# Poster session

Thursday, October 20, 2005 - 4:30pm - 6:00pm

Lind 400

**Regularization and Prior Error Distributions in Ill-posed**

Problems

Jodi Mead (Boise State University)

We will examine the validity of parameter estimates in ill-posed

problems when errors in data and initial parameter estimates are

from normal and non-normal distributions. Given appropriate

initial parameter estimates and the data error covariance matrix, the

covariance matrix for errors in initial parameter estimates can be

recovered and highly accurate parameter estimates can be found. This

approach allows the regularization to be varied with each parameter.**Ultrasound Breast Tomography with Full-Wave Non-Linear Inverse**

Scattering

Michael Andre (University of California, San Diego)

The fundamentals of medical ultrasound imaging have not changed since its

inception 60 or more years ago. 180-degree pulse-echo backscatter is used for

image formation without accounting for refraction, diffraction, multiple

scattering, etc. Various forms of ultrasound computed tomography that

incorporate the transmitted wave component have been proposed and investigated

for many years with mixed success. These methods apply approximations of

inverse scattering tomography: time-of-flight, Born, Rytov, Diffraction

Tomography, etc. Techniscan Medical Systems (Salt Lake City, Utah) and the

University of California, San Diego are beginning pre-clinical evaluation of a

new system for breast imaging that applies novel inverse scattering methods to

provide a unique method for calculating ultrasound characteristics of speed and

attenuation of sound traveling through human tissue. We have developed an

efficient inversion method for the coefficients of the partial differential

equation that governs wave propagation in the human breast. The procedure is

based on nonlinear minimization, fast computation of the forward problem and

analytic computational formulas for actions of the Jacobian of the forward

operator and its Hermitian adjoint. The goal of the development is to provide

quantitative, high-resolution two- and three-dimensional ultrasonic imaging

combined with unique information about tissue properties at sub millimeter

resolution in an effort to improve diagnosis of breast cancer. Details of the

imaging system design and the inversion method will be summarized. Sample

images from human subjects and preliminary results in 26 patients with known

breast masses will be presented.**Imaging of Physiological Properties of Human Skin from Spectral Reflectance Data**

Jakob Stamnes (University of Bergen)

Joint with K.P. Nielsen, M. Biryulina, G.Ryzhikov, K.

Stamnes, and L. Zhao.

We present a new method, based on inverse radiative transfer

modelling, for retrieving physiological parameters of human skin

tissue from multi-spectral reflectance data. Whereas previous

attempts of such retrievals have been based either on empirical

formulas or simplified, inaccurate forward models, such as the

Kubelka-Munk theory, our forward model is based on the

discrete-ordinate solution of the radiative transfer equation, which

is both fast and accurate. Examples are given of retrievals based on

simulated reflectance data or in-vivo measurements.**Convergence of Approximations of Solutions to First-order Pseudodifferential Wave Equations with Products of Fourier Integral Operators**

Jerome Le Rousseau (Université d'Aix-Marseille I (Université de Provence))

An approximation of the solution to a hyperbolic equation with

a damping term is introduced. It is built as the composition of Fourier

integral operators (FIO). We prove the convergence of this

approximation in the sense of Sobolev norms as well as for the

wavefront set of the solution. We apply the introduced method to

numerically image seismic data.**Imaging Cardiac Activity by the D-bar Method for**

Electrical Impedance Tomography

Jennifer Mueller (Colorado State University)

Electrical Impedance Tomography (EIT) is an imaging technique that uses the

propagation of electromagnetic waves through a medium to form an image. In

medical EIT, current is applied through electrodes on the surface of the body,

the resulting voltages are measured on the electrodes, and the inverse

conductivity problem is solved numerically to reconstruct the conductivity

distribution in the interior. Here results are shown from EIT data taken on

electrodes placed around the circumference of a human chest to reconstruct a 2-D

cross-section of the torso. The images show changes in conductivity during a

cardiac cycle made from the D-bar reconstruction algorithm based on the 1996

uniqueness proof of A. Nachman [Ann.Math. 143].**Ultrawideband Microwave Breast Cancer Detection: Beamforming for 3-D MRI-derived Numerical Phantoms**

Shakti Davis (University of Wisconsin, Madison)

