# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

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## Inverse Problems & Imaging

January 2018 , Volume 12 , Issue 1

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2018, 12(1): 1-28 doi: 10.3934/ipi.2018001 +[Abstract](2755) +[HTML](242) +[PDF](538.76KB)
Abstract:

We consider a magnetic Schrödinger operator \begin{document} $(\nabla^X)^*\nabla^X+q$ \end{document} on a compact Riemann surface with boundary and prove a \begin{document} $\log\log$ \end{document}-type stability estimate in terms of Cauchy data for the electric potential and magnetic field under the assumption that they satisfy appropriate a priori bounds. We also give a similar stability result for the holonomy of the connection 1-form \begin{document} $X$ \end{document}.

2018, 12(1): 29-57 doi: 10.3934/ipi.2018002 +[Abstract](3308) +[HTML](317) +[PDF](4980.26KB)
Abstract:

Region-of-Interest (ROI) tomography aims at reconstructing a region of interest \begin{document} $C$ \end{document} inside a body using only x-ray projections intersecting \begin{document} $C$ \end{document} and it is useful to reduce overall radiation exposure when only a small specific region of a body needs to be examined. We consider x-ray acquisition from sources located on a smooth curve \begin{document} $Γ$ \end{document} in \begin{document} $\mathbb R^3$ \end{document} verifying the classical Tuy condition. In this generic situation, the non-trucated cone-beam transform of smooth density functions \begin{document} $f$ \end{document} admits an explicit inverse \begin{document} $Z$ \end{document} as originally shown by Grangeat. However \begin{document} $Z$ \end{document} cannot directly reconstruct \begin{document} $f$ \end{document} from ROI-truncated projections. To deal with the ROI tomography problem, we introduce a novel reconstruction approach. For densities \begin{document} $f$ \end{document} in \begin{document} $L^{∞}(B)$ \end{document} where \begin{document} $B$ \end{document} is a bounded ball in \begin{document} $\mathbb R^3$ \end{document}, our method iterates an operator \begin{document} $U$ \end{document} combining ROI-truncated projections, inversion by the operator \begin{document} $Z$ \end{document} and appropriate regularization operators. Assuming only knowledge of projections corresponding to a spherical ROI \begin{document} $C \subset B$ \end{document}, given \begin{document} $ε >0$ \end{document}, we prove that if \begin{document} $C$ \end{document} is sufficiently large our iterative reconstruction algorithm converges at exponential speed to an \begin{document} $ε$ \end{document}-accurate approximation of \begin{document} $f$ \end{document} in \begin{document} $L^{∞}$ \end{document}. The accuracy depends on the regularity of \begin{document} $f$ \end{document} quantified by its Sobolev norm in \begin{document} $W^5(B)$ \end{document}. Our result guarantees the existence of a critical ROI radius ensuring the convergence of our ROI reconstruction algorithm to an \begin{document} $ε$ \end{document}-accurate approximation of \begin{document} $f$ \end{document}. We have numerically verified these theoretical results using simulated acquisition of ROI-truncated cone-beam projection data for multiple acquisition geometries. Numerical experiments indicate that the critical ROI radius is fairly small with respect to the support region \begin{document} $B$ \end{document}.

2018, 12(1): 59-90 doi: 10.3934/ipi.2018003 +[Abstract](2741) +[HTML](247) +[PDF](543.55KB)
Abstract:

Inverse transport theory concerns the reconstruction of the absorption and scattering coefficients in a transport equation from knowledge of the albedo operator, which models all possible boundary measurements. Uniqueness and stability results are well known and are typically obtained for errors of the albedo operator measured in the \begin{document} $L^1$ \end{document} sense. We claim that such error estimates are not always very informative. For instance, arbitrarily small blurring and misalignment of detectors result in \begin{document} $O(1)$ \end{document} errors of the albedo operator and hence in \begin{document} $O(1)$ \end{document} error predictions on the reconstruction of the coefficients, which are not useful.

This paper revisit such stability estimates by introducing a more forgiving metric on the measurements errors, namely the \begin{document} $1-$ \end{document}Wasserstein distances, which penalize blurring or misalignment by an amount proportional to the width of the blurring kernel or to the amount of misalignment. We obtain new stability estimates in this setting.

We also consider the effect of errors, still measured in the \begin{document} $1-$ \end{document} Wasserstein distance, on the generation of the probing source. This models blurring and misalignment in the design of (laser) probes and allows us to consider discretized sources. Under appropriate assumptions on the coefficients, we quantify the effect of such errors on the reconstructions.

2018, 12(1): 91-123 doi: 10.3934/ipi.2018004 +[Abstract](3141) +[HTML](288) +[PDF](583.43KB)
Abstract:

We detect an inclusion with infinite conductivity from boundary measurements represented by the Dirichlet-to-Neumann map for the conductivity equation. We use both the enclosure method and the probe method. We use the enclosure method to prove partial results when the underlying equation is the quasilinear \begin{document} $p$ \end{document}-Laplace equation. Further, we rigorously treat the forward problem for the partial differential equation \begin{document} $\operatorname{div}(σ\lvert\nabla u\rvert^{p-2}\nabla u) = 0$ \end{document} where the measurable conductivity \begin{document} $σ\colonΩ\to[0,∞]$ \end{document} is zero or infinity in large sets and \begin{document} $1<p<∞$ \end{document}.

