November 7 - 11, 2005
Imaging sensors, hardware, and algorithms are under increasing
pressure to accommodate ever larger and higher-dimensional data sets;
ever faster capture, sampling, and processing rates; ever lower power
consumption; communication over ever more difficult channels; and
radically new sensing modalities. Fortunately, over the past few
decades, there has been an enormous increase in computational power
and data storage capacity, which provides a new angle to tackle these
challenges. We could be on the verge of moving from a "digital signal
processing" (DSP) paradigm, where analog signals (including light
fields) are sampled periodically to create their digital counterparts
for processing, to a "computational signal processing" (CSP) paradigm,
where analog signals are converted directly to any of a number of
intermediate, "condensed" representations for processing using
optimization techniques. At the foundation of CSP lie new uncertainty
principles that generalize Heisenberg's between the time and frequency
domains and the concept of compressibility. As an example of CSP, I
will overview "Compressive Imaging", an emerging field based on the
revelation that a small number of linear projections of a compressible
image contain enough information for image reconstruction and
processing. The implications of compressive imaging are promising for
many applications and enable the design of new kinds of imaging
systems and cameras. For more information, a compressive imaging
resource page is available on the web at
dsp.rice.edu/cs
3D optical elements modulate light through interaction with an entire
volume of variable refractive index (as opposed to a sequence of surfaces
used in traditional optics.) One commonly used form of 3D optics is
gradient-index (GRIN) where the modulation is base-band. Instead, we have
emphasized use of modulations on a spatial carrier (grating.) We have
demonstrated that the resulting controllable shift variance and dispersion
can be used for optical slicing, real-time optical tomography, and
hyper-spectral imaging in three spatial dimensions. The extended degrees of
freedom available in defining the optical response of 3D optics with a
carrier makes this kind of optical elements suitable for computational
imaging. We will discuss examples where over-constrained least-squares
(pseudo-inverse) and maximum likelihood (Viterbi) algorithms were used to
maximize the image information extracted from the raw images.
Image normalization refers to eliminating
image variations (such as
noise, illumination, or occlusion) that are related to
conditions of
image acquisition and are irrelevant to object
identity. Image normalization can be used as a
preprocessing stage to
assist computer
or human object perception. In this paper, a
class-based image
normalization method is
proposed. Objects in this method are represented in
the PCA basis, and
mutual information is used to identify irrelevant
principal
components. These components are then discarded to
obtain a normalized
image which is not affected by the specific conditions
of image
acquisition.
The method is demonstrated to produce visually
pleasing
results and to improve significantly the accuracy of
known recognition algorithms.
The use of mutual information is a significant
advantage over the
standard method of discarding components according to
the eigenvalues,
since eigenvalues
correspond to variance and have no direct relation to
the relevance of
components to representation.
An additional advantage of the
proposed algorithm is that many types of image
variations are handled in a unified framework.
COMP-I is a program under the DARPA MONTAGE program focusing on the
construction of thin digital imaging systems. COMP-I uses optical
prefilters to encode the impulse response of multiple aperture imaging
systems. The COMP-I program is near the completion of phase I
development and has produced both visible and IR imaging systems based
on focal plane coding and diffractive coding elements. This talk will
review the design philosophy of COMP-I and describe recent experimental
results. The talk will focus in particular on image inference strategies
from compressively sampled measurements.
Network tomography has been regarded as one of the most promising
methodologies for performance evaluation and diagnosis of the massive
and decentralized Internet. It can be used to infer unobservable network
behaviors from directly measurable metrics and does not require
cooperation between network internal elements and the end users. For
instance, the Internet users may estimate link level characteristics
such as loss and delay from end-to-end measurements, whereas the network
operators can evaluate the Internet path-level traffic intensity based
on link-level traffic measurements.
In this paper, we present a novel estimation approach for the network
tomography problem. Unlike previous methods, we do not work with the
model distribution directly, but rather we work with its
characteristic function that is the Fourier transform of the
distribution. In addition, we also obtain some identifiability
results that apply not only to specific distribution models such as
discrete distributions but also to general distributions. We focus on
network delay tomography and develop a Fourier domain inference
algorithm based on flexible mixture models of link delays. Through
extensive model simulation and simulation using real Internet trace,
we are able to demonstrate that the new algorithm is computationally
more efficient and yields more accurate estimates than previous
methods especially for a network with heterogeneous link delays.
