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Inverse Problems & Imaging

2012 , Volume 6 , Issue 1

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Small volume asymptotics for anisotropic elastic inclusions
Elena Beretta, Eric Bonnetier, Elisa Francini and Anna L. Mazzucato
2012, 6(1): 1-23 doi: 10.3934/ipi.2012.6.1 +[Abstract](221) +[PDF](406.3KB)
Abstract:
We derive asymptotic expansions for the displacement at the boundary of a smooth, elastic body in the presence of small inhomogeneities. Both the body and the inclusions are allowed to be anisotropic. This work extends prior work of Capdeboscq and Vogelius (Math. Modeling Num. Anal. 37, 2003) for the conductivity case. In particular, we obtain an asymptotic expansion of the difference between the displacements at the boundary with and without inclusions, under Neumann boundary conditions, to first order in the measure of the inclusions. We impose no geometric conditions on the inclusions, which need only be measurable sets. The first-order correction contains a moment or polarization tensor $\mathbb{M}$ that encodes the effect of the inclusions. We also derive some basic properties of this tensor $\mathbb{M}$. In the case of thin, strip-like, planar inhomogeneities we obtain a formula for $\mathbb{M}$ only in terms of the elasticity tensors, which we assume strongly convex, their inverses, and a frame on the curve that supports the inclusion. We prove uniqueness of $\mathbb{M}$ in this setting and recover the formula previously obtained by Beretta and Francini (SIAM J. Math. Anal., 38, 2006).
On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach
Roman Chapko and B. Tomas Johansson
2012, 6(1): 25-38 doi: 10.3934/ipi.2012.6.25 +[Abstract](266) +[PDF](693.3KB)
Abstract:
We investigate the problem of determining the stationary temperature field on an inclusion from given Cauchy data on an accessible exterior boundary. On this accessible part the temperature (or the heat flux) is known, and, additionally, on a portion of this exterior boundary the heat flux (or temperature) is also given. We propose a direct boundary integral approach in combination with Tikhonov regularization for the stable determination of the temperature and flux on the inclusion. To determine these quantities on the inclusion, boundary integral equations are derived using Green's functions, and properties of these equations are shown in an $L^2$-setting. An effective way of discretizing these boundary integral equations based on the Nyström method and trigonometric approximations, is outlined. Numerical examples are included, both with exact and noisy data, showing that accurate approximations can be obtained with small computational effort, and the accuracy is increasing with the length of the portion of the boundary where the additionally data is given.
Identification of obstacles using only the scattered P-waves or the scattered S-waves
Drossos Gintides and Mourad Sini
2012, 6(1): 39-55 doi: 10.3934/ipi.2012.6.39 +[Abstract](173) +[PDF](409.7KB)
Abstract:
In this work, we are concerned with the inverse scattering by obstacles for the linearized, homogeneous and isotropic elastic model. We study the uniqueness issue of detecting smooth obstacles from the knowledge of elastic far field patterns. We prove that the 'pressure' parts of the far field patterns over all directions of measurements corresponding to all 'pressure' (or all 'shear') incident plane waves are enough to guarantee uniqueness. We also establish that the shear parts of the far field patterns corresponding to all the 'shear' (or all 'pressure') incident waves are also enough. This shows that any of the two different types of waves is enough to detect obstacles at a fixed frequency. The proof is reconstructive and it can be used to set up an algorithm to detect the obstacle from the mentioned data.
Positive definiteness of Diffusion Kurtosis Imaging
Shenglong Hu, Zheng-Hai Huang, Hong-Yan Ni and Liqun Qi
2012, 6(1): 57-75 doi: 10.3934/ipi.2012.6.57 +[Abstract](209) +[PDF](1184.9KB)
Abstract:
Diffusion Kurtosis Imaging (DKI) is a new Magnetic Resonance Imaging (MRI) model to characterize the non-Gaussian diffusion behavior in tissues. In reality, the term $bD_{app}-\frac{1}{6}b^2D_{app}^2K_{app}$ in the extended Stejskal and Tanner equation of DKI should be positive for an appropriate range of $b$-values to make sense physically. The positive definiteness of the above term reflects the signal attenuation in tissues during imaging. Hence, it is essential for the validation of DKI.
    In this paper, we analyze the positive definiteness of DKI. We first characterize the positive definiteness of DKI through the positive definiteness of a tensor constructed by diffusion tensor and diffusion kurtosis tensor. Then, a conic linear optimization method and its simplified version are proposed to handle the positive definiteness of DKI from the perspective of numerical computation. Some preliminary numerical tests on both synthetical and real data show that the method discussed in this paper is promising.
