# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

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## Inverse Problems & Imaging

August 2019 , Volume 13 , Issue 4

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2019, 13(4): 703-720 doi: 10.3934/ipi.2019032 +[Abstract](1637) +[HTML](258) +[PDF](413.93KB)
Abstract:

This article studies the approximate recovery of low-rank matrices acquired through binary measurements. Two types of recovery algorithms are considered, one based on hard singular value thresholding and the other one based on semidefinite programming. In case no thresholds are introduced before binary quantization, it is first shown that the direction of the low-rank matrices can be well approximated. Then, in case nonadaptive thresholds are incorporated, it is shown that both the direction and the magnitude can be well approximated. Finally, by allowing the thresholds to be chosen adaptively, we exhibit a recovery procedure for which low-rank matrices are fully approximated with error decaying exponentially with the number of binary measurements. In all cases, the procedures are robust to prequantization error. The underlying arguments are essentially deterministic: they rely only on an unusual restricted isometry property of the measurement process, which is established once and for all for Gaussian measurement processes.

2019, 13(4): 721-744 doi: 10.3934/ipi.2019033 +[Abstract](1999) +[HTML](218) +[PDF](419.77KB)
Abstract:

A rigorous mathematical model and an efficient computational method are proposed to solving the inverse elastic surface scattering problem which arises from the near-field imaging of periodic structures. We demonstrate how an enhanced resolution can be achieved by using more easily measurable far-field data. The surface is assumed to be a small and smooth perturbation of an elastically rigid plane. By placing a rectangular slab of a homogeneous and isotropic elastic medium with larger mass density above the surface, more propagating wave modes can be utilized from the far-field data which contributes to the reconstruction resolution. Requiring only a single illumination, the method begins with the far-to-near (FtN) field data conversion and utilizes the transformed field expansion to derive an analytic solution for the direct problem, which leads to an explicit inversion formula for the inverse problem. Moreover, a nonlinear correction scheme is developed to improve the accuracy of the reconstruction. Results show that the proposed method is capable of stably reconstructing surfaces with resolution controlled by the slab's density.

2019, 13(4): 745-754 doi: 10.3934/ipi.2019034 +[Abstract](1798) +[HTML](247) +[PDF](382.08KB)
Abstract:

Retrieval of magnetization parameters using magnetic tensor gradient measurements receives attention in recent years. Determination of subsurface properties from the observed potential field measurements is referred to as inversion. Little regularizing inversion results using full tensor magnetic gradient modeling so far has been reported in the literature. Traditional magnetic inversion is based on the total magnetic intensity (TMI) data and solving the corresponding mathematical physical model. In recent years, with the development of the advanced technology, acquisition of the full tensor gradient magnetic data becomes available. In this paper, we study invert the magnetic parameters using the full tensor magnetic gradient data. A sparse Tikhonov regularization model is established. In solving the minimization model, the conjugate gradient method is addressed. Numerical and field data experiments are performed to show feasibility of our algorithm.

2019, 13(4): 755-786 doi: 10.3934/ipi.2019035 +[Abstract](1883) +[HTML](263) +[PDF](532.38KB)
Abstract:

We propose a second-order total generalized variation (TGV) regularization for the reconstruction of the initial condition in variational data assimilation problems. After showing the equivalence between TGV regularization and a Bayesian MAP estimator, we focus on the detailed study of the inviscid Burgers' data assimilation problem. Due to the difficult structure of the governing hyperbolic conservation law, we consider a discretize–then–optimize approach and rigorously derive a first-order optimality condition for the problem. For the numerical solution, we propose a globalized reduced Newton-type method together with a polynomial line-search strategy, and prove convergence of the algorithm to stationary points. The paper finishes with some numerical experiments where, among others, the performance of TGV–regularization compared to TV–regularization is tested.

2019, 13(4): 787-803 doi: 10.3934/ipi.2019036 +[Abstract](1720) +[HTML](219) +[PDF](556.21KB)
Abstract:

The proposal of efficient distributions is a crucial step for decision making in practice. Mixture models are adjustment tools which are often used to describe complex phenomena. However, as one disadvantage, such models impose hard inference procedures, submitted to a large number of parameters. To solve this issue, this paper proposes a new model which is able to describe multimodal, symmetric and asymmetric behaviors with only three parameters, called compound truncated Poisson normal (CTPN) distribution. Some properties of the CTPN law are derived and discussed: characteristic and cumulant functions and ordinary moments. A moment estimation procedure for CTPN parameters is also provided. This procedure consists of solving one nonlinear equation in terms of a single parameter. An application with images of synthetic aperture radar (SAR) is made. The results present evidence that the CTPN can outperform the \begin{document}$\mathcal{G}^0$\end{document}, \begin{document}$\mathcal{K}$\end{document} and BGN (laws commonly used in SAR literature), as well as GBGL models.