2018, 12(1): 125-152 doi: 10.3934/ipi.2018005 +[Abstract](2486) +[HTML](253) +[PDF](1660.87KB)
Abstract:

We investigate in this paper the diffusion magnetic resonance imaging (MRI) in deformable organs such as the living heart. The difficulty comes from the hight sensitivity of diffusion measurement to tissue motion. Commonly in literature, the diffusion MRI signal is given by the complex magnetization of water molecules described by the Bloch-Torrey equation. When dealing with deformable organs, the Bloch-Torrey equation is no longer valid. Our main contribution is then to introduce a new mathematical description of the Bloch-Torrey equation in deforming media. In particular, some numerical simulations are presented to quantify the influence of cardiac motion on the estimation of diffusion. Moreover, based on a scaling argument and on an asymptotic model for the complex magnetization, we derive a new apparent diffusion coefficient formula. Finally, some numerical experiments illustrate the potential of this new version which gives a better reconstruction of the diffusion than using the classical one.

2018, 12(1): 153-174 doi: 10.3934/ipi.2018006 +[Abstract](2751) +[HTML](361) +[PDF](444.47KB)
Abstract:

In this paper, we consider the block orthogonal matching pursuit (BOMP) algorithm and the block orthogonal multi-matching pursuit (BOMMP) algorithm respectively to recover block sparse signals from an underdetermined system of linear equations. We first introduce the notion of block restricted orthogonality constant (ROC), which is a generalization of the standard restricted orthogonality constant, and establish respectively the sufficient conditions in terms of the block RIC and ROC to ensure the exact and stable recovery of any block sparse signals in both noiseless and noisy cases through the BOMP and BOMMP algorithm. We finally show that the sufficient condition on the block RIC and ROC is sharp for the BOMP algorithm.

2018, 12(1): 175-204 doi: 10.3934/ipi.2018007 +[Abstract](3755) +[HTML](315) +[PDF](5194.5KB)
Abstract:

We present a reconstruction method for estimating the pulse-wave velocity in the brain from dynamic MRI data. The method is based on solving an inverse problem involving an advection equation. A space-time discretization is used and the resulting largescale inverse problem is solved using an accelerated Landweber type gradient method incorporating sparsity constraints and utilizing a wavelet embedding. Numerical example problems and a real-world data test show a significant improvement over the results obtained by the previously used method.

2018, 12(1): 205-227 doi: 10.3934/ipi.2018008 +[Abstract](2966) +[HTML](319) +[PDF](420.82KB)
Abstract:

Some scattering problems for the multidimensional biharmonic operator are studied. The operator is perturbed by first and zero order perturbations, which maybe complex-valued and singular. We show that the solutions to direct scattering problem satisfy a Lippmann-Schwinger equation, and that this integral equation has a unique solution in the weighted Sobolev space \begin{document}$H_{-δ}^2$\end{document}. The main result of this paper is the proof of Saito's formula, which can be used to prove a uniqueness theorem for the inverse scattering problem. The proof of Saito's formula is based on norm estimates for the resolvent of the direct operator in \begin{document}$H_{-δ}^1$\end{document}.

2018, 12(1): 229-237 doi: 10.3934/ipi.2018009 +[Abstract](2669) +[HTML](235) +[PDF](353.3KB)
Abstract:

We consider restricted light ray transforms arising from an inverse problem of finding cosmic strings. We construct a relative left parametrix for the transform on two tensors, which recovers the space-like and some light-like singularities of the two tensor.

2018, 12(1): 239-259 doi: 10.3934/ipi.2018010 +[Abstract](3723) +[HTML](381) +[PDF](779.26KB)
Abstract:

Digital tomographic image reconstruction uses multiple x-ray projections obtained along a range of different incident angles to reconstruct a 3D representation of an object. For example, computed tomography (CT) generally refers to the situation when a full set of angles are used (e.g., 360 degrees) while tomosynthesis refers to the case when only a limited (e.g., 30 degrees) angular range is used. In either case, most existing reconstruction algorithms assume that the x-ray source is monoenergetic. This results in a simplified linear forward model, which is easy to solve but can result in artifacts in the reconstructed images. It has been shown that these artifacts can be reduced by using a more accurate polyenergetic assumption for the x-ray source, but the polyenergetic model requires solving a large-scale nonlinear inverse problem. In addition to reducing artifacts, a full polyenergetic model can be used to extract additional information about the materials of the object; that is, to provide a mechanism for quantitative imaging. In this paper, we develop an approach to solve the nonlinear image reconstruction problem by incorporating total variation (TV) regularization. The corresponding optimization problem is then solved by using a scaled gradient descent method. The proposed algorithm is based on KKT conditions and Nesterov's acceleration strategy. Experimental results on reconstructed polyenergetic image data illustrate the effectiveness of this proposed approach.

2018  Impact Factor: 1.469