# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

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## Inverse Problems & Imaging

November 2011 , Volume 5 , Issue 4

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2011, 5(4): 745-773 doi: 10.3934/ipi.2011.5.745 +[Abstract](1824) +[PDF](452.8KB)
Abstract:
In this article we seek stability estimates in the inverse problem of determining the potential or the velocity in a wave equation in an anisotropic medium from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichlet-to-Neumann map associated to the wave equation. We prove in dimension $n\geq 2$ that the knowledge of the Dirichlet-to-Neumann map for the wave equation uniquely determines the electric potential and we prove Hölder-type stability in determining the potential. We prove similar results for the determination of velocities close to 1.
2011, 5(4): 775-791 doi: 10.3934/ipi.2011.5.775 +[Abstract](1436) +[PDF](374.1KB)
Abstract:
We prove that it takes at most two measurements on the boundary to recover the heat coefficient of a one dimensional heat equation if its lower bound is known. Otherwise a finite number of measurements is needed. We also provide a new constructive algorithm for its recovery. Using asymptotics of eigenfunctions of the associated Sturm-Liouville problem we show that a hot spot initial condition generates all, except maybe a finite number of boundary spectral data. Then a counting argument based on the method of false position helps search for the number of missing boundary spectral data which is then unraveled by a finite number of measurements. Finally, we show how the boundary spectral data is converted into spectral data, and the well known Gelfand-Levitan-Gasymov inverse spectral theory of Sturm-Liouville operators yields the reconstruction of the heat coefficient uniquely.
2011, 5(4): 793-813 doi: 10.3934/ipi.2011.5.793 +[Abstract](1894) +[PDF](484.6KB)
Abstract:
Assume a time-harmonic electromagnetic wave is scattered by an infinitely long cylindrical conductor surrounded by an unknown piecewise homogenous medium remaining invariant along the cylinder axis. We prove that, in TM mode, the far field patterns for all incident and observation directions at a fixed frequency uniquely determine the unknown surrounding medium as well as the shape of the cylindrical conductor. A similar uniqueness result is obtained for the scattering by multilayered penetrable periodic structures in a piecewise homogeneous medium. The periodic interfaces and refractive indices can be uniquely identified from the near field data measured only above (or below) the structure for all quasi-periodic incident waves with a fixed phase-shift. The proofs are based on the singularity of the Green function to a two dimensional elliptic equation with piecewise constant leading coefficients.
2011, 5(4): 815-841 doi: 10.3934/ipi.2011.5.815 +[Abstract](2036) +[PDF](2509.2KB)
Abstract:
The first contribution of this paper is the comparison of learned dictionary based approaches to inpainting and denoising of images in natural scenes, where emphasis is given on the use of complete and overcomplete dictionary learned by independent component analysis. The second contribution of the paper relates to the formulation of a problem of denoising an image corrupted by a salt and pepper type of noise (this problem is equivalent to estimating saturated pixel values), as a noiseless inpainting problem, whereupon noise corrupted pixels are treated as missing pixels. A maximum a posteriori (MAP) approach to image denoising is not applicable in such a case due to the fact that variance of the impulsive noise is infinite and the MAP based estimation relies on solving an optimization problem with an inequality constraint that depends on the variance of the additive noise. Through extensive comparative performance analysis of the inpainting task, it is demonstrated that ICA-learned basis outperforms K-SVD and morphological component analysis approaches in terms of visual quality. It yielded similar performance as a field of experts method but with more than two orders of magnitude lower computational complexity. On the same problems, Fourier and wavelet bases as representatives of fixed bases, exhibited the poorest performance. It is also demonstrated that noiseless inpainting-based approach to image denoising (estimation of the saturated pixel values) greatly outperforms denoising based on two-dimensional myriad filtering that is a theoretically optimal solution for this class of additive impulsive noise.
2011, 5(4): 843-858 doi: 10.3934/ipi.2011.5.843 +[Abstract](1429) +[PDF](401.0KB)
Abstract:
In this paper the notion of the Krein spectral shift function is extended to the radial Schrödinger operator with fixed energy. Then we analyze the connections between the tail of the potential and the decay rate of the fixed-energy phase shifts. Finally we extend former results on the uniqueness of the fixed-energy inverse scattering problem to a general class of potentials.
2011, 5(4): 859-877 doi: 10.3934/ipi.2011.5.859 +[Abstract](1330) +[PDF](439.5KB)
Abstract:
We prove that a potential $q$ can be reconstructed from the Dirichlet-to-Neumann map for the Schrödinger operator $-\Delta_g + q$ in a fixed admissible $3$-dimensional Riemannian manifold $(M,g)$. We also show that an admissible metric $g$ in a fixed conformal class can be constructed from the Dirichlet-to-Neumann map for $\Delta_g$. This is a constructive version of earlier uniqueness results by Dos Santos Ferreira et al. [10] on admissible manifolds, and extends the reconstruction procedure of Nachman [31] in Euclidean space. The main points are the derivation of a boundary integral equation characterizing the boundary values of complex geometrical optics solutions, and the development of associated layer potentials adapted to a cylindrical geometry.
2011, 5(4): 879-891 doi: 10.3934/ipi.2011.5.879 +[Abstract](1661) +[PDF](992.9KB)
Abstract:
Computed tomography (CT) has become a common analysis method in the materials sciences. It allows the internal visualisation of the complete volume of an object, providing 3D information about the internal structures. One field where CT is applied is the examination of fibre-reinforced composite structures. Fibre-reinforced composites are typically composed of two types of material, mainly of high strength fibres embedded in a surrounding matrix. In this material class, the fibres typically determine the strength of the composite materials, which is largely dependent on the orientation of the fibres. Knowledge of the fibre orientation is therefore essential for the evaluation of maximal loading or for the prediction of failure. The easiest way to determine the fibre orientation is to compute it from the reconstructions received from the tomograph. A different method to determine fibre orientation is to compute it directly from Radon data using the combination of reconstruction and image analysis introduced by Louis [A. K. Louis, Combining Image Reconstruction and Image Analysis with an Application to 2D - Tomography, SIAM J. Imaging Sciences 1 (2008), 188--208]. This can be achieved by adapting the reconstruction process in computed tomography by the use of anisotropic, elongated convolution filters, leading to a set of reconstruction kernels that are dependent on the angle of the projections, thereby reflecting the anisotropy of the filters. In this paper, the two-dimensional case of computing fibre orientation directly from simulated Radon data is presented.
2011, 5(4): 893-913 doi: 10.3934/ipi.2011.5.893 +[Abstract](1490) +[PDF](603.9KB)
Abstract:
We present a new direct algorithm aiming at the reconstruction of the optical wavefront from the Shack-Hartmann sensor measurements in Single Conjugate Adaptive Optics (SCAO) systems. The objective of an adaptive optics system designed for a large telescope can be only achieved if the wavefront reconstruction is sufficiently fast. Our scheme does not contain any explicit regularization for the reconstruction process but is still able to provide a good quality of reconstruction. The analysis of quality is given for three varying parameters: the diameter of the telescope, the number of subapertures and the level of photon noise. It has been shown both analytically and numerically that the quality of the reconstruciton, measured by the Strehl ratio, is reasonable for the small photon noise level and increases with the increasing number of subapertures for the same telescope size. The impact of the photon noise on the reconstruction gets higher with the increasing telescope diameter. The computational complexity of the method is linear in the number of unkowns. Counting all summation and multiplication steps the scaling factor is $14$. Moreover, due to its simple structure, the cumulative reconstructor is pipelinable and parallelizable, which makes the effective computation even faster.

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