# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

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## Inverse Problems & Imaging

April 2020 , Volume 14 , Issue 2

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2020, 14(2): 171-203 doi: 10.3934/ipi.2020009 +[Abstract](1064) +[HTML](200) +[PDF](4046.3KB)
Abstract:

In this paper, we present a patch based weighted means filter for removing an impulse noise by adapting the fundamental idea of the non-local means filter to the random-valued impulse noise. Our approach is to give a weight to a pixel in order to evaluate the probability that the pixel is contaminated by the impulse noise, which we call Reliable Weight of the pixel. With the help of the Reliable Weights we introduce the similarity function to measure the similarity among patches of the image contaminated by a random impulse noise. It turns out that the similarity function has significant anti impulse noise interference ability. We then incorporate the Reliable Weights and the similarity function into a filter designed to remove the random impulse noise. Under suitable conditions, we establish two convergence theorems to demonstrate that our method is feasible. Simulation results confirm that our filter is competitive compared to recently proposed methods.

2020, 14(2): 205-232 doi: 10.3934/ipi.2020010 +[Abstract](856) +[HTML](143) +[PDF](17507.44KB)
Abstract:

We consider the inverse problem of denoising an image where each point (pixel) is an element of a target set, which we refer to as a target-valued image. The target sets considered are either (ⅰ) a closed convex set of Euclidean space or (ⅱ) a closed subset of the sphere. The energy for the denoising problem consists of an \begin{document}$L^2$\end{document}-fidelity term which is regularized by the Dirichlet energy. A relaxation of this energy, based on the heat kernel, is introduced and the associated minimization problem is proven to be well-posed. We introduce a diffusion generated method which can be used to efficiently find minimizers of this energy. We prove results for the stability and convergence of the method for both types of target sets. The method is demonstrated on a variety of synthetic and test problems, with associated target sets given by the semi-positive definite matrices, the cube, spheres, the orthogonal matrices, and the real projective line.

2020, 14(2): 233-265 doi: 10.3934/ipi.2020011 +[Abstract](1157) +[HTML](197) +[PDF](922.69KB)
Abstract:

We establish the exact recovery guarantees for a class of Riemannian optimization methods based on the embedded manifold of low rank matrices for matrix completion. Assume \begin{document}$m$\end{document} entries of an \begin{document}$n\times n$\end{document} rank \begin{document}$r$\end{document} matrix are sampled independently and uniformly with replacement. We first show that with high probability the Riemannian gradient descent and conjugate gradient descent algorithms initialized by one step hard thresholding are guaranteed to converge linearly to the measured matrix provided

where \begin{document}$C_\kappa$\end{document} is a numerical constant depending on the condition number of the measured matrix. Then the sampling complexity is further improved to

via the resampled Riemannian gradient descent initialization. The analysis of the new initialization procedure relies on an asymmetric restricted isometry property of the sampling operator and the curvature of the low rank matrix manifold. Numerical simulation shows that the algorithms are able to recover a low rank matrix from nearly the minimum number of measurements.

2020, 14(2): 267-290 doi: 10.3934/ipi.2020012 +[Abstract](819) +[HTML](432) +[PDF](2820.82KB)
Abstract:

Low rank approximation has been extensively studied in the past. It is most suitable to reproduce rectangular like structures in the data. In this work we introduce a generalization using "shifted" rank-\begin{document}$1$\end{document} matrices to approximate \begin{document}$\mathit{\boldsymbol{{A}}}\in \mathbb{C}^{M\times N}$\end{document}. These matrices are of the form \begin{document}$S_{\mathit{\boldsymbol{{\lambda}}}}(\mathit{\boldsymbol{{u}}}\mathit{\boldsymbol{{v}}}^*)$\end{document} where \begin{document}$\mathit{\boldsymbol{{u}}}\in \mathbb{C}^M$\end{document}, \begin{document}$\mathit{\boldsymbol{{v}}}\in \mathbb{C}^N$\end{document} and \begin{document}$\mathit{\boldsymbol{{\lambda}}}\in \mathbb{Z}^N$\end{document}. The operator \begin{document}$S_{\mathit{\boldsymbol{{\lambda}}}}$\end{document} circularly shifts the \begin{document}$k$\end{document}-th column of \begin{document}$\mathit{\boldsymbol{{u}}}\mathit{\boldsymbol{{v}}}^*$\end{document} by \begin{document}$\lambda_k$\end{document}.

