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Inverse Problems & Imaging

August 2020 , Volume 14 , Issue 4

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Image restoration from noisy incomplete frequency data by alternative iteration scheme
Xiaoman Liu and Jijun Liu
2020, 14(4): 583-606 doi: 10.3934/ipi.2020027 +[Abstract](598) +[HTML](173) +[PDF](1793.83KB)
Abstract:

Consider the image restoration from incomplete noisy frequency data with total variation and sparsity regularizing penalty terms. Firstly, we establish an unconstrained optimization model with different smooth approximations on the regularizing terms. Then, to weaken the amount of computations for cost functional with total variation term, the alternating iterative scheme is developed to obtain the exact solution through shrinkage thresholding in inner loop, while the nonlinear Euler equation is appropriately linearized at each iteration in exterior loop, yielding a linear system with diagonal coefficient matrix in frequency domain. Finally the linearized iteration is proven to be convergent in generalized sense for suitable regularizing parameters, and the error between the linearized iterative solution and the one gotten from the exact nonlinear Euler equation is rigorously estimated, revealing the essence of the proposed alternative iteration scheme. Numerical tests for different configurations show the validity of the proposed scheme, compared with some existing algorithms.

Joint reconstruction in low dose multi-energy CT
Jussi Toivanen, Alexander Meaney, Samuli Siltanen and Ville Kolehmainen
2020, 14(4): 607-629 doi: 10.3934/ipi.2020028 +[Abstract](942) +[HTML](247) +[PDF](4143.61KB)
Abstract:

Multi-energy CT takes advantage of the non-linearly varying attenuation properties of elemental media with respect to energy, enabling more precise material identification than single-energy CT. The increased precision comes with the cost of a higher radiation dose. A straightforward way to lower the dose is to reduce the number of projections per energy, but this makes tomographic reconstruction more ill-posed. In this paper, we propose how this problem can be overcome with a combination of a regularization method that promotes structural similarity between images at different energies and a suitably selected low-dose data acquisition protocol using non-overlapping projections. The performance of various joint regularization models is assessed with both simulated and experimental data, using the novel low-dose data acquisition protocol. Three of the models are well-established, namely the joint total variation, the linear parallel level sets and the spectral smoothness promoting regularization models. Furthermore, one new joint regularization model is introduced for multi-energy CT: a regularization based on the structure function from the structural similarity index. The findings show that joint regularization outperforms individual channel-by-channel reconstruction. Furthermore, the proposed combination of joint reconstruction and non-overlapping projection geometry enables significant reduction of radiation dose.

Uniqueness results in the inverse spectral Steklov problem
Germain Gendron
2020, 14(4): 631-664 doi: 10.3934/ipi.2020029 +[Abstract](457) +[HTML](142) +[PDF](473.52KB)
Abstract:

This paper is devoted to an inverse Steklov problem for a particular class of \begin{document}$ n $\end{document}-dimensional manifolds having the topology of a hollow sphere and equipped with a warped product metric. We prove that the knowledge of the Steklov spectrum determines uniquely the associated warping function up to a natural invariance.

Thermoacoustic Tomography with circular integrating detectors and variable wave speed
Chase Mathison
2020, 14(4): 665-682 doi: 10.3934/ipi.2020030 +[Abstract](421) +[HTML](144) +[PDF](464.17KB)
Abstract:

We explore Thermoacoustic Tomography with circular integrating detectors assuming variable, smooth wave speed. We show that the measurement operator in this case is a Fourier Integral Operator and examine how the singularities in initial data and measured data are related through the canonical relation of this operator. We prove which of those singularities in the initial data are visible from a fixed open subset of the set on which measurements are taken. In addition, numerical results are shown for both full and partial data.

Learnable Douglas-Rachford iteration and its applications in DOT imaging
Jiulong Liu, Nanguang Chen and Hui Ji
2020, 14(4): 683-700 doi: 10.3934/ipi.2020031 +[Abstract](489) +[HTML](243) +[PDF](1174.97KB)
Abstract:

How to overcome the ill-posed nature of inverse problems is a pervasive problem in medical imaging. Most existing solutions are based on regularization techniques. This paper proposed a deep neural network (DNN) based image reconstruction method, the so-called DR-Net, that leverages the interpretability of existing regularization methods and adaptive modeling capacity of DNN. Motivated by a Douglas-Rachford fixed-point iteration for solving \begin{document}$ \ell_1 $\end{document}-norm relating regularization model, the proposed DR-Net learns the prior of the solution via a U-Net based network, as well as other important regularization parameters. The DR-Net is applied to solve image reconstruction problem in diffusion optical tomography (DOT), a non-invasive imaging technique with many applications in medical imaging. The experiments on both simulated and experimental data showed that the proposed DNN based image reconstruction method significantly outperforms existing regularization methods.

Reconstruction of the derivative of the conductivity at the boundary
Felipe Ponce-Vanegas
2020, 14(4): 701-718 doi: 10.3934/ipi.2020032 +[Abstract](414) +[HTML](138) +[PDF](420.95KB)
Abstract:

We describe a method to reconstruct the conductivity and its normal derivative at the boundary from the knowledge of the potential and current measured at the boundary. The method of reconstruction works for isotropic conductivities with low regularity. This boundary determination for rough conductivities implies the uniqueness of the conductivity in the whole domain \begin{document}$ \Omega $\end{document} when it lies in \begin{document}$ W^{1+\frac{n-5}{2p}+, p}(\Omega) $\end{document}, for dimensions \begin{document}$ n\ge 5 $\end{document} and for \begin{document}$ n\le p<\infty $\end{document}.

Extended sampling method for interior inverse scattering problems
Fang Zeng
2020, 14(4): 719-731 doi: 10.3934/ipi.2020033 +[Abstract](449) +[HTML](145) +[PDF](1648.53KB)
Abstract:

We consider an interior inverse scattering problem of reconstructing the shape of a cavity. The measurements are the scattered fields on a curve inside the cavity due to only one point source. In this paper, we employ the extending sampling method to reconstruct the cavity based on limited data. Numerical examples are provided to show the effectiveness of the method.

Saturation-Value Total Variation model for chromatic aberration correction
Wei Wang, Ling Pi and Michael K. Ng
2020, 14(4): 733-755 doi: 10.3934/ipi.2020034 +[Abstract](392) +[HTML](187) +[PDF](16373.12KB)
Abstract:

Chromatic aberration generally occurs in the regions of sharp edges in captured digital color images. The main aim of this paper is to propose and develop a novel Saturation-Value Total Variation (SVTV) model for chromatic aberration correction. In the proposed optimization model, there are three terms for the correction purpose. The SVTV regularization term is to model the target color image in HSV color space instead of RGB color space, and to avoid oscillations in the recovering process. In correction process, the gradient matching terms based on the green component are used to govern both red and blue components, and the intensity terms are employed to fit red, green and blue data components. The existence of the minimizer of the optimization model is analyzed and an efficient optimization algorithm is also developed for solving the resulting variational problem. Experimental results are presented to illustrate the effectiveness of the proposed model and to show that the correction results are better than those by using the other testing methods.

2019  Impact Factor: 1.373

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