \epsilon^{\gamma}$\end{document}, where \begin{document}$\epsilon$\end{document} is the noise level.

2019, 13(3): 597-633 doi: 10.3934/ipi.2019028 +[Abstract](918) +[HTML](191) +[PDF](679.32KB)
Abstract:

For a given smooth compact manifold \begin{document}$M$\end{document}, we introduce an open class \begin{document}$\mathcal G(M)$\end{document} of Riemannian metrics, which we call metrics of the gradient type. For such metrics \begin{document}$g$\end{document}, the geodesic flow \begin{document}$v^g$\end{document} on the spherical tangent bundle \begin{document}$SM \to M$\end{document} admits a Lyapunov function (so the \begin{document}$v^g$\end{document}-flow is traversing). It turns out, that metrics of the gradient type are exactly the non-trapping metrics.

For every \begin{document}$g \in \mathcal G(M)$\end{document}, the geodesic scattering along the boundary \begin{document}$\partial M$\end{document} can be expressed in terms of the scattering map \begin{document}$C_{v^g}: \partial_1^+(SM) \to \partial_1^-(SM)$\end{document}. It acts from a domain \begin{document}$\partial_1^+(SM)$\end{document} in the boundary \begin{document}$\partial(SM)$\end{document} to the complementary domain \begin{document}$\partial_1^-(SM)$\end{document}, both domains being diffeomorphic. We prove that, for a boundary generic metric \begin{document}$g \in \mathcal G(M)$\end{document}, the map \begin{document}$C_{v^g}$\end{document} allows for a reconstruction of \begin{document}$SM$\end{document} and of the geodesic foliation \begin{document}$\mathcal F(v^g)$\end{document} on it, up to a homeomorphism (often a diffeomorphism).

Also, for such \begin{document}$g$\end{document}, the knowledge of the scattering map \begin{document}$C_{v^g}$\end{document} makes it possible to recover the homology of \begin{document}$M$\end{document}, the Gromov simplicial semi-norm on it, and the fundamental group of \begin{document}$M$\end{document}. Additionally, \begin{document}$C_{v^g}$\end{document} allows to reconstruct the naturally stratified topological type of the space of geodesics on \begin{document}$M$\end{document}.

We aim to understand the constraints on \begin{document}$(M, g)$\end{document}, under which the scattering data allow for a reconstruction of \begin{document}$M$\end{document} and the metric \begin{document}$g$\end{document} on it, up to a natural action of the diffeomorphism group \begin{document}$\mathsf{Diff}(M, \partial M)$\end{document}. In particular, we consider a closed Riemannian \begin{document}$n$\end{document}-manifold \begin{document}$(N, g)$\end{document} which is locally symmetric and of negative sectional curvature. Let \begin{document}$M$\end{document} is obtained from \begin{document}$N$\end{document} by removing an \begin{document}$n$\end{document}-domain \begin{document}$U$\end{document}, such that the metric \begin{document}$g|_M$\end{document} is boundary generic, of the gradient type, and the homomorphism \begin{document}$\pi_1(U) \to \pi_1(N)$\end{document} of the fundamental groups is trivial. Then we prove that the scattering map \begin{document}$C_{v^{g|_M}}$\end{document} makes it possible to recover \begin{document}$N$\end{document} and the metric \begin{document}$g$\end{document} on it.

2019, 13(3): 635-652 doi: 10.3934/ipi.2019029 +[Abstract](1048) +[HTML](255) +[PDF](903.05KB)
Abstract:

The establishment of relevant model and solving an inverse random source problem are one of the main tools for analyzing mechanical properties of elastic materials. In this paper, we study an inverse random source problem for biharmonic equation in two dimension. Under some regularity assumptions on the structure of random source, the well-posedness of the forward problem is established. Moreover, based on the explicit solution of the forward problem, we can solve the corresponding inverse random source problem via two transformed integral equations. Numerical examples are presented to illustrate the validity and effectiveness of the proposed inversion method.

2019, 13(3): 653-677 doi: 10.3934/ipi.2019030 +[Abstract](1206) +[HTML](293) +[PDF](3011.76KB)
Abstract:

A dual expectation-maximization (EM) algorithm for total variation (TV) regularized Gaussian mixture model (GMM) is proposed in this paper. The algorithm is built upon the EM algorithm with TV regularization (EM-TV) model which combines the statistical and variational methods together for image segmentation. Inspired by the projection algorithm proposed by Chambolle, we give a dual algorithm for the EM-TV model. The related dual problem is smooth and can be easily solved by a projection gradient method, which is stable and fast. Given the parameters of GMM, the proposed algorithm can be seen as a forward-backward splitting method which converges. This method can be easily extended to many other applications. Numerical results show that our algorithm can provide high quality segmentation results with fast computation speed. Compared with the well-known statistics based methods such as hidden Markov random field with EM method (HMRF-EM), the proposed algorithm has a better performance. The proposed method could also be applied to MRI segmentation such as SPM8 software and improve the segmentation results.

2019, 13(3): 679-701 doi: 10.3934/ipi.2019031 +[Abstract](967) +[HTML](221) +[PDF](499.91KB)
Abstract:

The momentum ray transform \begin{document}$I^k$\end{document} integrates a rank \begin{document}$m$\end{document} symmetric tensor field \begin{document}$f$\end{document} over lines in \begin{document}$\mathbb{R}^n$\end{document} with the weight \begin{document}$t^k$\end{document}: \begin{document}$(I^k\!f)(x,\xi) = \int_{-\infty}^\infty t^k\langle f(x+t\xi),\xi^m\rangle\, \mathrm{d} t.$\end{document} In particular, the ray transform \begin{document}$I = I^0$\end{document} was studied by several authors since it had many tomographic applications. We present an algorithm for recovering \begin{document}$f$\end{document} from the data \begin{document}$(I^0\!f,I^1\!f,\dots, I^m\!f)$\end{document}. In the cases of \begin{document}$m = 1$\end{document} and \begin{document}$m = 2$\end{document}, we derive the Reshetnyak formula that expresses \begin{document}$\|f\|_{H^s_t({\mathbb R}^n)}$\end{document} through some norm of \begin{document}$(I^0\!f,I^1\!f,\dots, I^m\!f)$\end{document}. The \begin{document}$H^{s}_{t}$\end{document}-norm is a modification of the Sobolev norm weighted differently at high and low frequencies. Using the Reshetnyak formula, we obtain a stability estimate.

2018  Impact Factor: 1.469