These kind of shifts naturally appear in applications, where an object \begin{document}$\mathit{\boldsymbol{{u}}}$\end{document} is observed in \begin{document}$N$\end{document} measurements at different positions indicated by the shift \begin{document}$\mathit{\boldsymbol{{\lambda}}}$\end{document}. The vector \begin{document}$\mathit{\boldsymbol{{v}}}$\end{document} gives the observation intensity. This model holds for seismic waves that are recorded at \begin{document}$N$\end{document} sensors at different times \begin{document}$\mathit{\boldsymbol{{\lambda}}}$\end{document}. Other examples are a car that moves through a video changing its position \begin{document}$\mathit{\boldsymbol{{\lambda}}}$\end{document} in each of the \begin{document}$N$\end{document} frames, or non-destructive testing based on ultrasonic waves that are reflected by defects inside the material.

The main difficulty of the above stated problem lies in finding a suitable shift vector \begin{document}$\mathit{\boldsymbol{{\lambda}}}$\end{document}. Once the shift is known, a simple singular value decomposition can be applied to reconstruct \begin{document}$\mathit{\boldsymbol{{u}}}$\end{document} and \begin{document}$\mathit{\boldsymbol{{v}}}$\end{document}. We propose a greedy method to reconstruct \begin{document}$\mathit{\boldsymbol{{\lambda}}}$\end{document}. By using the formulation of the problem in Fourier domain, a shifted rank-\begin{document}$1$\end{document} approximation can be calculated in \begin{document}$O(NM\log M)$\end{document}. Convergence to a locally optimal solution is guaranteed. Furthermore, we give a heuristic initial guess strategy that shows good results in the numerical experiments.

We validate our approach in several numerical experiments on different kinds of data. We compare the technique to shift-invariant dictionary learning algorithms. Furthermore, we provide examples from application including object segmentation in non-destructive testing and seismic exploration as well as object tracking in video processing.

2020, 14(2): 291-316 doi: 10.3934/ipi.2020013 +[Abstract](824) +[HTML](170) +[PDF](5080.28KB)
Abstract:

Region-of-interest computed tomography (ROI CT) aims at reconstructing a region within the field of view by using only ROI-focused projections. The solution of this inverse problem is challenging and methods of tomographic reconstruction that are designed to work with full projection data may perform poorly or fail when applied to this setting. In this work, we study the ROI CT problem in the presence of measurement noise and formulate the reconstruction problem by relaxing data fidelity and consistency requirements. Under the assumption of a robust width prior that provides a form of stability for data satisfying appropriate sparsity-inducing norms, we derive reconstruction performance guarantees and controllable error bounds. Based on this theoretical setting, we introduce a novel iterative reconstruction algorithm from ROI-focused projection data that is guaranteed to converge with controllable error while satisfying predetermined fidelity and consistency tolerances. Numerical tests on experimental data show that our algorithm for ROI CT is competitive with state-of-the-art methods especially when the ROI radius is small.

