Campuses:

Reception and Poster Session

Monday, October 17, 2005 - 4:30pm - 6:00pm
Lind 400
  • A New Reconstruction Algorithm for Radon Data
    Yuan Xu (University of Oregon)
    A new reconstruction algorithm for Radon data is introduced. We call the
    new algorithm OPED as it is based on Orthogonal Polynomial Expansion on the
    Disk. OPED is fundamentally different from the filtered back projection (FBP)
    method. It allows one to use fan geometry directly without the additional
    procedures such as interpolation or rebinning. It reconstructs high degree
    polynomials exactly and converges unifomly for smooth functions without the
    assumption that functions are band-limited. Our initial test indicates that
    the algorithm is stable, provides high resolution images, and has a small
    global error. Working with the geometry specified by the algorithm and
    a new mask, OPED could also lead to a reconstruction method working with
    reduced x-ray dose.
  • A Newton-Type Method for 3D Inverse Obstacle Scattering Problems
    Thorsten Hohage (Georg-August-Universität zu Göttingen)
    We consider the inverse problem to reconstruct the shape of an obstacle from measurements of scattered fields. The forward
    problem is solved by a wavelet boundary element method. For the inverse problem we use a preconditioned Newton method. Particular
    emphasis is put on the reconstruction of non star-shaped obstacles.
  • Wave Propagation in Optical Waveguides with Imperfections
    Giulio Ciraolo (Università di Firenze)
    The problem of electromagnetic wave propagation in a 2-D infinite optical
    waveguide will be presented.
    We give a description on how to construct a solution to the electromagnetic wave
    propagation problem in a 2-D and 3-D rectilinear optical waveguide. Numerical
    simulations will also be shown.
    Furthermore, in the 2-D case, we will present a mathematical framework which
    allows us to study waveguides with imperfections. In this case, some numerical
    result concerning the far field of the solution and the coupling between guided
    modes will be shown.
  • Marching Schemes for Inverse Acoustic Scattering Problems
    Frank Wuebbeling (Westfälische Wilhelms-Universität Münster)
    The solution of time-harmonic inverse scattering problems usually involves solving the Helmholtz equation many times. On the other hand,
    these boundary value problems with radiation condition at infinity are notoriously hard to solve. In the context of inverse scattering,
    however, boundary value problems can be rewritten as initial value problems.

    We develop an efficient marching scheme for computing a filtered version of the solution of the initial value problem for the Helmholtz
    equation in 2D and 3D. Stability and error estimates are developped, a numerical example is given.
  • Sparsity-Driven Feature-Enhanced Imaging
    Mujdat Cetin (Massachusetts Institute of Technology)
    same abstract as the regular talk
  • Uniqueness, Stability and Numerical Methods for some Inverse and

    Ill-posed Cauchy Problems

    Michael Klibanov (University of North Carolina)
    Some new results concerning global uniqueness theorems and stability
    estimates for coefficient inverse problems will be presented. In
    addition, the presentation will cover some new and previous results
    about the stability of the Cauchy problem for hyperbolic equations with
    the data at the lateral surface. This problem is almost equivalent with
    the inverse problem of determining initial conditions in hyperbolic
    equations. Therefore, stability estimates for this Cauchy problem
    actually imply refocusing of time reversed wave fields. Our recent
    numerical studies confirming this statement will be presented. In
    addition, a globally convergent algorithm for a class of coefficient
    inverse problems will be discussed. The main tool of all these studies
    is the method of Carleman estimates.
  • Nonlinear Integral Equations in Inverse Obstacle Scattering
    Olha Ivanyshyn (Georg-August-Universität zu Göttingen)
    We present a novel solution method for
    inverse obstacle scattering problems for time-harmonic waves
    based on a pair of nonlinear and
    ill-posed integral equations for the unknown boundary
    that arises from the reciprocity gap principle.
    This integral equations can
    be solved by linearization, i.e., by
    regularized Newton iterations. We present
    a mathematical foundation of the method and illustrate
    its feasibility by numerical examples.
  • Some Recent Developments of the Analytical Reconstruction Techniques for Inverse Scattering and Inverse Boundary Value Problems in Multidimension
    Roman Novikov (Université de Nantes)
    Some recent developments of the analytical reconstruction
    techniques for inverse scattering and inverse boundary value problems
    in multidemension
    Abstract: This poster presents (the abstracts and references of)
    the recent works
    1. R.G.Novikov, The d-bar approach to approximate inverse scattering
    at fixed energy in three dimensions, International Mathematics
    Research Papers 2005:6 (2005) 287-349;
    2. R.G.Novikov, Formulae and equations for finding scattering data
    from the Dirichlet-to-Neumann map with nonzero background potential,
    Inverse Problems 21 (2005) 257-270.

