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Inverse Problems and Imaging

April 2018 , Volume 12 , Issue 2

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Reconstruction of cloud geometry from high-resolution multi-angle images
Guillaume Bal, Jiaming Chen and Anthony B. Davis
2018, 12(2): 261-280 doi: 10.3934/ipi.2018011 +[Abstract](4352) +[HTML](257) +[PDF](709.18KB)

We consider the reconstruction of the interface of compact, connected "clouds" from satellite or airborne light intensity measurements. In a two-dimensional setting, the cloud is modeled by an interface, locally represented as a graph, and an outgoing radiation intensity that is consistent with a diffusion model for light propagation in the cloud. Light scattering inside the cloud and the internal optical parameters of the cloud are not modeled explicitly. The main objective is to understand what can or cannot be reconstructed in such a setting from intensity measurements in a finite (on the order of 10) number of directions along the path of a satellite or an aircraft. Numerical simulations illustrate the theoretical predictions. Finally, we explore a kinematic extension of the algorithm for retrieving cloud motion (wind) along with its geometry.

On recovery of an inhomogeneous cavity in inverse acoustic scattering
Fenglong Qu and Jiaqing Yang
2018, 12(2): 281-291 doi: 10.3934/ipi.2018012 +[Abstract](4312) +[HTML](274) +[PDF](484.18KB)

Consider the time-harmonic acoustic scattering of an incident point source inside an inhomogeneous cavity. By constructing an equivalent integral equation, the well-posedness of the direct problem is proved in $L^p$ with using the classical Fredholm theory. Motivated by the previous work [10], a novel uniqueness result is then established for the inverse problem of recovering the refractive index of piecewise constant function from the wave fields measured on a closed surface inside the cavity.

Support theorem for the Light-Ray transform of vector fields on Minkowski spaces
Siamak RabieniaHaratbar
2018, 12(2): 293-314 doi: 10.3934/ipi.2018013 +[Abstract](4576) +[HTML](194) +[PDF](724.39KB)

We study the Light-Ray transform of integrating vector fields on the Minkowski time-space $\boldsymbol{{\rm R}}^{1+n}$, $n≥ 2$, with the Minkowski metric. We prove a support theorem for vector fields vanishing on an open set of light-like lines. We provide examples to illustrate the application of our results to the inverse problem for the hyperbolic Dirichlet-to-Neumann map.

Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation
Atsushi Kawamoto
2018, 12(2): 315-330 doi: 10.3934/ipi.2018014 +[Abstract](5206) +[HTML](281) +[PDF](381.52KB)

We consider the first and half order time fractional equation with the zero initial condition. We investigate an inverse source problem of determining the time-independent source factor by the spatial data at an arbitrarily fixed time and we establish the conditional stability estimate of Hölder type in our inverse problem. Our method is based on the Bukhgeim-Klibanov method by means of the Carleman estimate. We also derive the Carleman estimate for the first and half order time fractional diffusion equation.

Mumford-Shah-TV functional with application in X-ray interior tomography
Zhenhua Zhao, Yining Zhu, Jiansheng Yang and Ming Jiang
2018, 12(2): 331-348 doi: 10.3934/ipi.2018015 +[Abstract](5093) +[HTML](306) +[PDF](1324.77KB)

Both total variation (TV) and Mumford-Shah (MS) functional are broadly used for regularization of various ill-posed problems in the field of imaging and image processing. Incorporating MS functional with TV, we propose a new functional, named as Mumford-Shah-TV (MSTV), for the object image of piecewise constant. Both the image and its edge can be reconstructed by MSTV regularization method. We study the regularizing properties of MSTV functional and present an Ambrosio-Tortorelli type approximation for it in the sense of Γ-convergence. We apply MSTV regularization method to the interior problem of X-ray CT and develop an algorithm based on split Bregman and OS-SART iterations. Numerical and physical experiments demonstrate that high-quality image and its edge within the ROI can be reconstructed using the regularization method and algorithm we proposed.

The factorization method for cracks in elastic scattering
Jun Guo, Qinghua Wu and Guozheng Yan
2018, 12(2): 349-371 doi: 10.3934/ipi.2018016 +[Abstract](4602) +[HTML](236) +[PDF](1232.66KB)

This paper is concerned with the scattering problems of a crack with Dirichlet or mixed impedance boundary conditions in two dimensional isotropic and linearized elasticity. The well posedness of the direct scattering problems for both situations are studied by the boundary integral equation method. The inverse scattering problems we are dealing with are the shape reconstruction of the crack from the knowledge of far field patterns due to the incident plane compressional and shear waves. We aim at extending the well known factorization method to crack determination in inverse elastic scattering, although it has been proved valid in acoustic and electromagnetic scattering, electrical impedance tomography and so on. The numerical examples are presented to illustrate the feasibility of this method.

