Showing papers in "Inverse Problems in 1995"

Identification of the permeability distribution in soil by hydraulic tomography

Abstract: A tomographical method is presented that allows the reconstruction of the structure of soil layers between two boreholes. If the soil is water saturated, the transient field equation for pore water pressure S(x) delta tu(x,t)- Del .(k(x) Del u(x,t))=Q can be used to reconstruct the soil structure from its permeability k(x). The transient potential field u(x, t) is controlled by hydraulic dipoles (point source and point sink). The positions of source and sink are varied over both boreholes. Pore water pressure observations are used as data for the inverse problem. Uniqueness problems are discussed. The forward problem is discretized in a FE algorithm, the inverse problem solved by a standard output least-squares method.

Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by a control method

Abstract: Let u(f) be the solution to a hyperbolic equation in a bounded domain Omega subset R':u"(x, t) = Delta u(x,t) + sigma (t)f(x) (x in Omega, 0 0, we will show the stability estimate of ||f||L2(Omega) by ||partial u(f)/partial n||H1(0,T;L2(Gamma)), a reconstruction formula of f from partial u(f)/partial n and a Tikhonov regularization. Our methodology is based on exact boundary controllability and a Volterra integral equation of the first kind with kernel sigma.

A propagation-backpropagation method for ultrasound tomography

Abstract: Ultrasound tomography is modelled by the inverse problem of a 2D Helmholtz equation at fixed frequency with plane-wave irradiation. It is assumed that the field is measured outside the support of the unknown potential f for finitely many incident waves. Starting out from an initial guess f0 for f we propagate the measured field through the object f0 to yield a computed held whose difference to the measurements is in turn backpropagated. The backpropagated field is used to update f0. The propagation as well as the backpropagation are done by a finite difference marching scheme. The whole process is carried out in a single-step fashion, i.e. the updating is done immediately after backpropagating a single wave. It is very similar to the well known ART method in X-ray tomography, with the projection and backprojection step replaced by propagation and backpropagation.

The phase retrieval problem

Abstract: In the phase retrieval problem one seeks to recover an unknown function g(t) from the amplitude mod g(k) mod of its Fourier transform. Since phase and amplitude are, in general, independent of each other, it is necessary to make use of other kinds of information which implicitly or explicitly constrain the admissible solutions g(t). In this paper we survey a variety of results explaining circumstances under which g(t) may be uniquely recovered from mod g(k) mod and supplementary information. A number of explicit formulae for the phase are discussed. We pay particular attention to the phase retrieval problem as it arises in certain inverse-scattering applications.

Frechet derivatives in inverse obstacle scattering

Abstract: We are concerned with inverse obstacle scattering problems. For a fixed incident wave we consider the operator mapping an obstacle onto the far-field pattern of the scattered wave. The existence and characterizations of the Frechet derivatives for the exterior Robin problem and the transmission problem are proved.

Approximate inverse for linear and some nonlinear problems

Abstract: In this paper we present a method for solving problems such as Af = g by constructing an approximate inverse which maps the data g to a regularized solution of this equation of the first kind. No discretization for f is needed. The solution operator can be precomputed independently of the data. This works for linear problems and for nonlinear problems with a special structure. The regularization is achieved by computing mollified versions of the (minimum-norm) solution. It is shown that this class of regularization operators contains, as special cases, the classical methods such as Tikhonov - Phillips, iteration methods and also discretization methods. In the case where the operator has some invariance properties the storage needs are dramatically reduced.

On local tomography

Abstract: In this paper we explain how the local tomography approach to tomographic problems can be extended to a wide range of situations including limited data problems, attenuated transforms, and generalized radon transforms. Numerical examples illustrate the use of local tomography applied to complete and limited data problems. Our analytic results are obtained through the use of microlocal analysis.

