# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

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## Inverse Problems & Imaging

October 2017 , Volume 11 , Issue 5

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2017, 11(5): 783-798 doi: 10.3934/ipi.2017037 +[Abstract](1527) +[HTML](5) +[PDF](4267.6KB)
Abstract:

Surface denoising is a fundamental problem in geometry processing and computer graphics. In this paper, we propose a wavelet frame based variational model to restore surfaces which are corrupted by mixed Gaussian and impulse noise, under the assumption that the region corrupted by impulse noise is unknown. The model contains a universal \begin{document}$\ell_1 + \ell_2$\end{document} fidelity term and an \begin{document}$\ell_1$\end{document}-regularized term which makes additional use of the wavelet frame transform on surfaces in order to preserve key features such as sharp edges and corners. We then apply the augmented Lagrangian and accelerated proximal gradient methods to solve this model. In the end, we demonstrate the efficacy of our approach with numerical experiments both on surfaces and functions defined on surfaces. The experimental results show that our method is competitive relative to some existing denoising methods.

2017, 11(5): 799-823 doi: 10.3934/ipi.2017038 +[Abstract](2419) +[HTML](5) +[PDF](1873.6KB)
Abstract:

This paper examines issues of data completion and location uncertainty, popular in many practical PDE-based inverse problems, in the context of option calibration via recovery of local volatility surfaces. While real data is usually more accessible for this application than for many others, the data is often given only at a restricted set of locations. We show that attempts to "complete missing data" by approximation or interpolation, proposed and applied in the literature, may produce results that are inferior to treating the data as scarce. Furthermore, model uncertainties may arise which translate to uncertainty in data locations, and we show how a model-based adjustment of the asset price may prove advantageous in such situations. We further compare a carefully calibrated Tikhonov-type regularization approach against a similarly adapted EnKF method, in an attempt to fine-tune the data assimilation process. The EnKF method offers reassurance as a different method for assessing the solution in a problem where information about the true solution is difficult to come by. However, additional advantage in the latter approach turns out to be limited in our context.

2017, 11(5): 825-855 doi: 10.3934/ipi.2017039 +[Abstract](2860) +[HTML](4) +[PDF](1873.6KB)
Abstract:

Curvilinear surfaces in 3D Euclidean spaces are commonly represented by triangular meshes. The structure of the triangulation is important, since it affects the accuracy and efficiency of the numerical computation on the mesh. Remeshing refers to the process of transforming an unstructured mesh to one with desirable structures, such as the subdivision connectivity. This is commonly achieved by parameterizing the surface onto a simple parameter domain, on which a structured mesh is built. The 2D structured mesh is then projected onto the surface via the parameterization. Two major tasks are involved. Firstly, an effective algorithm for parameterizing, usually conformally, surface meshes is necessary. However, for a highly irregular mesh with skinny triangles, computing a folding-free conformal parameterization is difficult. The second task is to build a structured mesh on the parameter domain that is adaptive to the area distortion of the parameterization while maintaining good shapes of triangles. This paper presents an algorithm to remesh a highly irregular mesh to a structured one with subdivision connectivity and good triangle quality. We propose an effective algorithm to obtain a conformal parameterization of a highly irregular mesh, using quasi-conformal Teichmüller theories. Conformality distortion of an initial parameterization is adjusted by a quasi-conformal map, resulting in a folding-free conformal parameterization. Next, we propose an algorithm to obtain a regular mesh with subdivision connectivity and good triangle quality on the conformal parameter domain, which is adaptive to the area distortion, through the landmark-matching Teichmüller map. A remeshed surface can then be obtained through the parameterization. Experiments have been carried out to remesh surface meshes representing real 3D geometric objects using the proposed algorithm. Results show the efficacy of the algorithm to optimize the regularity of an irregular triangulation.

2017, 11(5): 857-874 doi: 10.3934/ipi.2017040 +[Abstract](2059) +[HTML](4) +[PDF](990.3KB)
Abstract:

This article extends the framework of Bayesian inverse problems in infinite-dimensional parameter spaces, as advocated by Stuart (Acta Numer. 19:451–559,2010) and others, to the case of a heavy-tailed prior measure in the family of stable distributions, such as an infinite-dimensional Cauchy distribution, for which polynomial moments are infinite or undefined. It is shown that analogues of the Karhunen–Loéve expansion for square-integrable random variables can be used to sample such measures on quasi-Banach spaces. Furthermore, under weaker regularity assumptions than those used to date, the Bayesian posterior measure is shown to depend Lipschitz continuously in the Hellinger metric upon perturbations of the misfit function and observed data.

2017, 11(5): 875-900 doi: 10.3934/ipi.2017041 +[Abstract](1679) +[HTML](4) +[PDF](668.4KB)
Abstract:

In this work, we consider a FDE (fractional diffusion equation)

with a time-dependent diffusion coefficient \begin{document}$a(t)$\end{document}. This is an extension of [13], which deals with this FDE in one-dimensional space. For the direct problem, given an \begin{document}$a(t),$\end{document} we establish the existence, uniqueness and some regularity properties with a more general domain \begin{document}$Ω$\end{document} and right-hand side \begin{document}$F(x,t)$\end{document}. For the inverse problem–recovering \begin{document}$a(t),$\end{document} we introduce an operator \begin{document}$K$\end{document} one of whose fixed points is \begin{document}$a(t)$\end{document} and show its monotonicity, uniqueness and existence of its fixed points. With these properties, a reconstruction algorithm for \begin{document}$a(t)$\end{document} is created and some numerical results are provided to illustrate the theories.

2017, 11(5): 901-916 doi: 10.3934/ipi.2017042 +[Abstract](2133) +[HTML](10) +[PDF](864.5KB)
Abstract:

We propose a direct imaging method based on the reverse time migration method for finding extended obstacles with phaseless total field data in the half space. We prove that the imaging resolution of the method is essentially the same as the imaging results using the scattering data with full phase information when the obstacle is far away from the surface of the half-space where the measurement is taken. Numerical experiments are included to illustrate the powerful imaging quality.

2018  Impact Factor: 1.469