    These works give some new developments of the analytical reconstruction
    techiques for inverse scattering and inverse boundary value problems
    in multidimension.
  • Towards Effective Seismic Imaging in Anisotropic Elastic Media
    Murthy Guddati (North Carolina State University)
    [joint work with A.H. Heidari]

    A critical ingredient in high-frequency imaging is the
    migration
    operator that back-propagates the surface response to the
    hidden
    reflectors. Migration is often performed using one-way wave
    equations
    (OWWEs) that allow wave propagation in a preferred direction
    while
    suppressing the propagation in the opposite direction. OWWEs
    are
    typically obtained by approximating the factorized full-wave
    equation;
    this process is well-developed for the acoustic wave equation,
    but not
    for elastic wave equations, especially when the material is
    anisotropic.
    Furthermore, existing elastic OWWEs are computationally
    expensive. For
    these reasons, in spite of the existence of strongly coupled
    elastic
    waves, seismic migration is performed routinely using acoustic
    OWWEs,
    naturally resulting in significant errors in the image.

    With the ultimate goal of developing accurate and efficient
    imaging
    algorithms for anisotropic elastic media, we develop new
    approximations
    of elastic OWWEs. Named the arbitrarily wide-angle wave
    equations
    (AWWEs), these approximations appear to be effective for
    isotropic as
    well as anisotropic media. The implementation of AWWE-migration
    in
    isotropic (heterogeneous) elastic media is complete, while
    further work
    remains to be done to incorporate the effects of anisotropy.
    This poster
    outlines (a) the basic idea behind AWWEs, (b) the
    implementation of
    AWWE-migration along with some results, and (c) future
    challenges
    related to using AWWEs for imaging in anisotropic elastic
    media.
  • VISUAL-D: Backhoe Challenge Problem
    D. Gregory Arnold (US Air Force Research Laboratory)
    AFRL/SNA has several data sets and challenge problems for imaging. Specifically, I will highlight the data that was recently made
    available at IMA. This includes a radar Backhoe Data Dome, adaptive learning of MSTAR data, a 3-D Ladar ATR Challenge problem,
    Video EO and IR data for feature-aided tracking, and Thru-wall data. All of these data sets have been publicly released and are
    also available from https://www.sdms.afrl.af.mil/main.php.
  • Unique Determination of the Travel Time from Dynamic Boundary Measurements in Anisotropic Elastic Media
    Anna Mazzucato (The Pennsylvania State University)Lizabeth Rachele (Rensselaer Polytechnic Institute)
    We microlocally decouple the system of
    equations for anisotropic elastodynamics (in 3 dimensions) following a
    result of M. Taylor. We then show that the dynamic Dirichlet-to-Neumann
    map uniquely determines the travel time through a bounded elastic body for
    any wave mode that has disjoint light cone. We apply this result
    to cases of transversely isotropic media with rays that are geodesics with
    respect to Riemannian metrics, and conclude that certain material
    parameters are uniquely determined up to diffeomorphisms that fix the
    boundary. We have shown that material parameters of general
    anisotropic elastic media may be uniquely determined by the
    Dirichlet-to-Neumann map only up to pullback by
    diffeomorphisms fixing the boundary.
  • Towards 3D Least-squares Inversion of Prestack Seismic Data
    Patrick Lailly (Institut Français du Pétrole)
    (work
    in collaboration with Y. Pion (IFP), Jerome Le Rousseau and Thierry Gallouet
    (Univ. Aix-Marseille 1)