A total variation enhanced modified gradient algorithm for profile reconstruction

Abstract: The total variation minimization method for deblurring noisy data is shown to be effective in dramatically increasing the resolution in a modified gradient approach to index of refraction reconstruction from measured scattered field data. Numerical evidence is presented which shows that by including the total variation in the functional to be minimized the reconstructions of piecewise constant profiles are considerably sharpened. The stability of the modified gradient method with respect to noise is apparently also enhanced. Furthermore, the presence of the total variation does not appear to adversely effect the established effectiveness of the modified gradient method in reconstructing smooth profiles.

A Method for Imaging Corrosion Damage in Thin Plates from Electrostatic Data

Abstract: The problem of quantitative nondestructive evaluation of corrosion in plates is considered. The inpection method uses boundary measurements of currents and voltages to determine the material loss caused by corrosion. The development of the method is based on linearization and the assumption that the plate is thin. The behavior of the method is examined in numerical situations. Keywords: nondestructive evaluation, corrosion monitoring, electrical impedance tomography, numerical reconstruction methods

Numerical solution of the sideways heat equation by difference approximation in time

Abstract: We consider a Cauchy problem for the heat equation in the quarter plane, where data are given at x=1 and a solution is sought in the interval 0

A factorization method for the 3D X-ray transform

Abstract: We consider the inversion of the three-dimensional (3D) X-ray transform with a limited data set containing the line integrals which have two intersections with the lateral surface of a cylindrical detector The usual solution to this problem is based on 3D filtered-backprojection, but this method is slow This paper presents a new algorithm which factors the 3D reconstruction problem into a set of independent 2D radon transforms for a stack of parallel slices Each slice is then reconstructed using standard 2D filtered-backprojection The algorithm is based on the application of the stationary-phase approximation to the 2D Fourier transform of the data, and is an extension to three dimensions of the frequency-distance relation derived by Edholm et al(1986) for the 2D radon transform Error estimates are also obtained

Inverse problems for a perturbed dissipative half-space

Abstract: Addresses the scattering of acoustic and electromagnetic waves from a perturbed dissipative half-space. For simplicity, the perturbation is assumed to have compact support. Section 1 discusses the application that motivated this work and explains how the scalar model used here is related to Maxwell's equations. Section 2 introduces three formulations for direct and inverse problems for the half-space geometry. Two of these formulations relate to scattering problems, and the third to a boundary value problem. Section 3 shows how the scattering problems can be related to the boundary value problem. This shows that the three inverse problems are equivalent in a certain sense. In section 4, the boundary value problem is used to outline a simple way to formulate a multi-dimensional layer stripping procedure. This procedure is unstable and does not constitute a practical algorithm for solving the inverse problem. The paper concludes with three appendices, the first two of which carry out a careful construction of solutions of the direct problems and the third of which contains a discussion of some properties of the scattering operator.

A wave equation in 2+1: Painleve analysis and solutions

Abstract: In this paper the nonlinear equation mty=(myxx+mxmy)x is thoroughly analysed The Painleve test is performed yielding a positive result The Backlund transformations are found and the Darboux-Moutard-Matveev formalism arises in the context of this analysis The singular manifold method, based upon the Painleve analysis, is proved to be a useful tool for generating solutions Some interesting explicit expressions for one and two solitons are obtained and analysed in such a way

Optimal radiation beam profiles considering the stochastic process of patient positioning in fractionated radiation therapy

Abstract: We present a solution to the problem of finding optimal beam profiles in fractionated radiation therapy when taking the uncertainty in beam patient alignment into account. The problem was previously solved for the special cases of one single dose fraction and infinitely many fractions. For few fractions (

Doppler tomography for vector fields

Abstract: The problem considered is to reconstruct a vector field in the case when, for each line, the distribution of the vector field components along the line is known. Such data are obtained from spectral analysis of signals, affected by Doppler shifts, caused by reflections on moving particles ('velocity spectra'). In the general setting, the problem is unsolved. Using only integrated data along lines, one encounters the same problem as when reconstructing vector fields by means of time-of-flight measurements. Reconstruction formulae are given in this case, establishing the known fact that only the curl of the vector field can be reconstructed without additional information. Inspired by the application to which the paper is most devoted, i.e. ultrasound Doppler measurements on blood flows, the case where the flow takes place in narrow channels is investigated in simulations and experiments.

