# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

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## Inverse Problems & Imaging

February 2017 , Volume 11 , Issue 1

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2017, 11(1): 1-23 doi: 10.3934/ipi.2017001 +[Abstract](3074) +[HTML](149) +[PDF](1114.3KB)
Abstract:

In this paper, we propose an image segmentation model where an $L^1$ variant of the Euler's elastica energy is used as boundary regularization. An interesting feature of this model lies in its preference for convex segmentation contours. However, due to the high order and non-differentiability of Euler's elastica energy, it is nontrivial to minimize the associated functional. As in recent work on the ordinary $L^2$-Euler's elastica model in imaging, we propose using an augmented Lagrangian method to tackle the minimization problem. Specifically, we design a novel augmented Lagrangian functional that deals with the mean curvature term differently than in previous works. The new treatment reduces the number of Lagrange multipliers employed, and more importantly, it helps represent the curvature more effectively and faithfully. Numerical experiments validate the efficiency of the proposed augmented Lagrangian method and also demonstrate new features of this particular segmentation model, such as shape driven and data driven properties.

2017, 11(1): 25-45 doi: 10.3934/ipi.2017002 +[Abstract](1944) +[HTML](121) +[PDF](5035.9KB)
Abstract:

In this work, we present a modified Time-Reversal Mirror (TRM) Method, called Source Time Reversal (STR), to find the spatial distribution of a seismic source induced by mining activity. This methodology is based on a known full description of the temporal dependence of the source, the Duhamel's principle, and the time-reverse property of the wave equation. We also provide an error estimate of the reconstruction when the measurements are acquired over the entire boundary, and we show experimentally the influence of measuring on a subdomain of the boundary. Numerical results indicate that the methodology is able to recover continuous and discontinuous sources, and it remains stable for partial boundary measurements.

2017, 11(1): 47-64 doi: 10.3934/ipi.2017003 +[Abstract](1482) +[HTML](104) +[PDF](380.4KB)
Abstract:

In the paper [1] it is shown that the set of Riemannian metrics which do not admit global limiting Carleman weights is open and dense, by studying the conformally invariant Weyl and Cotton tensors. In the paper [7] it is shown that the set of Riemannian metrics which do not admit local limiting Carleman weights at any point is residual, showing that it contains the set of metrics for which there are no local conformal diffeomorphisms between any distinct open subsets. This paper is a continuation of [1], in order to prove that the set of Riemannian metrics which do not admit local limiting Carleman weights at any point is open and dense.

2017, 11(1): 65-85 doi: 10.3934/ipi.2017004 +[Abstract](3190) +[HTML](113) +[PDF](2330.4KB)
Abstract:

In this paper, we focus on the challenging problem of removing the spatially varying out-of-focus blur from a single natural image. We first propose an effective method to estimate the blur map by the total variation refinement on Hölder coefficient, then discuss the properties of the corresponding kernel matrix. A tight-frame based energy functional, whose minimizer is related to the optimal defocus result, is thus built. For tackling functional more efficiently, we describe the numerical procedure based on an accelerated primal-dual scheme. To verify the effectiveness of our method, we compare it with some state-of-the-art schemes using both synthesized and natural images. Experimental results demonstrate that the proposed method performs better than the compared methods.

2017, 11(1): 87-97 doi: 10.3934/ipi.2017005 +[Abstract](1778) +[HTML](89) +[PDF](339.1KB)
Abstract:

We describe new mathematical structures associated with the scattering of plane waves in piecewise constant layered media, a basic model for acoustic imaging of laminated structures and in geophysics. Using explicit formulas for the reflection Green's function it is shown that the measurement operator satisfies a system of quasilinear PDE with smooth coefficients, and that the sum of the amplitude data has a simple expression in terms of inverse hyperbolic tangent of the reflection coefficients. In addition we derive a simple geometric description of the measured data, which, in the generic case, yields a natural factorization of the inverse problem.