Microwave imaging has the potential to be a highly sensitive modality

for breast cancer detection due to the dielectric-properties

contrast that exists between malignant and normal breast tissue at

microwave frequencies. One microwave imaging approach is to transmit

ultrawideband (UWB) microwave pulses into the breast, record the

scattered fields, and use radar methods such as beamforming to detect

and localize significant scatterers such as tumors. We previously

proposed a beamforming technique and demonstrated its accuracy and

robustness for tumor detection using 2-D MRI-derived numerical breast

phantoms (Davis, et. al, JEMWA, 17(2):357-381, 2003) and simple 3-D

physical phantoms (Li, et. al, IEEE T-MTT, 52(8):1856-1865, 2004). In

this poster we extend our investigation to 3-D MRI-derived numerical

breast phantoms. These anatomically realistic breast phantoms

represent a prone patient with an antenna array surrounding the

breast. Small (properties in a region to represent a specified malignant-to-normal

tissue contrast. We solve for backscattered fields at each antenna

position using the FDTD-method and construct a 3-D image of scattered

energy in the breast using our beamforming technique. The resulting

images exhibit localized high-energy peaks within a few mm of the

true tumor locations as expected. This work represents our first

successful demonstration of detecting and localizing very small

tumors in 3-D MRI-derived numerical breast models.**Iterative Solver for the Wave Equation in the Frequency Domain**

Rene-Edouard Plessix (The Shell Group)

Joint work with Wim Mulder.

To retrieve the long and short spatial frequencies of the velocity model from seismic data,

several authors have proposed to work in the frequency-domain. The data are inverted per

frequency going from the low to the high. This approach has been used for long

offset data in two dimensional space. It relies on the solution of the wave equation in the

frequency domain (Helmholtz equation). Whereas in two dimensional space, a direct solver of the frequency-domain

wave equation provides an efficient method, in three dimensional space, this approach

is not feasible because the linear system becomes too large. This difficulty may be

overcome with an iterative solver for the Helmholtz equation.

During his Ph. D work, Y. Erlangga has studied an iterative approach based on a

preconditioned bicgstab (conjugate-gradient type) method. The efficiency of the method

depends on the preconditioner. It was proposed to use a damped wave equation

as a preconditioner and to approximate the inverse of the damped equation with a multigrid

method. Strong damping is required for the preconditioner, otherwise the

multigrid method does not convergence. Two-dimensional examples show that this approach is robust and that the number of iterations

depends linearly on the frequency when the number of grid points

per wavelength is kept constant. Thus, this approach provides a sub-optimal solution.

In the poster, several numerical examples will be

presented to assess the efficiency of the iterative approach.

Its relevance for migration in two and three dimensions and for

inversion algorithms will also be discussed.**Wideband Through-The-Wall Radar Imaging Experimentations**

Uttam Majumder (US Air Force Research Laboratory)

The Center for Advanced Communications (CAC) at Villanova

University

along with Air Force Research Laboratory (AFRL) has conducted

several

preliminary experimentations on through-the-wall imaging and

collected

real data on different settings behind the wall using a

newly-integrated

RF instrumentation suite. The full-polarization, 2D aperture

data measurements

are taken using an Agilent network analyzer, Model ENA 5071B,

implementing a

step frequency waveform over a 2-3 GHz frequency range. The

imaging room is

a typical computer lab that has been lined with radar absorbing

material.

Three different arrangements of the room's contents are

considered: empty scene,

calibration scene, and populated scene. The empty scene allows

measurement

of the noise/clutter background and supports coherent

subtraction with the other two scenes. The calibration scene

contains

isolated reflectors that may be used to determine a

fully-polarimetric

radiometric calibration solution for the experimental system.

The populated scene contains a number of common objects such as

a phone, computer, tables, chair and filing cabinet and a jug of

saline solution.

Data was collected each scene with and without a wall. The wall

is composed of plywood and gypsum board on a wood frame.

The antennas are mounted on a 2D scanner that moves the

antennas along and adjacent to the wall and is

controlled by the network analyzer.

Two additional antennas are fixed to the scanner frame and act

as bistatic receivers.**Progress in Quantitative Biomechanical Imaging**

Paul Barbone (Boston University)

Joint work with Michael S. Richards, Nachiket H. Gokhale,

Carlos Rivas Aroni, Ricardo Leiderman, Jeffrey C. Bamber,

and Assad A. Oberai.