Uniqueness theorems for an inverse problem in a doubly periodic structure

Abstract: Consider an electromagnetic plane wave incident on a doubly periodic structure in R3. The inverse problem is to determine the shape of the structure from the scattered field. Uniqueness theorems are proved by applying the uniqueness theorem of Cauchy-Kowalewska, by extending Isakov's approach and using a result on local injectivity of maps between finite-dimensional spaces.

Phase retrieval of radiated fields

Abstract: The problem of determining radiated electromagnetic fields from phaseless distributions on one or more surfaces surrounding the source is considered We first examine the theoretical aspects and basic points of an appropriate formulation and show the advantage of tackling the problem as the inversion of the quadratic operator, which, by acting on the real and imaginary parts of the field, provides square amplitude distributions Next, useful properties and representations of both fields and square amplitude distributions are introduced, thus making it possible to come to a convenient finite-dimensional model of the problem, to recognize its ill-posed nature and, finally, to define an appropriate generalized solution Novel uniqueness conditions for the solution of the problem and questions regarding the attainment of the generalized solution are discussed The geometrical properties of the functional set corresponding to the range of the quadratic operator relating the unknowns to the data are examined The question of avoiding local minima problems in the search for the generalized solution is carefully discussed and the crucial role of the ratio between the dimension of the data representation space and that of the unknowns is emphasized

The inverse problem of dynamics: basic facts

Abstract: Finding potentials or force fields from given families of orbits is the type of inverse problem of dynamics discussed in this paper. We present the pertinent partial differential equations for various versions of the problem such as, for instance, for conservative or autonomous non-conservative fields in two or three dimensions, for inertial or relating frames, for one material point or, more, generally, for holonomic systems with n degrees of freedom. The notion of the family boundary curves is introduced. The role of the homogeneity of the given family or of the required potential, as well as the question of multitude of compatible pairs of orbits and potentials is discussed. Comments on the relation of the problem to problems of astronomical interest are made, at appropriate places, throughout the text.

Global uniqueness for inverse parabolic problems with final observation

Abstract: It is shown that the unknown time-dependent right-hand side with explicitly bounded growth rate in a general parabolic equation is uniquely determined by one additional final measurement. As an important application we derive the corresponding uniqueness results for different coefficient identification problems with no smallness assumptions imposed.

Image reconstruction from truncated, two-dimensional, parallel projections

Abstract: Full three-dimensional scanning allows a significant improvement in image quality in X-ray transmission computerized tomography (CT), in single-photon emission computerized tomography (SPECT) and in positron emission tomography (PET). Increased detection efficiency is obtained by increasing the solid angle seen by the detectors and by detecting photons which are no longer confined to a set of parallel slices as in the standard 2D scanning mode. Consequently, 3D image reconstruction cannot be factored as usual into a set of independent 2D reconstructions, and hence one has to invert the 3D X-ray transform with limited data. Assuming a basic knowledge of standard 2D tomography, this paper presents a review of analytic methods for the reconstruction of a 3D image from a set of 2D parallel projections along some limited set of directions. This inverse problem is overdetermined, i.e., the projection data are redundant, and the consequences of this property are analysed. Redundancy is used to generate classes of exact filters for 3D filtered-backprojection, thereby allowing considerable versatility in the design of inversion algorithms tailored to specific applications. The review also covers the inversion of the 3D X-ray transform when the 2D parallel projections are incompletely measured (truncated), a situation which arises for example in PET, In view of data redundancy, it is possible to build convolution kernels for filtered-backprojection which have a limited support and are not, therefore, affected by truncation. A similar analysis and utilization of data redundancy could be proposed in any application where, instead of trying to define the smallest data set from which the problem can be solved, one attempts to optimize the signal-to-noise ratio by measuring and by incorporating into the reconstruction as much data as practically feasible.