    Migration is the standard tool used in seismic imaging. It consists in
    applying to the data the adjoint of the linearized forward map. 3D
    least-squares inversion would consist in solving a huge linear system, a
    tremendous task at first glance. However physical intuition and numerical
    evidence (i. e. visualization of the Hessian) indicate that solving this
    linear system should not be that difficult. By doing so, we expect to
    improve the spatial resolution and to remove the acquisition footprint.
  • Fast, High-Order Integral Equation Methods for Scattering by

    Inhomogeneous Media

    Mckay Hyde (Rice University)
    Integral equation methods for the time-harmonic scattering problem
    are attractive since the radiation condition at infinity is
    automatically satisfied (no absorbing boundary condition is
    required), only the scattering obstacle itself needs to be
    discretized, and the integral operator is compact, leading to better
    conditioned linear systems than for differential operators. However,
    there has been limited success in developing integral equation
    methods which are both efficient and high-order accurate.

    We will present recent work on integral equation methods that are
    both efficient (O(N log N) complexity) and high-order accurate in
    computing the time-harmonic scattering by inhomogeneous media. The
    efficiency of our methods relies on the use of fast Fourier
    transforms (FFTs) while the high-order accuracy results from
    systematic use of partitions of unity, regularizing changes of
    variables, and Fourier smoothing of the refractive index.
  • Image Preconditioning for a SAR Image Reconstruction Algorithm for Multipath Scattering
    David Garren (Science Applications International Corporation)
    Recent analysis has resulted in an innovative technique for forming synthetic aperture radar (SAR) images without the multipath ghost artifacts that arise in traditional methods. This technique separates direct-scatter echoes in an image from echoes that are the result of multipath, and then maps each set of reflections to a metrically correct image space. Current processing schemes place the multipath echoes at incorrect (i.e., ghost) locations due to fundamental assumptions implicit in conventional array processing. Two desired results are achieved by use of this Image Reconstruction Algorithm for Multipath Scattering (IRAMS). First, the intensities of the ghost returns are reduced in the primary image space, thereby improving the relationship between the image pattern and the physical distribution of the scatterers. Second, a higher dimensional image space that enhances the intensities of the multipath echoes is created which offers the potential of dramatically improving target detection and identification capabilities. This paper develops techniques in order to precondition the input images at each level and each offset in the IRAMS architecture in order to reduce multipath false alarms.
  • A Compactly Supported Approximate Wavefield Extrapolator for Seismic Imaging
    Gary Margrave (University of Calgary)
    Seismic imaging in highly heterogeneous media requires an adaptive, robust, and efficient wavefield extrapolator. The homgeneous
    medium wavefield extrapolator has no spatial adaptivity but the locally homogeneous approximate extrapolator (LHA) is a highly
    accurate Fourier integral operator that adapts rapidly in space. Efficient application of either wavefield extrapolator is
    complicated by the fact that they have impulse responses that are not compactly supported, though they decay rapidly. Simple
    locaization methods, such as windowing, result in compactly supported approximations that are unstable in a recursive marching
    scheme. I present an analysis of this instability effect and a localization scheme that can design compactly supported approximate
    extrapolators that are sufficiently stable for hundreds of marching steps. I illustrate the method with seismic images from the
    Marmousi synthetic dataset.
  • Image Reconstruction in Thermoacoustic Tomography
    Gaik Ambartsoumian (Texas A & M University)
    Thermoacoustic tomography (TCT or TAT) is a new and promising method
    of medical imaging. It is based on a so-called hybrid imaging technique,
    where the input and output signals have different physical nature. In TCT
    a radiofrequency (RF) electromagnetic pulse is sent through the biological
    object triggering an acoustic wave measured on the edge of that object.
    The obtained data is then used to recover the RF absorption function.

    The poster addresses several problems of image reconstruction in
    thermoacoustic tomography. The presented results include injectivity
    properties of the related spherical Radon transform, its range description,
    reconstruction formulas and their implementation as well as some other
    results.