# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

All Issues

## Inverse Problems & Imaging

2009 , Volume 3 , Issue 3

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2009, 3(3): 373-382 doi: 10.3934/ipi.2009.3.373 +[Abstract](432) +[PDF](178.3KB)
Abstract:
The paper is devoted to the range description of the Radon type transform that averages a function over all spheres centered on a given sphere. Such transforms arise naturally in thermoacoustic tomography, a novel method of medical imaging. Range descriptions have recently been obtained for such transforms, and consisted of smoothness and support conditions, moment conditions, and some additional orthogonality conditions of spectral nature. It has been noticed that in odd dimensions, surprisingly, the moment conditions are superfluous and can be eliminated. It is shown in this text that in fact the same happens in any dimension.
2009, 3(3): 383-387 doi: 10.3934/ipi.2009.3.383 +[Abstract](373) +[PDF](119.9KB)
Abstract:
This paper deals with the application of non-convex, sublinear penalty terms to the regularization of possibly non-linear inverse problems the solutions of which are assumed to have a sparse expansion with respect to some given basis or frame. It is shown that this type of regularization is well-posed and yields sparse results. Moreover, linear convergence rates are derived under the additional assumption of a certain range condition.
2009, 3(3): 389-403 doi: 10.3934/ipi.2009.3.389 +[Abstract](413) +[PDF](273.5KB)
Abstract:
In this work we consider a modified MUSIC method for determining the positions of a collection of small perfectly conducting buried objects from measurements of time-harmonic electromagnetic fields on the surface of ground. This method is based on an asymptotic analysis of certain integrals of electric and magnetic fields, so-called reciprocity gap functionals, as the buried objects shrink to points. Unlike standard MUSIC reconstruction methods our algorithm avoids the computation of the Green's function for the background medium during the reconstruction process, but on the other hand it requires more measurement data. After describing the theoretical foundation of this reconstruction method, we provide numerical results showing its performance. We also compare these results to reconstructions obtained by a standard MUSIC algorithm.
2009, 3(3): 405-452 doi: 10.3934/ipi.2009.3.405 +[Abstract](442) +[PDF](549.3KB)
Abstract:
The equations for the electromagnetic field in an anisotropic media are written in a form containing only the transverse field components relative to a half plane boundary. The operator corresponding to this formulation is the electromagnetic system's matrix. A constructive proof of the existence of directional wave-field decomposition with respect to the normal of the boundary is presented.
In the process of defining the wave-field decomposition (wave-splitting), the resolvent set of the time-Laplace representation of the system's matrix is analyzed. This set is shown to contain a strip around the imaginary axis. We construct a splitting matrix as a Dunford-Taylor type integral over the resolvent of the unbounded operator defined by the electromagnetic system's matrix. The splitting matrix commutes with the system's matrix and the decomposition is obtained via a generalized eigenvalue-eigenvector procedure. The decomposition is expressed in terms of components of the splitting matrix. The constructive solution to the question of the existence of a decomposition also generates an impedance mapping solution to an algebraic Riccati operator equation. This solution is the electromagnetic generalization in an anisotropic media of a Dirichlet-to-Neumann map.
2009, 3(3): 453-464 doi: 10.3934/ipi.2009.3.453 +[Abstract](477) +[PDF](211.3KB)
Abstract:
Let $(M,g)$ be a simple Riemannian manifold with boundary and consider the geodesic ray transform of symmetric 2-tensor fields. Let the integral of such a field $f$ along maximal geodesics vanish on an appropriate open subset of the space of geodesics in $M$. Under the assumption that the metric $g$ is real-analytic, it is shown that there exists a vector field $v$ satisfying $f=dv$ on the set of points lying on these geodesics and $v=0$ on the intersection of this set with the boundary ∂$M$ of the manifold $M$. Using this result, a Helgason's type of a support theorem for the geodesic ray transform is proven. The approach is based on analytic microlocal techniques.
2009, 3(3): 465-486 doi: 10.3934/ipi.2009.3.465 +[Abstract](523) +[PDF](888.2KB)
Abstract:
It is well known that a matched filter gives the maximum possible output signal-to-noise ratio (SNR) when the input is a scattering signal from a point like radar target in the presence of white noise. However, a matched filter produces unwanted sidelobes that can mask vital information. Several researchers have presented various methods of dealing with this problem. They have employed different kinds of less optimal filters in terms of the output SNR from a point-like target than that of the matched filter. In this paper we present a method of designing codes, called perfect and almost perfect pulse compression codes, that do not create unwanted sidelobes when they are convolved with the corresponding matched filter. We present a method of deriving these types of codes from any binary phase radar codes that do not contain zeros in the frequency domain. Also, we introduce a heuristic algorithm that can be used to design almost perfect codes, which are more suitable for practical implementation in a radar system. The method is demonstrated by deriving some perfect and almost perfect pulse compression codes from some binary codes. A rigorous method of comparing the performance of almost perfect codes (truncated) with that of perfect codes is presented.
2009, 3(3): 487-503 doi: 10.3934/ipi.2009.3.487 +[Abstract](645) +[PDF](682.0KB)
Abstract:
We propose a fast algorithm for solving the Basis Pursuit problem, minu $\{|u|_1\: \Au=f\}$, which has application to compressed sensing. We design an efficient method for solving the related unconstrained problem minu $E(u) = |u|_1 + \lambda \||Au-f\||^2_2$ based on a greedy coordinate descent method. We claim that in combination with a Bregman iterative method, our algorithm will achieve a solution with speed and accuracy competitive with some of the leading methods for the basis pursuit problem.
2009, 3(3): 505-536 doi: 10.3934/ipi.2009.3.505 +[Abstract](391) +[PDF](311.4KB)
Abstract:
In this paper we determine a (possibly) non-continuous scalar relaxation kernel of bounded variation in an integrodifferential equation related to a Banach space when a nonlocal additional measurement involving the state function is available. We prove a result concerning global existence and uniqueness.
An application is given, in the framework of space of continuous functions, to the case of one-dimensional hyperbolic second-order integrodifferential equations endowed with initial and Dirichlet boundary conditions.
2009, 3(3): 537-550 doi: 10.3934/ipi.2009.3.537 +[Abstract](411) +[PDF](202.5KB)
Abstract:
We study inverse scattering for $\Delta_g+V$ on $(X,g)$ a conformally compact manifold with metric $g,$ with variable sectional curvature -α2(y) at the boundary and $V\in C^\infty(X)$ not vanishing at the boundary. We prove that the scattering matrices at two fixed energies $\lambda_1,$ $\lambda_2$ in a suitable subset of c , determines α, and the Taylor series of both the potential and the metric at the boundary.

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