Are our parameter estimators biased? The significance of finite-difference regularization operators

Abstract: The resolution of a particular solution to a given inverse problem may be quantified by resorting to a posteriori resolution indicators. In constrained least-squares inversions, for weakly nonlinear problems, the authors compute resolution kernels that relate the true model parameters to the estimated ones. These kernels reflect the significant effect on resolution that regularizing operators have. Since these operators are normally chosen to control the effect of noise, the kernels predict the degradation of the solutions in the presence of noise. For real problems with data errors, the knowledge of the resolving kernels enables us to quantify confidence in any parameter estimate, and gives a clear physical picture of the resolution; the resolution (indirectly) depends on the noise level in the data. The resolving kernels are an indication of the averaging of the true model parameters. The question arises as to whether these averages are biased or not. Popular methods, such as SVD truncation, or the use of a priori covariance operators, produce biased averages. In contrast, finite-difference regularizing operators systematically produce unbiased averages and prove to be a superior regularization technique in this respect. The authors tested the use of finite-difference regularizing operators using a simulation case study, in which they synthesized full waveforms for a simulated multi-offset 'VSP' survey, and inverted the picked traveltimes for 1D anisotropic parameters. The model resolution matrix for this case study was shown to be a reliable resolution indicator and it gave invaluable information on the essence of this inverse problem.

On a uniform approximation of the density function of a string equation using eigenvalues and nodal points and some related inverse nodal problems

Abstract: Using the eigenvalues and the nodal points of the corresponding eigenfunctions of a string equation with density function rho(x), we construct a sequence of piecewise linear continuous functions which converges to rho-1/2(x) uniformly We also obtain formulae for derivatives of rho-1/2(x) in terms of these spectral data, and using these formulae we find a necessary and sufficient condition for the convexity of rho-1/2(x) in terms of the nodal points of the eigenfunctions of the string equation

On finding a surface crack from boundary measurements

Abstract: We prove the uniqueness of the determination of a surface crack from one special boundary measurement of an electrical or elastic field. Then we suggest and test a numerical algorithm for identification of a polygonal plane crack based on the Schwarz-Christoffel formula. The numerical experiments with cracks consisting of one or two intervals show the high stability and precision of this algorithm.

Bayesian inference and Gibbs sampling in spectral analysis and parameter estimation: II

Abstract: Bayesian inference theory and Gibbs sampling techniques are introduced and applied to spectral analysis and parameter estimation for both single- and multiple-frequency signals. Specifically, the marginal posterior probabilities for amplitudes and frequencies are obtained by using Gibbs sampling without performing the integrations, no matter whether the variance of the noise is known or unknown. The best estimates of the parameters can be inferred from these probabilities together with the corresponding variances. In addition, when the variance of the noise is unknown, an estimate about the variance of the noise can also be made. Comparisons of our results have been made with results using the FFT method as well as with Bretthorst's (1990) method. The approach outlined shows several advantages.

Regularization of a non-characteristic Cauchy problem for a parabolic equation

Abstract: In this paper the non-characteristic Cauchy problem ut- alpha (x)uxx-b(x)ux-c(x)u=0, x in (0,l), t in R; u(0,t)= phi (t), t in R; ux(0,t)=0, t in R; is considered. The problem is well known to be severely ill-posed: a small perturbation in the Cauchy data may cause a dramatically large error in the solution. In this paper the following mollification method is suggested for this problem: if the Cauchy data are given inexactly then we mollify them by elements of well-posedness classes of the problem, namely by elements of an appropriate co-regular multiresolution approximation {Vj}j in Z of L2(R) which is generated by the father wavelet of Meyer (1992). Within VJ the problem is well posed, and we can find a mollification parameter J depending on the noise level epsilon in the Cauchy data such that the error estimation between the exact solution and the mollified solution is of Holder type. The method can be numerically implemented using fundamental results by Beylkin, Coifman and Rokhlin (1991) on representing (pseudo)differential operators in wavelet bases. A stable marching difference scheme based on this method is suggested. Several numerical examples are given.

