# 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.