Thursday, October 20, 2005 - 4:30pm - 6:00pm
- Regularization and Prior Error Distributions in Ill-posed
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
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
along with Air Force Research Laboratory (AFRL) has conducted
preliminary experimentations on through-the-wall imaging and
real data on different settings behind the wall using a
RF instrumentation suite. The full-polarization, 2D aperture
are taken using an Agilent network analyzer, Model ENA 5071B,
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
Three different arrangements of the room's contents are
considered: empty scene,
calibration scene, and populated scene. The empty scene allows
of the noise/clutter background and supports coherent
subtraction with the other two scenes. The calibration scene
isolated reflectors that may be used to determine a
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
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
- 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
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
- 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
- Direct Reconstruction-Segmentation, as Motivated by Electron
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
- 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.