Shape reconstruction using diffracted waves and canonical solutions

Abstract: This work deals with the determination of the shape of a generally-non-circular impenetrable cylinder from the way it scatters incident sound. A complete family (of generally non-orthogonal functions) representation of the scattered field is employed to match the total measured field. The resolution of the direct problem during the inversion is bypassed by assuming a priori that the coefficients in the field representation are locally those of an impenetrable circular cylinder. These coefficients are known explicitly to within a single parameter which is determined by resolution of a nonlinear equation. This parameter is none other than the length of the position vector joining the origin to the given point on the boundary of the cylinder, so that by varying the locations of the field measurement point and boundary point one generates a discrete form of the polar coordinate parametric equation of the boundary. Numerical examples of the results of the inversion scheme are given for cylinders with both convex (circular, elliptical) and non-convex boundaries.

On the Frechet differentiability of boundary integral operators in the inverse elastic scattering problem

Abstract: This paper is concerned with the study of the Frechet differentiability properties of the operator connecting the scattered field with scatterer`s surface in the framework of the inverse elastic scattering problem. We adopt the integral equation approach, which transfers the solution of the inverse problem to the solution of a boundary integral equation of the second kind. We study the behaviour of the appeared integral operators and prove that they constitute Frechet differentiable operators. As we show, this result leads to the conclusion that the scattered elastic field is Frechet differentiable with respect to the boundary of the scatterer. Finally we present a characterization of the Frechet derivative of the scattered field as the solution of a direct scattering elastic problem with suitable Dirichlet boundary conditions.

On isospectral rods, horns and strings

Abstract: The free undamped vibrations of rods, horns and taut strings are governed by second-order differential equations. It is known that the inverse problem, namely the reconstruction of such a system, e.g. the reconstruction of the cross-sectional profile of a rod, requires the knowledge of two free vibration spectra corresponding to two different sets of end conditions. This paper is concerned with families of second-order systems which have one spectrum in common. The analysis is based on the reduction of the governing equation to the standard Sturm-Liouville form, the use of the Darboux lemma, and the research of Trubowitz, Poschel (1987), Deift (1978) and others. In particular the paper establishes necessary and sufficient conditions for isospectral flow from one rod to another rod with the same end conditions, using double Darboux transformations.

The detection and monitoring of leukemia using electromagnetic waves: numerical analysis

Abstract: The presence of leukemia in bone marrow causes an increase in the electric permittivity and a decrease in the conductivity of the marrow. This suggests the possibility of detecting leukemia or monitoring the treatment of leukemia by electromagnetic imaging. In a previous paper we developed a dual-space method for solving this inverse scattering problem. In this paper we examine the viability of this method through the use of numerical experiments on synthetic data.

Propagation of transient electromagnetic waves in time-varying media-direct and inverse scattering problems

Abstract: Wave propagation of transient electromagnetic waves in time-varying media is considered. The medium, which is assumed to be inhomogeneous and dispersive, lacks invariance under time translations. The spatial variation of the medium is assumed to be in the depth coordinate, i.e. it is stratified. The constitutive relations of the medium are a time integral of a generalized susceptibility kernel and the field. The generalized susceptibility kernel depends on one spatial and two time coordinates. The concept of wave splitting is introduced. The direct and inverse scattering problems are solved by the use of an imbedding or a Green function approach. The direct and the inverse scattering problems are solved for a homogeneous semi-infinite medium. Explicit algorithms are developed. In this inverse scattering problem, a function depending on two time coordinates is reconstructed. Several numerical computations illustrate the performance of the algorithms.