2020, 14(2): 317-337 doi: 10.3934/ipi.2020014 +[Abstract](1038) +[HTML](182) +[PDF](736.0KB)
Abstract:

In the context of international nuclear safeguards, the International Atomic Energy Agency (IAEA) has recently approved passive gamma emission tomography (PGET) as a method for inspecting spent nuclear fuel assemblies (SFAs). The PGET instrument is essentially a single photon emission computed tomography (SPECT) system that allows the reconstruction of axial cross-sections of the emission map of an SFA. The fuel material heavily self-attenuates its gamma-ray emissions, so that correctly accounting for the attenuation is a critical factor in producing accurate images. Due to the nature of the inspections, it is desirable to use as little a priori information as possible about the fuel, including the attenuation map, in the reconstruction process. Current reconstruction methods either do not correct for attenuation, assume a uniform attenuation throughout the fuel assembly, or assume an attenuation map based on an initial filtered back-projection reconstruction. We propose a method to simultaneously reconstruct the emission and attenuation maps by formulating the reconstruction as a constrained minimization problem with a least squares data fidelity term and regularization terms. Using simulated data, we show that our approach produces clear reconstructions which allow for a highly reliable classification of spent, missing, and fresh fuel rods.

2020, 14(2): 339-361 doi: 10.3934/ipi.2020015 +[Abstract](1123) +[HTML](179) +[PDF](5305.61KB)
Abstract:

Hyperspectral image (HSI) super-resolution is a technique to improve the spatial resolution of a HSI for better visual perception and down stream applications. This is a very ill-posed inverse problem and is often solved by fusing the low-resolution (LR) HSI with a high-resolution (HR) multispectral image (MSI). It is more challenging for blind HSI super-resolution, i.e., when the spatial degradation operators are completely unknown. In this paper, we propose a novel sparse tensor factorization model for the task of blind HSI super-resolution using the spatial non-local self-similarity and spectral global correlation of HSIs. Image clustering method is employed to collect some similar 3D cubes of HSIs which can be formed as some 4D image clusters with high correlation. We conduct cluster wise computation to not only save computation time but also to introduce a non-local regularity originated from the redundancy of cubes. By using the sparsity of tensor decomposition and the low-rank in non-local self-similarity direction underlying 4D similar clusters, we design a sparse tensor regularization term, which preserves the spatial-spectral structural correlation of HSIs. In addition, we present a proximal alternating direction method of multipliers (ADMM) based algorithm to efficiently solve the proposed model. Numerical experiments demonstrate that the proposed model outperforms many state-of-the-art HSI super-resolution methods.

2020, 14(2): 363-384 doi: 10.3934/ipi.2020016 +[Abstract](666) +[HTML](140) +[PDF](1690.34KB)
Abstract:

This paper addresses the problem of identifying impenetrable obstacles in a Kirchhoff-Love infinite plate from multistatic near-field data. The Linear Sampling Method is introduced in this context. We firstly prove a uniqueness result for such an inverse problem. We secondly provide the classical theoretical foundation of the Linear Sampling Method. We lastly show the feasibility of the method with the help of numerical experiments.

2020, 14(2): 385-398 doi: 10.3934/ipi.2020017 +[Abstract](687) +[HTML](172) +[PDF](3289.28KB)
Abstract:

We model electrical impedance tomography (EIT) based on the minimum energy principle. It results in a constrained minimization problem in terms of current density. The new formulation is proved to have a unique solution within appropriate function spaces. By characterizing its solution with the Lagrange multiplier method, we relate the new formulation to the so-called shunt model and the complete electrode model (CEM) of EIT. Based on the new formulation, we also propose a new numerical method to solve the forward problem of EIT. The new solver is formulated in terms of current. It was shown to give similar results to that of the traditional finite element method, with simulations on a 2D EIT model.

2020, 14(2): 399-399 doi: 10.3934/ipi.2020018 +[Abstract](560) +[HTML](104) +[PDF](136.09KB)
Abstract:

The copyright of the paper entitled "Incorporating structural prior information and sparsity into EIT using parallel level sets" [1] was originally owned by American Institute of Mathematical Sciences (AIMS, LLC) when it was published online in IPI Volume 13, Issue 2, April 2019 regular issue. The authors of this paper [1] paid the Open Access fee after the paper was published.

The paper entitled "Incorporating structural prior information and sparsity into EIT using parallel level sets" [1] is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

2019  Impact Factor: 1.373