2017, 11(1): 99-123 doi: 10.3934/ipi.2017006 +[Abstract](1913) +[HTML](843) +[PDF](528.2KB)
Abstract:

An inverse obstacle scattering problem for the wave governed by the Maxwell system in the time domain, in particular, over a finite time interval is considered. It is assumed that the electric field \begin{document}$\boldsymbol{E}$\end{document} and magnetic field \begin{document}$\boldsymbol{ H}$\end{document} which are solutions of the Maxwell system are generated only by a current density at the initial time located not far a way from an unknown obstacle. The obstacle is embedded in a medium like air which has constant electric permittivity \begin{document}$ε$\end{document} and magnetic permeability \begin{document}$μ$\end{document}. It is assumed that the fields on the surface of the obstacle satisfy the Leontovich boundary condition \begin{document}$\boldsymbol{ ν}×\boldsymbol{H}-λ\,\boldsymbol{ ν}×(\boldsymbol{ E}×\boldsymbol{ ν})=\boldsymbol{ 0}$\end{document} with admittance \begin{document}$λ$\end{document} an unknown positive function and \begin{document}$\boldsymbol{ ν}$\end{document} the unit outward normal. The observation data are given by the electric field observed at the same place as the support of the current density over a finite time interval. It is shown that an indicator function computed from the electric fields corresponding two current densities enables us to know: the distance of the center of the common spherical support of the current densities to the obstacle; whether the value of the admittance \begin{document}$λ$\end{document} is greater or less than the special value \begin{document}$\sqrt{ε/μ}$\end{document}.

2017, 11(1): 125-149 doi: 10.3934/ipi.2017007 +[Abstract](2331) +[HTML](133) +[PDF](533.1KB)
Abstract:

An inverse problem to determine a space-dependent factor in a semilinear time-fractional diffusion equation is considered. Additional data are given in the form of an integral with the Borel measure over the time. Uniqueness of the solution of the inverse problem is studied. The method uses a positivity principle of the corresponding differential equation that is also proved in the paper.

2017, 11(1): 151-176 doi: 10.3934/ipi.2017008 +[Abstract](2578) +[HTML](90) +[PDF](996.2KB)
Abstract:

We consider Tikhonov and sparsity-promoting regularization in Banach spaces for inverse scattering from penetrable anisotropic media. To this end, we equip an admissible set of material parameters with the \begin{document}$L^p$\end{document}-topology and use Meyers' gradient estimate for solutions of elliptic equations to analyze the dependence of scattered fields and their Fréchet derivatives on the material parameter. This allows to show convergence of a non-linear Tikhonov regularization against a minimum-norm solution to the inverse problem, but also to set up sparsity-promoting versions of that regularization method. For both approaches, the discrepancy is defined via a \begin{document}$q$\end{document}-Schatten norm or an \begin{document}$L^q$\end{document}-norm with \begin{document}$1 < q < ∞$\end{document}. Numerical reconstruction examples indicate the reconstruction quality of the method, as well as the qualitative dependence of the reconstructions on \begin{document}$q$\end{document}.

2017, 11(1): 177-202 doi: 10.3934/ipi.2017009 +[Abstract](2195) +[HTML](115) +[PDF](7208.1KB)
Abstract:

We describe a foveated compressive sensing approach for image analysis applications that utilizes knowledge of the task to be performed to reduce the number of required sensor measurements and sensor size, weight, and power (SWAP) compared to conventional Nyquist sampling and compressive sensing-based approaches. Our Compressive Optical Foveated Architecture (COFA) adapts the dictionary and compressive measurements to structure and sparsity in the signal, task, and scene by reducing measurement and dictionary mutual coherence and increasing sparsity using principles of actionable information and foveated compressive sensing. Actionable information is used to extract task-relevant regions of interest (ROIs) from a low-resolution scene analysis by eliminating the effects of nuisances for occlusion and anomalous motion detection. From the extracted ROIs, preferential measurements are taken using foveation as part of the compressive sensing adaptation process. The task-specific measurement matrix is optimized by using a novel saliency-weighted coherence minimization with respect to the learned signal dictionary. This incorporates the relative usage of the atoms in the dictionary. We utilize a patch-based method to learn the signal priors. A tree-structured dictionary of image patches using K-SVD is learned which can sparsely represent any given image patch with the tree structure. We have implemented COFA in an end-to-end simulation of a vehicle fingerprinting task for aerial surveillance using foveated compressive measurements adapted to hierarchical ROIs consisting of background, roads, and vehicles. Our results show 113× reduction in measurements over conventional sensing and 28× reduction over compressive sensing using random measurements.

2017, 11(1): 203-220 doi: 10.3934/ipi.2017010 +[Abstract](2523) +[HTML](129) +[PDF](509.7KB)
Abstract:

This paper is concerned with the inverse problem to recover the scalar, complex-valued refractive index of a medium from measurements of scattered time-harmonic electromagnetic waves at a fixed frequency. The main results are two variational source conditions for near and far field data, which imply logarithmic rates of convergence of regularization methods, in particular Tikhonov regularization, as the noise level tends to 0. Moreover, these variational source conditions imply conditional stability estimates which improve and complement known stability estimates in the literature.

2018  Impact Factor: 1.469