Table of contents

By cross-correlating successive ultrasonic scans of the biological tissue that is excited by a transient source on the surface, recent experiments show that elastic wave displacement during the interval between the two scans can be measured everywhere in the tissue. Assuming that we know the displacement history, w(t,x), of a propagating shear wave everywhere, we identify the stiffness change inside the tissue in terms of the Lamé coefficient, μ. So, the goal is: find μ from w. In the experiment, we consider that there is a dominant frequency, τ, called a central frequency. We then take the Fourier transform of w (in time) and evaluate this transform at τ. Using this transform of the data, our method starts with the asymptotic expansion of geometrical optics. By tracing the amplitude change of the displacement along the geometrical rays as they travel into the medium, we are able to recover μ without directly taking derivatives of the displacement data.

We consider the question: what can be determined about the stiffness distribution in biological tissue from indirect measurements? This leads us to consider an inverse problem for the identification of coefficients in the second-order hyperbolic system that models the propagation of elastic waves. The measured data for our inverse problem are the time-dependent interior vector displacements. In the isotropic case, we establish sufficient conditions for the unique identifiability of wave speeds and the simultaneous identifiability of both density and the Lamé parameters. In the anisotropic case, counterexamples are presented to exhibit the nonuniqueness and to show the structure of the set of shear tensors corresponding to the same given data.

We establish some global stability results together with logarithmic estimates in Sobolev norms for the inverse problem of recovering a Robin coefficient on part of the boundary of a smooth 2D domain from overdetermined measurements on the complementary part of a solution to the Laplace equation in the domain, using tools from analytic function theory.

We describe in some detail the structure of the set of the infinitely many monoparametric families f(x,y) = c to which one, two or three single orbits f₀(x,y) = 0 (possibly resulting from astronomical observations) may be classified. By a heuristic argument we show how families including more than three preassigned single orbits can be constructed. The motivation for this study stems from the fact that the inverse problem of dynamics relates potentials V (x,y) to monoparametric families of orbits (and not to single orbits). So, as an application of the geometrical considerations of the present study we find a partial differential equation, the solution of which helps us to gain a full family out of one observed orbit in a known potential.

Phase contrast tomography is a developing area in imaging. It is an extension of conventional tomography additionally employing phase information. However, a strong coherence condition is needed for the measurement set-up. Applications at synchrotron devices have been performed since 1998 with convincing success. Devices for small laboratories are about to be implemented in the near future. In this paper several models for phase contrast tomography are derived from the wave equation. Error estimates show the area of validity of these models. Using results from tomography, such as approximate inverse and fast backprojection, efficient reconstruction algorithms are worked out. Numerical tests with synthetic and real data are included.

Let T be a (possibly nonlinear) continuous operator on Hilbert space . If, for some starting vector x, the orbit sequence {T^kx,k = 0,1,...} converges, then the limit z is a fixed point of T; that is, Tz = z. An operator N on a Hilbert space is nonexpansive (ne) if, for each x and y in , Even when N has fixed points the orbit sequence {N^kx} need not converge; consider the example N = −I, where I denotes the identity operator. However, for any the iterative procedure defined by converges (weakly) to a fixed point of N whenever such points exist. This is the Krasnoselskii–Mann (KM) approach to finding fixed points of ne operators.

A wide variety of iterative procedures used in signal processing and image reconstruction and elsewhere are special cases of the KM iterative procedure, for particular choices of the ne operator N. These include the Gerchberg–Papoulis method for bandlimited extrapolation, the SART algorithm of Anderson and Kak, the Landweber and projected Landweber algorithms, simultaneous and sequential methods for solving the convex feasibility problem, the ART and Cimmino methods for solving linear systems of equations, the CQ algorithm for solving the split feasibility problem and Dolidze's procedure for the variational inequality problem for monotone operators.

Bar code reconstruction involves recovering a clean signal from an observed one that is corrupted by convolution with a kernel and additive noise. The precise form of the convolution kernel is also unknown, making reconstruction harder than in the case of standard deblurring. On the other hand, bar codes are functions that have a very special form—this makes reconstruction feasible. We develop and analyse a total variation based variational model for the solution of this problem. This new technique models systematically the interaction of neighbouring bars in the bar code under convolution with a kernel, as well as the estimation of the unknown parameters of the kernel from global information contained in the observed signal.

We are concerned with the possibility of identifying the real parameters a and b on the right-hand side of the equation for a function u satisfying the boundary conditions with any fixed sufficiently smooth function .

In the case of a smooth curve , we provide a sufficient condition, under which the pair (a,b) can be uniquely reconstructed through the specified function Φ. On the basis of this sufficient condition, we show that there are at most finitely many pairs (a,b) if ω is (simply connected and) different from a disc.

If ω has a corner we prove that the pair (a,b) is unique and can be calculated explicitly if Φ is known on one side near the corner.

A prescription for finding the densities of circular membranes which are isospectral to a uniform circular membrane is extended to cover isospectrality to any circular membrane density, thereby also revealing the underlying group-theoretical structure which is further elucidated. Membranes isospectral to radial-density membranes no longer possess the radial symmetry, and some examples are discussed.