It has been noted that many realistic networks have a power law degree distribution
and exhibit the small world phenomenon. We consider graph drawing methods that take
advantage of recent developments in the modeling of such networks.
Our main approach is to partition the edge set of a graph into ``local'' edges
and "global" edges, and to use a force-directed method that emphasizes the local edges.
We show that our drawing method works well for networks that contain underlying geometric graphs
augmented with random edges, and demonstrate the method on a few examples.
We present fast approximation algorithms for the maximum short flow problem, and for testing
whether a short flow of a certain size exists between given vertices.
Using these algorithms, we give a fast approximation algorithm for determining
local subgraphs of a given graph. The drawing algorithm we present can be applied to
general graphs, but is particularly well-suited for numerous small-world networks with power
law degree distribution.
This is a joint work with Reid Andersen and Linyuan Lu.
Many anomalous network events do not manifest themselves as abrupt,
easily-detectable changes in the volume of traffic at a single switch.
Rather, the footprint they leave is a modification of the pattern of traffic
at a number of routers in this network. Anomaly detection is then a question
of whether the current traffic pattern is sufficiently divergent from
"normal" traffic patterns. In this talk, I will describe a technique for
sequentially constructing a sparse kernel dictionary that forms a map of
network normality and illustrate how this map can be used to identify
anomalous events.
Solving the phase retrieval problem, i.e. reconstructing a compact, multidimensional function from the modulus of its Fourier transform, has applications in astronomy, x-ray diffraction, optical wave-front sensing, and other areas of physics and engineering. To solve such problems, one must have constraints on the function in order to have a chance for a unique solution. For some of these problems, the most important constraint is S, the support of the function, i.e., the set of points outside of which the function has value zero. The autocorrelation of the function can computed from the Fourier modulus, and A, the support of the autocorrelation function, is the Minkowski sum of S with -S. Therefore, in order to solve the phase retrieval problem, we first want to determine S, or at least estimate the smallest upper bound on S, from A = S - S. This paper will describe methods we have discovered so far for performing this inversion with the hope that others will point us to, or discover, additional approaches.
We have developed a broad class of coded aperture spectrometer
designs
for spectroscopy of diffuse biological and chemical sources. In contrast
to traditional designs, these spectrometers do not force a tradeoff
between resolution and throughput. As a result, they are ideal for
precision chemometric studies of weak, diffuse sources. I will discuss
the nature of the coding design and present results showing
high-precision concentration estimation of metabolites at clinical levels.
Joint work with A. Almansa, V. Caselles and B. Rouge.
We propose an algorithm to solve a problem in image restoration
which considers several different aspects of it, namely: irregular
sampling, denoising, deconvolution, and zooming. Our algorithm is
based on an extension of a previous image denoising algorithm
proposed by A. Chambolle using total variation,
combined with irregular to regular sampling algorithms proposed by
H.G. Feichtinger, K. GrĂ¶chenig, M. Rauth and T. Strohmer. Finally we
present some experimental
results and we compare them with those obtained with the algorithm
proposed by K. GrĂ¶chenig et al.
Joint with Michael Ting and Raviv Raich.
In many imaging problems a sparse reonstruction is desired. This could
be due to natural domain of the image, e.g., in molecular imaging only a
few voxels are non-zero, or a desired sparseness property, e.g.,
detection of corner reflectors in radar imaging. We present several new
methods for sparse reconstruction that account for positivity
constraints, convolution kernels, and unknown sparsity factors. For
illustration we apply these methods to reconstructing magnetic force
resonance microscopy images of compounds such as Benzene and DNA.
We propose a new method for encoding the geometry of surfaces
embedded in three-dimensional space. For a compact surface
representing the boundary of a three-dimensional solid, the distance
function is used to construct a skeletal graph that is invariant
with respect to translations, rotations, and scaling. The skeletal
graph is then equipped with weights that capture the geometry of the
surface. The information stored in the weighted graph is sufficient
for the restoration of the original surface. This proposed approach
leads to robust modeling of surfaces; independent of their scale and
position in a three-dimensional space.
