# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

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## Inverse Problems and Imaging

June 2022 , Volume 16 , Issue 3

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2022, 16(3): 483-523 doi: 10.3934/ipi.2021059 +[Abstract](982) +[HTML](315) +[PDF](16064.83KB)
Abstract:

A primary interest in dynamic inverse problems is to identify the underlying temporal behaviour of the system from outside measurements. In this work, we consider the case, where the target can be represented by a decomposition of spatial and temporal basis functions and hence can be efficiently represented by a low-rank decomposition. We then propose a joint reconstruction and low-rank decomposition method based on the Nonnegative Matrix Factorisation to obtain the unknown from highly undersampled dynamic measurement data. The proposed framework allows for flexible incorporation of separate regularisers for spatial and temporal features. For the special case of a stationary operator, we can effectively use the decomposition to reduce the computational complexity and obtain a substantial speed-up. The proposed methods are evaluated for three simulated phantoms and we compare the obtained results to a separate low-rank reconstruction and subsequent decomposition approach based on the widely used principal component analysis.

2022, 16(3): 525-545 doi: 10.3934/ipi.2021060 +[Abstract](987) +[HTML](362) +[PDF](2842.85KB)
Abstract:

This paper introduces a novel generative encoder (GE) framework for generative imaging and image processing tasks like image reconstruction, compression, denoising, inpainting, deblurring, and super-resolution. GE unifies the generative capacity of GANs and the stability of AEs in an optimization framework instead of stacking GANs and AEs into a single network or combining their loss functions as in existing literature. GE provides a novel approach to visualizing relationships between latent spaces and the data space. The GE framework is made up of a pre-training phase and a solving phase. In the former, a GAN with generator \begin{document}$G$\end{document} capturing the data distribution of a given image set, and an AE network with encoder \begin{document}$E$\end{document} that compresses images following the estimated distribution by \begin{document}$G$\end{document} are trained separately, resulting in two latent representations of the data, denoted as the generative and encoding latent space respectively. In the solving phase, given noisy image \begin{document}$x = \mathcal{P}(x^*)$\end{document}, where \begin{document}$x^*$\end{document} is the target unknown image, \begin{document}$\mathcal{P}$\end{document} is an operator adding an addictive, or multiplicative, or convolutional noise, or equivalently given such an image \begin{document}$x$\end{document} in the compressed domain, i.e., given \begin{document}$m = E(x)$\end{document}, the two latent spaces are unified via solving the optimization problem

and the image \begin{document}$x^*$\end{document} is recovered in a generative way via \begin{document}$\hat{x}: = G(z^*)\approx x^*$\end{document}, where \begin{document}$\lambda>0$\end{document} is a hyperparameter. The unification of the two spaces allows improved performance against corresponding GAN and AE networks while visualizing interesting properties in each latent space.

2022, 16(3): 547-567 doi: 10.3934/ipi.2021061 +[Abstract](717) +[HTML](364) +[PDF](34016.21KB)
Abstract:

In this paper, we study to decompose a color image into the illumination and reflectance components in saturation-value color space. By considering the spatial smoothness of the illumination component, the total variation regularization of the reflectance component, and the data-fitting in saturation-value color space, we develop a novel variational saturation-value model for image decomposition. The main aim of the proposed model is to formulate the decomposition of a color image such that the illumination component is uniform with only brightness information, and the reflectance component contains the color information. We establish the theoretical result about the existence of the solution of the proposed minimization problem. We employ a primal-dual algorithm to solve the proposed minimization problem. Experimental results are shown to illustrate the effectiveness of the proposed decomposition model in saturation-value color space, and demonstrate the performance of the proposed method is better than the other testing methods.

2022, 16(3): 569-594 doi: 10.3934/ipi.2021062 +[Abstract](773) +[HTML](264) +[PDF](2237.74KB)
Abstract:

We consider a heat equation and a wave equation in one spatial dimension. This article deals with the inverse problem of determining the size of the spatial interval from some extra boundary information on the solution. Under several different circumstances, we prove uniqueness, non-uniqueness and some size estimates. Moreover, we numerically solve the inverse problems and compute accurate approximations of the size. This is illustrated with several satisfactory numerical experiments.