Inverse obstacle scattering with limited-aperture data
Masaru Ikehata, Esa Niemi and Samuli Siltanen
2012, 6(1): 77-94 doi: 10.3934/ipi.2012.6.77 +[Abstract](251) +[PDF](505.1KB)
Abstract:
Inverse obstacle scattering aims to extract information about distant and unknown targets using wave propagation. This study concentrates on a two-dimensional setting using time-harmonic acoustic plane waves as incident fields and taking the obstacles to be sound-hard with smooth or polygonal boundary. Measurement data is simulated by sending one incident wave towards the area of interest and computing the far field pattern (1) on the whole circle of observation directions, (2) only in directions close to backscattering, and (3) only in directions close to forward-scattering. A variant of the enclosure method is introduced, based on applying the far field operator to an explicitly constructed density, yielding information about the convex hull of the obstacle. The numerical evidence presented suggests that the convex hull of obstacles can be approximately recovered from noisy limited-aperture far field data.
A multiphase logic framework for multichannel image segmentation
Matthew S. Keegan, Berta Sandberg and Tony F. Chan
2012, 6(1): 95-110 doi: 10.3934/ipi.2012.6.95 +[Abstract](218) +[PDF](1567.5KB)
Abstract:
We propose a novel framework for energy-based multiphase segmentation over multiple channels. The framework allows the user to combine the information from each channel as the user sees fit, and thus allows the user to define how the information from each channel should influence the result. The framework extends the two-phase Logic Framework [J. Vis. Commun. Image R. 16 (2005) 333-358] model. The logic operators of the Logic Framework are used to define objective functions for multiple phases and a condition is defined that prevents conflict between energy terms. This condition prevents local minima that may occur using ad hoc methods, such as summing the objective functions of each region.
Fast reconstruction algorithms for the thermoacoustic tomography in certain domains with cylindrical or spherical symmetries
Leonid Kunyansky
2012, 6(1): 111-131 doi: 10.3934/ipi.2012.6.111 +[Abstract](177) +[PDF](1474.5KB)
Abstract:
We propose three fast algorithms for solving the inverse problem of the thermoacoustic tomography corresponding to certain acquisition geometries. Two of these methods are designed to process the measurements done with point-like detectors placed on a circle (in 2D) or a sphere (in 3D) surrounding the object of interest. The third inversion algorithm works with the data measured by the integrating line detectors arranged in a cylindrical assembly rotating around the object. The number of operations required by these techniques is equal to $\mathcal{O}(n^{3} \log n)$ and $\mathcal{O}(n^{3} \log^2 n)$ for the 3D techniques (assuming the reconstruction grid with $n^3$ nodes) and to $\mathcal{O}(n^{2} \log n)$ for the 2D problem with $n \times n$ discretizetion grid. Numerical simulations show that on large computational grids our methods are at least two orders of magnitude faster than the finite-difference time reversal techniques. The results of reconstructions from real measurements done by the integrating line detectors are also presented, to demonstrate the practicality of our algorithms.
The order of convergence for Landweber Scheme with $\alpha,\beta$-rule
Caifang Wang and Tie Zhou
2012, 6(1): 133-146 doi: 10.3934/ipi.2012.6.133 +[Abstract](205) +[PDF](416.9KB)
Abstract:
The Landweber scheme is widely used in various image reconstruction problems. In previous works, $\alpha,\beta$-rule is suggested to stop the Landweber iteration so as to get proper iteration results. The order of convergence of discrepancy principal (DP rule), which is a special case of $\alpha,\beta$-rule, with constant relaxation coefficient $\lambda$ satisfying $0<\lambda\sigma_1^2<1,~(\|A\|_{V,W}=\sigma_1>0)$ has been studied. A sufficient condition for convergence of Landweber scheme is that the value $\lambda_m\sigma_1^2$ should be lied in a closed interval, i.e. $0<\varepsilon\leq\lambda_m\sigma_1^2\leq2-\varepsilon$, $(0<\varepsilon<1)$. In this paper, we mainly investigate the order of convergence of the $\alpha,\beta$-rule with variable relaxation coefficient $\lambda_m$ satisfying $0 < \varepsilon\leq\lambda_m \sigma_1^2 \leq 2-\varepsilon$. According to the order of convergence, we can conclude that $\alpha,\beta$-rule is the optimal rule for the Landweber scheme.

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