It is widely recognized that tissue pathologies often change biomechanical properties.

For instance, neoplastic tissue is typically highly vascularized, contains abnormal

concentrations of extracellular proteins (i.e. collagen, proteoglycans) and has a high

interstitial fluid pressure compared to most normal tissues. These differences in tissue

microstructure effectively change a tissues response to mechanical stimuli. Our work

focuses on noninvasively measuring and thereby imaging in vivo distributions of the

biomechanical properties of soft tissues. The intended short term application of our

work is the detection and diagnosis of breast cancer and other soft tissue pathologies.

Our efforts include the development and computational implementation of

mathematical models to describe soft tissue behavior, developing novel ultrasound techniques

to accurately measure vector displacements of tissue deformation, the analysis of

inverse problems associated with quantitative inference of material properties from

measured displacements, and development of algorithms to solve those inverse problems.

We present a combined ultrasound and image registration technique to quantitatively

measure tissue response to mechanical manipulation. We further present several different mathematical models describing tissue responses

for different experimental stimuli.

Some of these models are motivated by microstructural considerations. Where possible,

these model parameters are compared to values determined by independent mechanical

testing.**Velocity Analysis in the Presence of Uncertainty**

Eric Dussaud (Rice University)

Velocity Analysis resolves relatively long scales of earth

structure, typically wavelengths larger than 500m. Migration produces

images with length scales (wavelengths) on the order of 10's of m. In

between these two scale regimes lies another, corresponding roughly to

structures between 60 to 300m in extent, in which the resolution of

velocity analysis is uncertain and the energy of images is small to

non-existent. This work aims at assessing the impact on velocity analysis

of uncertainty at these intermediate length scales, using ideas on time

reversal and imaging in randomly inhomogeneous media developed by

Papanicolaou and colleagues, in combination with velocity estimation

methods of differential semblance type.**Nonlinear Inverse Scattering and Velocity Analysis**

William Symes (Rice University)

Migration velocity analysis (MVA) can be viewed as a solution

method for the linearized (Born) inverse scattering problem, in its

reflection seismic incarnation. MVA is limited by the single scattering

assumption - for example, it misinterprets multiply scattered waves - but

it is capable of making large changes in the model, and moving estimated

locations of scatterers by many wavelengths. The salient features of MVA

is its use of an extended (nonphysical) scattering model. Nonlinear least

squares inversion (NLS), on the other hand, incorporates whatever

details of wave physics are built into its underlying modeling engine.

However success appears to require that the initial estimate of wave

velocity (in an iterative solution method) be accurate to within a

wavelength, i.e. have kinematic properties very close to that of the

optimal model.

This poster will describe a nonlinear extended scattering model and a

related optimization formulation of inverse scattering. I will present

the results of some preliminary numerical explorations which suggest that

this approach may combine the global nature of MVA with the capacity of

NLS to accomodate nonlinear wave phenomena.**Signal Restoration Through Deconvolution Applied to Deep Mantle Seismic Probes**

Wolfgang Stefan (Arizona State University)

We present a method of signal restoration to improve

the signal to noise ratio, sharpen seismic arrival onset, and act as

an empirical source deconvolution of specific seismic arrivals. The

method is used on the shear wave time window containing SKS and

S, whereby using a Gaussian PSF produces more impulsive, narrower,

signals in the wave train. The resulting restored time series

facilitates more accurate and objective relative travel time

estimation of the individual seismic arrivals. Clean and sharp

reconstructions are obtained with real data, even for signals with

relatively high noise content. Reconstructed signals are simpler,

more impulsive, and narrower, which allows highlighting of some details

of arrivals that are not readily apparent in raw waveforms.**Local Tikhonov Regularization in n Dimensions**

Tom Scofield (Calvin College)