The linear sampling method is a method to reconstruct the shape of an obstacle in time-harmonic inverse scattering without a priori knowledge of either the physical properties or the number of disconnected components of the scatterer. Although it has been proven numerically to be a fast and reliable method in many situations, no mathematical argument has yet been found to prove why this is so. Using results obtained by Kirsch in deriving the related factorization method, we show in this paper that for a large class of scattering problems, linear sampling can be interpreted rigorously as a numerical method to reconstruct the shape of an obstacle; or in other words that linear sampling must work for problems in this class.

We give formulae describing the simplest statistical properties of single-photon emission computed tomography (SPECT) data modelled as the attenuated ray transform with Poisson noise. To obtain some of these formulae we obtain and use an inequality relating the even and odd parts of the attenuated ray transform without noise. Precise equations relating the even and odd parts of this transform are also discussed. Using these results we propose new possibilities for improving the stability of SPECT imaging based on the explicit inversion formula for the attenuated ray transformation with respect to the Poisson noise in the emission data. Numerical examples illustrating some of the theoretical conclusions of the present work are given.

We consider the problem of determining the shape of an object immersed in an acoustic medium from measurements obtained at a distance from the object. We recast this problem as a shape optimization problem where we search for the domain that minimizes a cost function that quantifies the difference between the measured and expected signals. The measured and expected signals are assumed to satisfy a boundary-value problem given by the Helmholtz equation with the Sommerfeld condition imposed at infinity. Gradient-based algorithms are used to solve this optimization problem. At every step of the algorithm the derivative of the cost function with respect to the parameters that describe the shape of the object is calculated. We develop an efficient method based on the adjoint equations to calculate the derivative and show how this method is implemented in a finite element setting. The predominant cost of each step of the algorithm is equal to one forward solution and one adjoint solution and therefore is independent of the number of parameters used to describe the shape of the object. Numerical examples showing the efficacy of the proposed methodology are presented.

We propose an original method for determining suitable refracting profiles between two media to solve two related problems: to produce a given wavefront from a single point source after refraction at the refracting profile, and to focus a given wavefront at a fixed point. These profiles are obtained as envelopes of specific families of Cartesian ovals. We study the singularities of these profiles and give a method to construct them from the data of the associated caustic.

A new formula is given for the reconstruction of a function in 3D space from a family of its ray integrals. The cost of the reconstruction is two-fold integration.

A physical interpretation of the minimum principle of 1D inverse scattering theory is established from the recently developed mathematical theory of 'single-sided' focusing. In sum, minimization corresponds to reducing the blurring to zero. The connection of 'single-sided' focusing and blurring to the minimum principle requires: (1) the nontrivial extension of the orthonormality relation of quantum scattering theory to the plasma wave equation; and (2) the series solution for the wave field at the wavefront. The following new results are obtained for the minimum principle—and the functional P that it minimizes. First, the deviation of P from its minimum quantitatively measures the blurring. Secondly, the variational fields used to minimize P are identical to the tails of the incident fields of the focusing problem. Thirdly, the same incident field that leads to focusing also minimizes P. Finally P has a global minimum because the incident field that focuses is unique and all other incident fields lead to blurring and thus to a greater value of P. Numerical calculations for a 'square-barrier' potential are used to illustrate the results.

The aim of this paper is to construct Levenberg–Marquardt level set methods for inverse obstacle problems, and to discuss their numerical realization. Based on a recently developed framework for the construction of level set methods, we can define Levenberg–Marquardt level set methods in a general way by varying the function space used for the normal velocity. In the typical case of a PDE-constraint, the iterative method yields an indefinite linear system to be solved in each iteration step, which can be reduced to a positive definite problem for the normal velocity. We discuss the structure of this system and the possibilities for its iterative solution.

Moreover, we investigate the application and numerical discretization of the method for two model problems, a mildly ill-posed source reconstruction problem and a severely ill-posed identification problem from boundary data. The numerical results demonstrate a significant speed-up with respect to standard gradient-based level set methods, in particular if topology changes occur during the level set evolution.

We examine the uniqueness of an N-field generalization of a 2D inverse problem associated with elastic modulus imaging: given N linearly independent displacement fields in an incompressible elastic material, determine the shear modulus. We show that for the standard case, N=1, the general solution contains two arbitrary functions which must be prescribed to make the solution unique. In practice, the data required to evaluate the necessary functions are impossible to obtain. For N=2, on the other hand, the general solution contains at most four arbitrary constants, and so very few data are required to find the unique solution. For N=4, the general solution contains only one arbitrary constant. Our results apply to both quasistatic and dynamic deformations.

We introduce an extension of the finite Toda lattice, depending on a choice of time independent parameters , which also includes the finite relativistic Toda lattice as a limit case and comes from the theory of orthogonal rational functions. Generalizing the method Moser used in the case of the finite Toda lattice, we solve this system of nonlinear differential equations for some initial data with the aid of a spectral transform. In particular we study a generalized eigenvalue problem of a pair of matrices (J,I+JD) where J is a symmetric Jacobi matrix and D = diag (α₀⁻¹,α₁⁻¹,...,α_N−1⁻¹). The inverse spectral transform is described in terms of terminating continued fractions. Finally we compute the time evolution of the spectral data.

Using the connection of this spectral transform with the theory of orthogonal rational functions we prove that the matrix J tends to a diagonal matrix containing the generalized eigenvalues and the parameters α₀,α₁,...,α_N−1, thereby establishing the explicit long-time behaviour of the matrix entries.