# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

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## Inverse Problems & Imaging

June 2017 , Volume 11 , Issue 3

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2017, 11(3): 427-454 doi: 10.3934/ipi.2017020 +[Abstract](2226) +[HTML](10) +[PDF](1419.3KB)
Abstract:

Electrical Impedance Tomography (EIT) is a non-invasive imaging modality that uses surface electrical measurements to determine the internal conductivity of a body. The mathematical formulation of the EIT problem is a nonlinear and severely ill-posed inverse problem for which direct D-bar methods have proved useful in providing noise-robust conductivity reconstructions. Recent advances in D-bar methods allow for conductivity reconstructions using EIT measurement data from only part of the domain (e.g., a patient lying on their back could be imaged using only data gathered on the accessible part of the body). However, D-bar reconstructions suffer from a loss of sharp edges due to a nonlinear low-pass filtering of the measured data, and this problem becomes especially marked in the case of partial boundary data. Including a priori data directly into the D-bar solution method greatly enhances the spatial resolution, allowing for detection of underlying pathologies or defects, even with no assumption of their presence in the prior. This work combines partial data D-bar with a priori data, allowing for noise-robust conductivity reconstructions with greatly improved spatial resolution. The method is demonstrated to be effective on noisy simulated EIT measurement data simulating both medical and industrial imaging scenarios.

2017, 11(3): 455-476 doi: 10.3934/ipi.2017021 +[Abstract](1161) +[HTML](2) +[PDF](452.4KB)
Abstract:

We consider the problem of developing a method to reconstruct a potential \begin{document}$q$\end{document} from the partial data Dirichlet-to-Neumann map for the Schrödinger equation \begin{document}$(-Δ_g+q)u=0$\end{document} on a fixed admissible manifold \begin{document}$(M,g)$\end{document}. If the part of the boundary that is inaccessible for measurements satisfies a flatness condition in one direction, then we reconstruct the local attenuated geodesic ray transform of the one-dimensional Fourier transform of the potential \begin{document}$q$\end{document}. This allows us to reconstruct \begin{document}$q$\end{document} locally, if the local (unattenuated) geodesic ray transform is constructively invertible. We also reconstruct \begin{document}$q$\end{document} globally, if \begin{document}$M$\end{document} satisfies certain concavity condition and if the global geodesic ray transform can be inverted constructively. These are reconstruction procedures for the corresponding uniqueness results given by Kenig and Salo [7]. Moreover, the global reconstruction extends and improves the constructive proof of Nachman and Street [14] in the Euclidean setting. We derive a certain boundary integral equation which involves the given partial data and describes the traces of complex geometrical optics solutions. For construction of complex geometrical optics solutions, following [14] and improving their arguments, we use a certain family of Green's functions for the Laplace-Beltrami operator and the corresponding single layer potentials. The constructive inversion problem for local or global geodesic ray transforms is one of the major topics of interest in integral geometry.

2017, 11(3): 477-500 doi: 10.3934/ipi.2017022 +[Abstract](982) +[HTML](10) +[PDF](451.9KB)
Abstract:

Waves can be used to probe and image an unknown medium. Passive imaging uses ambient noise sources to illuminate the medium. This paper considers passive imaging with moving sensors. The motivation is to generate large synthetic apertures, which should result in enhanced resolution. However Doppler effects and lack of reciprocity significantly affect the imaging process. This paper discusses the consequences in terms of resolution and it shows how to design appropriate imaging functions depending on the sensor trajectory and velocity.

2017, 11(3): 501-519 doi: 10.3934/ipi.2017023 +[Abstract](2459) +[HTML](16) +[PDF](4112.8KB)
Abstract:

Time-invariant Radon transforms play an important role in many fields of imaging sciences, whereby a function is transformed linearly by integrating it along specific paths, e.g. straight lines, parabolas, etc. In the case of linear Radon transform, the Fourier slice theorem establishes a simple analytic relationship between the 2-D Fourier representation of the function and the 1-D Fourier representation of its Radon transform. However, the theorem can not be utilized for computing the Radon integral along paths other than straight lines. We generalize the Fourier slice theorem to make it applicable to general time-invariant Radon transforms. Specifically, we derive an analytic expression that connects the 1-D Fourier coefficients of the function to the 2-D Fourier coefficients of its general Radon transform. For discrete data, the model coefficients are defined over the data coefficients on non-Cartesian points. It is shown numerically that a simple linear interpolation provide satisfactory results and in this case implementations of both the inverse operator and its adjoint are fast in the sense that they run in \begin{document}$O(N \;\text{log}\; N)$\end{document} flops, where \begin{document}$N$\end{document} is the maximum number of samples in the data space or model space. These two canonical operators are utilized for efficient implementation of the sparse Radon transform via the split Bregman iterative method. We provide numerical examples showing high-performance of this method for noise attenuation and wavefield separation in seismic data.

