# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

All Issues

## Inverse Problems & Imaging

December 2018 , Volume 12 , Issue 6

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2018, 12(6): 1263-1291 doi: 10.3934/ipi.2018053 +[Abstract](4137) +[HTML](319) +[PDF](742.1KB)
Abstract:

In this paper, a new variational model with fractional-order regularization term arising in registration of diffusion tensor image(DTI) is presented. Moreover, the existence of its solution is proved to ensure that there is a regular solution for this model. Furthermore, three numerical tests are also performed to show the effectiveness of this model.

2018, 12(6): 1293-1308 doi: 10.3934/ipi.2018054 +[Abstract](3308) +[HTML](219) +[PDF](369.33KB)
Abstract:

Consider a three-edge star graph, made up of unknown Sturm-Liouville operators on each edge. By using the heat propagation through the graph and measuring the heat transfer occurring at its vertices, we show that we can extract enough spectral data to reconstruct the three Sturm-Liouville operators by using the Gelfand-Levitan theory. Furthermore this reconstruction is achieved by a single measurement provided we use a special initial condition.

2018, 12(6): 1309-1342 doi: 10.3934/ipi.2018055 +[Abstract](4018) +[HTML](223) +[PDF](553.67KB)
Abstract:

In this paper we study local stability estimates for a magnetic Schrödinger operator with partial data on an open bounded set in dimension \begin{document}$n≥3$\end{document}. This is the corresponding stability estimates for the identifiability result obtained by Bukhgeim and Uhlmann [2] in the presence of a magnetic field and when the measurements for the Dirichlet-Neumann map are taken on a neighborhood of the illuminated region of the boundary for functions supported on a neighborhood of the shadow region. We obtain log log-estimates for the magnetic fields and log log log-estimates for the electric potentials.

2018, 12(6): 1343-1364 doi: 10.3934/ipi.2018056 +[Abstract](4074) +[HTML](270) +[PDF](2889.55KB)
Abstract:

Electron micrography (EM) is a detection method for determining the structure of macromolecular complexes and biological specimens. However, some restrictions in the EM system, including poor signal-to-noise, limited range of tilt angles, only a sub-region subject to electron exposure and unintentional movements of the specimen, make the reconstruction procedure severely ill-posed. Because of these limitations, there may be severe artifacts in reconstructed images. In this paper, we first design an algorithm that can quickly calculate the radiological paths. Then we combine an iterative reconstruction algorithm using the Mumford-Shah model with an artifact reduction strategy. The combined method can not only regularize the ill-posedness and provide the reconstruction and segmentation simultaneously but also smooth additional artifacts due to the limited data. Also we improved the algorithm used for the calculation of radiological paths to accelerate the reconstruction. The proposed algorithm was translated into OpenCL programs and kernel functions to asynchronously and in parallel update the reconstructed image along rays by GPUs. We tested the method on both simulated and real EM data. The results show that our artifact reduced Mumford-Shah algorithm can reduce the noise and artifacts while preserving and enhancing the edges in the reconstructed image.

2018, 12(6): 1365-1387 doi: 10.3934/ipi.2018057 +[Abstract](3244) +[HTML](218) +[PDF](545.01KB)
Abstract:

In this paper we consider the lens rigidity problem with partial data for conformal metrics in the presence of a magnetic field on a compact manifold of dimension \begin{document}$≥ 3$\end{document} with boundary. We show that one can uniquely determine the conformal factor and the magnetic field near a strictly convex (with respect to the magnetic geodesics) boundary point where the lens data is accessible. We also prove a boundary rigidity result with partial data assuming the lengths of magnetic geodesics joining boundary points near a strictly convex boundary point are known. The local lens rigidity result also leads to a global rigidity result under some strictly convex foliation condition. A discussion of a weaker version of the lens rigidity problem with partial data for general smooth curves is given at the end of the paper.

2018, 12(6): 1389-1410 doi: 10.3934/ipi.2018058 +[Abstract](4320) +[HTML](286) +[PDF](2981.83KB)
Abstract:

In this paper, we propose a novel image blocks based inpainting model using group sparsity and TV regularization. The block matching method is employed to collect similar image blocks which can be formed as sparse image groups. By reducing the redundant information in these groups, we can well restore textures missing in the inpainting areas. We built a variational framework based on a local SVD operator for block matching and group sparsity. In addition, TV regularization is naturally integrated in the model to reduce artificial effects which are caused by image blocks stacking in the block matching method. Besides, enforcing the sparsity of the representation, the SVD operators in our method are iteratively updated and play the role of dictionary learning. Thus it can greatly improve the quality of the restoration. Moreover, we mathematically show the existence of a minimizer for the proposed inpainting model. Convergence results of the proposed algorithm are also given in the paper. Numerical experiments demonstrate that the proposed model outperforms many benchmark methods such as BM3D based image inpainting.

2018, 12(6): 1411-1428 doi: 10.3934/ipi.2018059 +[Abstract](5006) +[HTML](289) +[PDF](542.49KB)
Abstract:

This paper concerns inverse source problems for the time-dependent Maxwell equations. The electric current density is assumed to be the product of a spatial function and a temporal function. We prove uniqueness and stability in determining the spatial or temporal function from the electric field, which is measured on a sphere or at a point over a finite time interval.

2018, 12(6): 1429-1442 doi: 10.3934/ipi.2018060 +[Abstract](3355) +[HTML](257) +[PDF](1782.64KB)
Abstract:

X-ray computed tomography technique has been used in many different practical applications. Often after reconstruction we need segment or decompose objects into different components. In this paper, we propose two new reconstruction methods that can decompose objects at the same time. By incorporating direction information, the proposed methods can decompose objects into various directional components. Furthermore, we propose an algorithm to obtain the direction information of the object directly from its CT measurements. We demonstrate the proposed methods on simulated and real samples to show their practical applicability. The numerical results show the differences between the two methods and effectiveness as dealing with fibre-crack decomposition problems.

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