Enhancing D-bar reconstructions for electrical impedance tomography with conformal maps
Nuutti Hyvönen, Lassi Päivärinta and Janne P. Tamminen
2018, 12(2): 373-400 doi: 10.3934/ipi.2018017 +[Abstract](5749) +[HTML](243) +[PDF](1593.57KB)

We present a few ways of using conformal maps in the reconstruction of two-dimensional conductivities in electrical impedance tomography. First, by utilizing the Riemann mapping theorem, we can transform any simply connected domain of interest to the unit disk where the D-bar method can be implemented most efficiently. In particular, this applies to the open upper half-plane. Second, in the unit disk we may choose a region of interest that is magnified using a suitable Möbius transform. To facilitate the efficient use of conformal maps, we introduce input current patterns that are named conformally transformed truncated Fourier basis; in practice, their use corresponds to positioning the available electrodes close to the region of interest. These ideas are numerically tested using simulated continuum data in bounded domains and simulated point electrode data in the half-plane. The connections to practical electrode measurements are also discussed.

Numerical method for image registration model based on optimal mass transport
Yangang Chen and Justin W. L. Wan
2018, 12(2): 401-432 doi: 10.3934/ipi.2018018 +[Abstract](5449) +[HTML](315) +[PDF](2262.71KB)

This paper proposes a numerical method for solving a non-rigid image registration model based on optimal mass transport. The main contribution of this paper is to address two issues. One is that we impose a proper periodic boundary condition, such that when the reference and template images are related by translation, or a combination of translation and non-rigid deformation, the numerical solution gives the underlying transformation. The other is that we design a numerical scheme that converges to the optimal transformation between the two images. As an additional benefit, our approach can decompose the transformation into translation and non-rigid deformation. Our numerical results show that the numerical solutions yield good-quality transformations for non-rigid image registration problems.

Efficient tensor tomography in fan-beam coordinates. Ⅱ: Attenuated transforms
François Monard
2018, 12(2): 433-460 doi: 10.3934/ipi.2018019 +[Abstract](4232) +[HTML](194) +[PDF](557.75KB)

This article extends the author's past work [11] to attenuated X-ray transforms, where the attenuation is complex-valued and only depends on position. We give a positive and constructive answer to the attenuated tensor tomography problem on the Euclidean unit disc in fan-beam coordinates. For a tensor of arbitrary order, we propose an equivalent tensor of the same order which can be uniquely and stably reconstructed from its attenuated transform, as well as an explicit and efficient procedure to do so.

Cloaking for a quasi-linear elliptic partial differential equation
Tuhin Ghosh and Karthik Iyer
2018, 12(2): 461-491 doi: 10.3934/ipi.2018020 +[Abstract](4678) +[HTML](201) +[PDF](533.77KB)

In this article we consider cloaking for a quasi-linear elliptic partial differential equation of divergence type defined on a bounded domain in $\mathbb{R}^N$ for $N = 2, 3$. We show that a perfect cloak can be obtained via a singular change of variables scheme and an approximate cloak can be achieved via a regular change of variables scheme. These approximate cloaks, though non-degenerate, are anisotropic. We also show, within the framework of homogenization, that it is possible to get isotropic regular approximate cloaks. This work generalizes to quasi-linear settings previous work on cloaking in the context of Electrical Impedance Tomography for the conductivity equation.

A globally convergent numerical method for a 3D coefficient inverse problem with a single measurement of multi-frequency data
Michael V. Klibanov, Dinh-Liem Nguyen, Loc H. Nguyen and Hui Liu
2018, 12(2): 493-523 doi: 10.3934/ipi.2018021 +[Abstract](5243) +[HTML](292) +[PDF](1037.64KB)

The goal of this paper is to reconstruct spatially distributed dielectric constants from complex-valued scattered wave field by solving a 3D coefficient inverse problem for the Helmholtz equation at multi-frequencies. The data are generated by only a single direction of the incident plane wave. To solve this inverse problem, a globally convergent algorithm is analytically developed. We prove that this algorithm provides a good approximation for the exact coefficient without any a priori knowledge of any point in a small neighborhood of that coefficient. This is the main advantage of our method, compared with classical approaches using optimization schemes. Numerical results are presented for both computationally simulated data and experimental data. Potential applications of this problem are in detection and identification of explosive-like targets.

A note on "Anisotropic total variation regularized $L^1$-approximation and denoising/deblurring of 2D bar codes"
Nils Dabrock and Yves van Gennip
2018, 12(2): 525-526 doi: 10.3934/ipi.2018022 +[Abstract](5171) +[HTML](197) +[PDF](227.19KB)

This note addresses an error in [1].

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




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