Many ill-posed linear integral equations are solved using standard

Tikhonov regularization. When solutions have edges, as is usually the

case in the image deblurring problem, this procedure generally carries

with it a choice between capturing the near-discontinuities found at edges

at the expense of introducing oscillations in regions that should be

smooth, or preserving smooth regions but oversmoothing edges. More

recently, local Tikhonov regularization methods have been introduced,

attempting to make this choice a local rather than global one. We prove

the convergence of such methods in R^n for general n. We also carry out

a discrete numerical implementation of such methods and provide examples

in 1 and 2 dimensions of results using both these methods and standard

Tikhonov regularization.**Seismic Velocity Analysis: In Time or Depth Domain?**

Herve Chauris (Mines-ParisTech)

jointly with Gilles Lambare (Ecole des Mines de Paris)

Seismic velocity analysis is a crucial step needed to obtain consistent

images of the subsurface. Several new methods appeared in the last 10 years, among them Slope Tomography and Differential Semblance

Optimization. We want to discuss here the link between these a priori different methods.

Slope Tomography is formulated in the prestack unmigrated time domain

and uses not only time information picked on seismic gathers,

but also associated slopes that better constrain the inversion scheme.

On the other side, Differential Semblance Optimization is formulated

in the depth migrated domain where adjacent images are compared

to obtain a final consistent image of the subsurface.

We analyse these two types of methods to show that they are in fact

equivalent from a theoretical point of view despite the different

formulation.**Direct Reconstruction-Segmentation, as Motivated by Electron**

Microscopy

Hstau Liao (University of Minnesota, Twin Cities)

Quite often in electron microscopy it is desired to segment the

reconstructed volumes of biological macromolecules, whose 3D structural

inference is crucial for the understanding of biological functions. We

propose approaches that directly produce a label (segmented) image from

the tomograms (projections).

Knowing that there are only a finitely many possible labels and by

postulating Gibbs priors on the underlying distribution of label images,

it is possible to recover the unknown image from only a few noisy

projections.**Problems in Sub-salt Imaging due to Layered-Earth Assumptions**

Scott Morton (Amerada Hess Corporation)

The standard approach to seismic imaging is rife with limitations due

to the assumption that the earth is approximately a layered medium.

Unfortunately much of the current petroleum exploration in the Gulf of

Mexico is around or beneath salt bodies which have complex 3-D shapes.

We illustrate several problems attributable to the layered-earth approach

in the standard model building process, state-of-the-art imaging algorithms

and available data interpretation tools used in sub-salt imaging.**Adjoint Method in Time Domain Ultrasound Tomography**

Frank Natterer (Westfälische Wilhelms-Universität Münster)

We model ultrasound tomography by the wave equation. Adjoint methods can

be used for the inversion. Unfortunately, due to the large number of

sources, adjoint methods are very time consuming. By preprocessing of the

data (wavefront synthesizing, plane wave stacking), adjoint methods can be

sped up by orders of magnitude. We analyse the preprocessed data in

Fourier domain. We present numerical results for the Salt Lake City breast

phantom and for the Marmousi data.**Texture Discrimination, Nonlinear Filtering, and Segmentation in Mammography**

Walter Richardson Jr. (University of Texas)

There are two primary signs used by the radiologist to

detect lesions. The first is mass: a benign neoplasm is smoothly

marginated whereas a malignancy is characterized by an indistinct border

which becomes more spiculated with time.

The second sign is microcalcification.

An essential ingredient of these indicators is

texture, used by the radiologist in many subtle ways to discriminate

between normal and cancerous tissue.

The irregular boundaries of suspect lesions suggest that they

could be identified by their local fractal signature.

Any real image is corrupted by some noise and it is necessary

to prefilter the data. Results are presented for two

edge-enhancing filters: the Weighted Majority - Minimum Range

filter and the mean-curvature dependent PDE filter of

Morel. Once the image has been filtered/transformed, the

Mumford-Shah approach is used for segmentation.**Exponential Radon Transform Inversion Based on Harmonic Analysis of the Euclidean Motion Group**

Can Yarman (Rensselaer Polytechnic Institute)

This paper presents a new method for the exponential Radon transform inversion based on harmonic analysis of the Euclidean motion

group (M(2)). The exponential Radon transform is modified to be formulated as a convolution over M(2). The convolution

representation leads to a block diagonalization of the modified exponential Radon transform in the Euclidean motion group Fourier

domain, which provides a deconvolution type inversion for the exponential Radon transform. Numerical examples are presented to show

the viability of the proposed method.