2017, 11(3): 521-538 doi: 10.3934/ipi.2017024 +[Abstract](1096) +[HTML](4) +[PDF](378.7KB)
Abstract:

We study detecting a boundary corrosion damage in the inaccessible part of a rectangular shaped electrostatic conductor from a one set of Cauchy data specified on an accessible boundary part of conductor. For this nonlinear ill-posed problem, we prove the uniqueness in a very general framework. Then we establish the conditional stability of Hölder type based on some a priori assumptions on the unknown impedance and the electrical current input specified in the accessible part. Finally a regularizing scheme of double regularizing parameters, using the truncation of the series expansion of the solution, is proposed with the convergence analysis on the explicit regularizing solution in terms of a practical average norm for measurement data.

2017, 11(3): 539-551 doi: 10.3934/ipi.2017025 +[Abstract](1622) +[HTML](7) +[PDF](498.9KB)
Abstract:

High-dimensional data often lie in low-dimensional subspaces instead of the whole space. Subspace clustering is a problem to analyze data that are from multiple low-dimensional subspaces and cluster them into the corresponding subspaces. In this work, we propose a \begin{document}$(k,k)$\end{document}-sparse matrix factorization method for subspace clustering. In this method, data itself is considered as the "dictionary", and each data point is represented as a linear combination of the basis of its cluster in the dictionary. Thus, the coefficient matrix is low-rank and sparse. With an appropriate permutation, it is also blockwise with each block corresponding to a cluster. With an assumption that each block is no more than \begin{document}$k$\end{document}-by-\begin{document}$k$\end{document} in matrix recovery, we seek a low-rank and \begin{document}$(k,k)$\end{document}-sparse coefficient matrix, which will be used for the construction of affinity matrix in spectral clustering. The advantage of our proposed method is that we recover a coefficient matrix with \begin{document}$(k,k)$\end{document}-sparse and low-rank simultaneously, which is better fit for subspace clustering. Numerical results illustrate the effectiveness that it is better than SSC and LRR in real-world classification problems such as face clustering and motion segmentation.

2017, 11(3): 553-575 doi: 10.3934/ipi.2017026 +[Abstract](1367) +[HTML](5) +[PDF](443.5KB)
Abstract:

In this paper, we use the theory of symmetric Dirichlet forms to give a probabilistic interpretation of Calderón's inverse conductivity problem in terms of reflecting diffusion processes and their corresponding boundary trace processes. This probabilistic interpretation comes in three equivalent formulations which open up novel perspectives on the classical question of unique determinability of conductivities from boundary data. We aim to make this work accessible to both readers with a background in stochastic process theory as well as researchers working on deterministic methods in inverse problems.

2017, 11(3): 577-600 doi: 10.3934/ipi.2017027 +[Abstract](3598) +[HTML](8) +[PDF](7525.7KB)
Abstract:

Region-based image segmentation is well-addressed by the Chan-Vese (CV) model. However, this approach fails when images are affected by artifacts (outliers) and illumination bias that outweigh the actual image contrast. Here, we introduce a model for segmenting such images. In a single energy functional, we introduce 1) a dynamic artifact class preventing intensity outliers from skewing the segmentation, and 2), in Retinex-fashion, we decompose the image into a piecewise-constant structural part and a smooth bias part. The CV-segmentation terms then only act on the structure, and only in regions not identified as artifacts. The segmentation is parameterized using a phase-field, and efficiently minimized using threshold dynamics.

We demonstrate the proposed model on a series of sample images from diverse modalities exhibiting artifacts and/or bias. Our algorithm typically converges within 10-50 iterations and takes fractions of a second on standard equipment to produce meaningful results. We expect our method to be useful for damaged images, and anticipate use in applications where artifacts and bias are actual features of interest, such as lesion detection and bias field correction in medical imaging, e.g., in magnetic resonance imaging (MRI).

2017  Impact Factor: 1.465