2022, 16(3): 595-611 doi: 10.3934/ipi.2021063 +[Abstract](714) +[HTML](238) +[PDF](494.27KB)
Abstract:

Consider the transmission eigenvalue problem

It is shown in [16] that there exists a sequence of eigenfunctions \begin{document}$(w_m, v_m)_{m\in\mathbb{N}}$\end{document} associated with \begin{document}$k_m\rightarrow \infty$\end{document} such that either \begin{document}$\{w_m\}_{m\in\mathbb{N}}$\end{document} or \begin{document}$\{v_m\}_{m\in\mathbb{N}}$\end{document} are surface-localized, depending on \begin{document}$\mathbf{n}>1$\end{document} or \begin{document}$0<\mathbf{n}<1$\end{document}. In this paper, we discover a new type of surface-localized transmission eigenmodes by constructing a sequence of transmission eigenfunctions \begin{document}$(w_m, v_m)_{m\in\mathbb{N}}$\end{document} associated with \begin{document}$k_m\rightarrow \infty$\end{document} such that both \begin{document}$\{w_m\}_{m\in\mathbb{N}}$\end{document} and \begin{document}$\{v_m\}_{m\in\mathbb{N}}$\end{document} are surface-localized, no matter \begin{document}$\mathbf{n}>1$\end{document} or \begin{document}$0<\mathbf{n}<1$\end{document}. Though our study is confined within the radial geometry, the construction is subtle and technical.

2022, 16(3): 613-624 doi: 10.3934/ipi.2021064 +[Abstract](803) +[HTML](253) +[PDF](371.81KB)
Abstract:

We study the well-posedness of a semi-linear fractional diffusion equation and formulate an associated inverse problem. We determine fractional power type nonlinearities from the exterior partial measurements of the Dirichlet-to-Neumann map. Our arguments are based on a first order linearization as well as the parabolic Runge approximation property.

2022, 16(3): 625-646 doi: 10.3934/ipi.2021065 +[Abstract](795) +[HTML](253) +[PDF](1993.62KB)
Abstract:

In this paper, we propose a nonconvex regularization model for images damaged by Cauchy noise and blur. This model is based on the method of the total variational proposed by Federica, Dong and Zeng [SIAM J. Imaging Sci.(2015)], where a variational approach for restoring blurred images with Cauchy noise is used. Here we consider the nonconvex regularization, namely a weighted difference of \begin{document}$l_1$\end{document}-norm and \begin{document}$l_2$\end{document}-norm coupled with wavelet frame, the alternating direction method of multiplier is carried out for this minimization problem, we describe the details of the algorithm and prove its convergence. Numerical experiments are tested by adding different levels of noise and blur, results show that our method can denoise and deblur the image better.

2022, 16(3): 647-671 doi: 10.3934/ipi.2021066 +[Abstract](739) +[HTML](332) +[PDF](11565.94KB)
Abstract:

For patients undergoing mechanical ventilation due to respiratory failure, 2D electrical impedance tomography (EIT) is emerging as a means to provide functional monitoring of pulmonary processes. In EIT, electrical current is applied to the body, and the internal conductivity distribution is reconstructed based on subsequent voltage measurements. However, EIT images are known to often suffer from large systematic artifacts arising from various limitations and exacerbated by the ill-posedness of the inverse problem. The direct D-bar reconstruction method admits a nonlinear Fourier analysis of the EIT problem, providing the ability to process and filter reconstructions in the nonphysical frequency regime. In this work, a technique is introduced for automated Fourier-domain filtering of known systematic artifacts in 2D D-bar reconstructions. The new method is validated using three numerically simulated static thoracic datasets with induced artifacts, plus two experimental dynamic human ventilation datasets containing systematic artifacts. Application of the method is shown to significantly reduce the appearance of artifacts and improve the shape of the lung regions in all datasets.

2021 Impact Factor: 1.483
5 Year Impact Factor: 1.462
2021 CiteScore: 2.6