# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

All Issues

## Inverse Problems and Imaging

February 2020 , Volume 14 , Issue 1

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2020, 14(1): 1-26 doi: 10.3934/ipi.2019061 +[Abstract](1925) +[HTML](164) +[PDF](1127.94KB)
Abstract:

We study the integral transform over a general family of broken rays in \begin{document}$\mathbb{R}^2$\end{document}. One example of the broken rays is the family of rays reflected from a curved boundary once. There is a natural notion of conjugate points for broken rays. If there are conjugate points, we show that the singularities conormal to the broken rays cannot be recovered from local data and therefore artifacts arise in the reconstruction. As for global data, more singularities might be recoverable. We apply these conclusions to two examples, the V-line transform and the parallel ray transform. In each example, a detailed discussion of the local and global recovery of singularities is given and we perform numerical experiments to illustrate the results.

2020, 14(1): 27-52 doi: 10.3934/ipi.2019062 +[Abstract](2463) +[HTML](165) +[PDF](822.45KB)
Abstract:

Existing reconstruction methods for single photon emission computed tomography (SPECT) are most based on discrete models, leading to low accuracy in reconstruction. Reconstruction methods based on integral equation models (IEMs) with a higher order piecewise polynomial discretization on the pixel grid for SEPCT imaging were recently proposed to overcome the accuracy deficiency of the discrete models. Discretization of IEMs based on the pixel grid leads to a system of a large dimension, which may require higher computational costs to solve. We develop a SPECT reconstruction method which employs an IEM of the SPECT data acquisition process and discretizes it on a content-adaptive unstructured grid (CAUG) with the total variation (TV) regularization aiming at reducing computational costs of the integral equation method. Specifically, we design a CAUG of the image domain for the discretization of the IEM, and propose a TV regularization defined on the CAUG for the resulting ill-posed problem. We then apply a preconditioned fixed-point proximity algorithm to solve the resulting non-smooth optimization problem, and provide convergence analysis of the algorithm. Numerical experiments are presented to demonstrate the superiority of the proposed method over the competing methods in terms of suppressing noise, preserving edges and reducing computational costs.

2020, 14(1): 53-75 doi: 10.3934/ipi.2019063 +[Abstract](2784) +[HTML](152) +[PDF](409.27KB)
Abstract:

In this article we study the inverse problem of determining the convection term and the time-dependent density coefficient appearing in the convection-diffusion equation. We prove the unique determination of these coefficients from the knowledge of solution measured on a subset of the boundary.

2020, 14(1): 77-96 doi: 10.3934/ipi.2019064 +[Abstract](5045) +[HTML](425) +[PDF](1475.63KB)
Abstract:

Poisson noise is an important type of electronic noise that is present in a variety of photon-limited imaging systems. Different from the Gaussian noise, Poisson noise depends on the image intensity, which makes image restoration very challenging. Moreover, complex geometry of images desires a regularization that is capable of preserving piecewise smoothness. In this paper, we propose a Poisson denoising model based on the fractional-order total variation (FOTV). The existence and uniqueness of a solution to the model are established. To solve the problem efficiently, we propose three numerical algorithms based on the Chambolle-Pock primal-dual method, a forward-backward splitting scheme, and the alternating direction method of multipliers (ADMM), each with guaranteed convergence. Various experimental results are provided to demonstrate the effectiveness and efficiency of our proposed methods over the state-of-the-art in Poisson denoising.

2020, 14(1): 97-115 doi: 10.3934/ipi.2019065 +[Abstract](2244) +[HTML](184) +[PDF](944.67KB)
Abstract:

This paper addresses the problem of the electro-communication for weakly electric fish. In particular we aim at sheding light on how the fish circumvent the jamming issue for both electro-communication and active electro-sensing. Our main result is a real-time tracking algorithm, which provides a new approach to the communication problem. It finds a natural application in robotics, where efficient communication strategies are needed to be implemented by bio-inspired underwater robots.

2020, 14(1): 117-132 doi: 10.3934/ipi.2019066 +[Abstract](3132) +[HTML](414) +[PDF](2355.85KB)
Abstract:

Bayesian inference methods have been widely applied in inverse problems due to the ability of uncertainty characterization of the estimation. The prior distribution of the unknown plays an essential role in the Bayesian inference, and a good prior distribution can significantly improve the inference results. In this paper, we propose a hybrid prior distribution on combining the nonlocal total variation regularization (NLTV) and the Gaussian distribution, namely NLTG prior. The advantage of this hybrid prior is two-fold. The proposed prior models both texture and geometric structures present in images through the NLTV. The Gaussian reference measure also provides a flexibility of incorporating structure information from a reference image. Some theoretical properties are established for the hybrid prior. We apply the proposed prior to limited tomography reconstruction problem that is difficult due to severe data missing. Both maximum a posteriori and conditional mean estimates are computed through two efficient methods and the numerical experiments validate the advantages and feasibility of the proposed NLTG prior.

2020, 14(1): 133-152 doi: 10.3934/ipi.2019067 +[Abstract](2239) +[HTML](153) +[PDF](905.3KB)
Abstract:

We analyze the Factorization method to reconstruct the geometry of a local defect in a periodic absorbing layer using almost only incident plane waves at a fixed frequency. A crucial part of our analysis relies on the consideration of the range of a carefully designed far field operator, which characterizes the geometry of the defect. We further provide some validating numerical results in a two dimensional setting.

2020, 14(1): 153-169 doi: 10.3934/ipi.2019068 +[Abstract](3458) +[HTML](190) +[PDF](340.84KB)
Abstract:

The Sturm-Liouville pencil is studied with arbitrary entire functions of the spectral parameter, contained in one of the boundary conditions. We solve the inverse problem, that consists in recovering the pencil coefficients from a part of the spectrum satisfying some conditions. Our main results are 1) uniqueness, 2) constructive solution, 3) local solvability and stability of the inverse problem. Our method is based on the reduction to the Sturm-Liouville problem without the spectral parameter in the boundary conditions. We use a special vector-functional Riesz-basis for that reduction.

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