2019, 13(4): 805-826 doi: 10.3934/ipi.2019037 +[Abstract](1717) +[HTML](308) +[PDF](5550.49KB)
Abstract:

This paper provides a fast approach to apply the Earth Mover's Distance (EMD) (a.k.a optimal transport, Wasserstein distance) for supervised and unsupervised image segmentation. The model globally incorporates the transportation costs (original Monge-Kantorovich type) among histograms of multiple dimensional features, e.g. gray intensity and texture in image's foreground and background. The computational complexity is often high for the EMD between two histograms on Euclidean spaces with dimensions larger than one. We overcome this computational difficulty by rewriting the model into a \begin{document}$L_1$\end{document} type minimization with the linear dimension of feature space. We then apply a fast algorithm based on the primal-dual method. Compare to several state-of-the-art EMD models, the experimental results based on image data sets demonstrate that the proposed method has superior performance in terms of the accuracy and the stability of the image segmentation.

2019, 13(4): 827-862 doi: 10.3934/ipi.2019038 +[Abstract](1780) +[HTML](239) +[PDF](1129.01KB)
Abstract:

We introduce a new algorithm to solve the problem of detecting unknown cavities immersed in a stationary viscous fluid, using partial boundary measurements. The considered fluid obeys a steady Stokes regime, the cavities are inclusions and the boundary measurements are a single compatible pair of Dirichlet and Neumann data, available only on a partial accessible part of the whole boundary. This inverse inclusion Cauchy-Stokes problem is ill-posed for both the cavities and missing data reconstructions, and designing stable and efficient algorithms is not straightforward. We reformulate the problem as a three-player Nash game. Thanks to an identifiability result derived for the Cauchy-Stokes inclusion problem, it is enough to set up two Stokes boundary value problems, then use them as state equations. The Nash game is then set between 3 players, the two first targeting the data completion while the third one targets the inclusion detection. We used a level-set approach to get rid of the tricky control dependence of functional spaces, and we provided the third player with the level-set function as strategy, with a cost functional of Kohn-Vogelius type. We propose an original algorithm, which we implemented using Freefem++. We present 2D numerical experiments for three different test-cases.The obtained results corroborate the efficiency of our 3-player Nash game approach to solve parameter or shape identification for Cauchy problems.

2019, 13(4): 863-878 doi: 10.3934/ipi.2019039 +[Abstract](1509) +[HTML](202) +[PDF](1482.06KB)
Abstract:

We provide a reconstruction scheme for complex-valued potentials in \begin{document}$H^s (\mathbb{R}^2)$\end{document} for \begin{document}$s > 0$\end{document}. The procedure extends the method of Bukhgeim relying on quadratic exponential solutions.

2019, 13(4): 879-901 doi: 10.3934/ipi.2019040 +[Abstract](1435) +[HTML](204) +[PDF](1015.83KB)
Abstract:

We are concerned with a novel sensor-based gesture input/ instruction technology which enables human beings to interact with computers conveniently. The human being wears an emitter on the finger or holds a digital pen that generates a time harmonic point charge. The inputs/instructions are performed through moving the finger or the digital pen. The computer recognizes the instruction by determining the motion trajectory of the dynamic point charge from the collected electromagnetic field measurement data. The identification process is mathematically modelled as a dynamic inverse source problem for time-dependent Maxwell's equations. From a practical point of view, the point source should be assumed to move in an unknown inhomogeneous background medium, which models the human body and the surroundings. Moreover, a salient feature is that the electromagnetic radiated data are only collected in a limited aperture. For the inverse problem, we develop, from the respectively deterministic and stochastic viewpoints, a dynamic direct sampling method and a modified particle filter method. Both approaches can effectively recover the motion trajectory. Rigorous theoretical justifications are presented for the mathematical modelling and the proposed recovery methods. Extensive numerical experiments are conducted to illustrate the promising features of the two proposed recognition approaches.

2019  Impact Factor: 1.373