American Institute of Mathematical Sciences

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.

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.

An inverse obstacle problem for the wave governed by the wave equation in a two layered medium is considered under the framework of the time domain enclosure method. The wave is generated by an initial data supported on a closed ball in the upper half-space, and observed on the same ball over a finite time interval. The unknown obstacle is penetrable and embedded in the lower half-space. It is assumed that the propagation speed of the wave in the upper half-space is greater than that of the wave in the lower half-space, which is excluded in the previous study: Ikehata and Kawashita, Inverse Problems and Imaging 12 (2018), no.5, 1173-1198. In the present case, when the reflected waves from the obstacle enter the upper half-space, the total reflection phenomena occur, which give singularities to the integral representation of the fundamental solution for the reduced transmission problem in the background medium. This fact makes the problem more complicated. However, it is shown that these waves do not have any influence on the leading profile of the indicator function of the time domain enclosure method.

An inverse obstacle scattering problem for the electromagnetic wave governed by the Maxwell system over a finite time interval is considered. It is assumed that the wave satisfies the Leontovich boundary condition on the surface of an unknown obstacle. The condition is described by using an unknown positive function on the surface of the obstacle which is called the surface admittance. The wave is generated at the initial time by a volumetric current source supported on a very small ball placed outside the obstacle and only the electric component of the wave is observed on the same ball over a finite time interval. It is shown that from the observed data one can extract information about the value of the surface admittance and the curvatures at the points on the surface nearest to the center of the ball. This shows that a single shot contains a meaningful information about the quantitative state of the surface of the obstacle.

EIT is a non-linear ill-posed inverse problem which requires sophisticated regularisation techniques to achieve good results. In this paper we consider the use of structural information in the form of edge directions coming from an auxiliary image of the same object being reconstructed. In order to allow for cases where the auxiliary image does not provide complete information we consider in addition a sparsity regularization for the edges appearing in the EIT image. The combination of these approaches is conveniently described through the parallel level sets approach. We present an overview of previous methods for structural regularisation and then provide a variational setting for our approach and explain the numerical implementation. We present results on simulations and experimental data for different cases with accurate and inaccurate prior information. The results demonstrate that the structural prior information improves the reconstruction accuracy, even in cases when there is reasonable uncertainty in the prior about the location of the edges or only partial edge information is available.

An inverse obstacle problem for the wave equation in a two layered medium is considered. It is assumed that the unknown obstacle is penetrable and embedded in the lower half-space. The wave as a solution of the wave equation is generated by an initial data whose support is in the upper half-space and observed at the same place as the support over a finite time interval. From the observed wave an indicator function in the time domain enclosure method is constructed. It is shown that, one can find some information about the geometry of the obstacle together with the qualitative property in the asymptotic behavior of the indicator function.

In this paper, we investigate the relations between the Radon and weighted divergent beam and cone transforms. Novel inversion formulas are derived for the latter two. The weighted cone transform arises, for instance, in image reconstruction from the data obtained by Compton cameras, which have promising applications in various fields, including biomedical and homeland security imaging and gamma ray astronomy. The inversion formulas are applicable for a wide variety of detector geometries in any dimension. The results of numerical implementation of some of the formulas in dimensions two and three are also provided.

An inverse obstacle scattering problem for the wave governed by the Maxwell system in the time domain, in particular, over a finite time interval is considered. It is assumed that the electric field \begin{document}$\boldsymbol{E}$\end{document} and magnetic field \begin{document}$\boldsymbol{ H}$\end{document} which are solutions of the Maxwell system are generated only by a current density at the initial time located not far a way from an unknown obstacle. The obstacle is embedded in a medium like air which has constant electric permittivity \begin{document}$ε$\end{document} and magnetic permeability \begin{document}$μ$\end{document}. It is assumed that the fields on the surface of the obstacle satisfy the Leontovich boundary condition \begin{document}$\boldsymbol{ ν}×\boldsymbol{H}-λ\,\boldsymbol{ ν}×(\boldsymbol{ E}×\boldsymbol{ ν})=\boldsymbol{ 0}$\end{document} with admittance \begin{document}$λ$\end{document} an unknown positive function and \begin{document}$\boldsymbol{ ν}$\end{document} the unit outward normal. The observation data are given by the electric field observed at the same place as the support of the current density over a finite time interval. It is shown that an indicator function computed from the electric fields corresponding two current densities enables us to know: the distance of the center of the common spherical support of the current densities to the obstacle; whether the value of the admittance \begin{document}$λ$\end{document} is greater or less than the special value \begin{document}$\sqrt{ε/μ}$\end{document}.