We propose a new approach to classical detection problem of discrimination
of a true signal from an interferent signal. We show that the detection
performance, as quantified by the receiver operating curve (ROC), can be
substantially improved when the signal is represented by a multi-component
data set that is actively manipulated by a shaped probing pulse. In this
case, the signal sought (agent) and the interfering signal (interferent) are
visualized by vectors in a multi-dimensional detection space. Separation of
these vectors is achieved by adaptive modification of a probing laser pulse
to actively manipulate the Hamiltonian of the agent and interferent. We
demonstrate one implementation of the concept of adaptive rotation of signal
vectors to chemical agent detection by means of strong-field time-of-flight
mass-spectrometry.
Quite often in electron microscopy it is desired to segment the
reconstructed volumes of biological macromolecules. Knowledge of the 3D
structure of the molecules can be crucial for the understanding of their
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 a Gibbs prior on the underlying distribution of label images,
we show that it is possible to recover the unknown image from only a few
noisy projections.
Cheney and later Kirsch showed that the Factorization
Method of Kirsch is equivalent to Devaney's MUSIC algorithm
for the case of scattering from inhomogeneous
media. We demonstrate a similar correspondence between
the Linear Sampling Method of Colton and Kirsch
as well as the Point Source Method of Potthast
and the MUSIC Algorithm for scattering from extended perfect
conductors. We extract the most attractive aspects of each algorithm
for a robust and simple proceedure for determining the support
of extended scatterers from far field data.
Seismic imaging typically assumes that all recorded energy has
scattered only once in the subsurface. To satisfy this
assumption, attempts are made to attenuate waves which have
scattered more than once (multiples), before the image is formed.
We propose a method of estimating the image artifacts caused by
leading-order internal multiples directly in the image to reduce
the difficulties caused by inaccurately estimating the multiples.
We consider a very general inverse problem on directed graphs.
Surprisingly, this problem can actually be solved, explicitly, in a large
class of examples. I will describe the construction of these examples, as
well as the method used to produce the inversion formulas. This is joint
work with F. Alberto Grunbaum.
In this talk we examine a class of inverse problems that arise on
graphs. We provide a review of
recent developments, including design aspects for identifiability
purposes, inference issues and
applications to computer networks.
Traditional optical design typically exploits only limited
prior knowledge of the object space to be imaged (e.g., resolution, field of view, nominal range, etc.) It is possible
to include stronger object space constraints (e.g., specific
objects of interest, operational SNR, background characteristics, etc.) into the optical design and thus generate a
more photo-efficient solution. In this talk I will discuss
both passive and active feature-specific imaging systems
for this purpose.
Sensor networking is an emerging technology that promises an
unprecedented ability to monitor the physical world via a spatially
distributed network of small and inexpensive wireless sensor nodes.
The nodes can measure the physical environment with a wide variety of
sensors, including acoustic, seismic, thermal, and infrared. While the
practically unlimited range of applications of sensor networks is
quite evident, our current understanding of their design and
management is far from complete. Because sensor networks collect data
in a spatially distributed fashion, statistical inference problems in
sensor networks present a distinct new challenge. In addition to
common issues such as signal-to-noise and sampling considerations,
limited energy resources place a very high cost on the sharing of data
via wireless communications. Consequently, energy efficient methods
for processing and communicating information play a central role in
the theory and practice of sensor networks. This talk will describe
"imaging" using wireless sensor networks, and discuss how recently
proposed "compressed sensing" schemes may be very advantageous in such
systems.
Abstract is in pdf format.
With the increase in life expectancy of the general population, the incidence of Alzheimer's Diease
is growing rapidly and impacts the lives of those with the disease and their care givers,
as well as the entire medical infrastructure. Research associated with AD focuses on early diagnosis,
and effective treatment and prevention strategies using neuroimaging biomarkers
which have demonstrated high sensitivity and specificity.
Many studies use PET data to measure differences in cerebral metabolic rates for glucose
before onset of the disease in the carriers of APOE $epsilon 4$.
Researchers hope to rapidly evaluate various preventive strategies on healthy subjects
which requires refining and extending technologies for reliable
detection of small scale features indicating functional or structural change.
Appropriate computational techniques must be developed and validated.
The PET working group of the National Institute of Aging
recently published recommendations for studies on aging that utilize imaging data,
acknowledging prior limitations of PET studies, while providing guidelines and protocols for
future neuroimaging research. We present initial results of restoration and registration
techniques for quantifying functional PET images.
Complex systems are ubiquitous in mathematics, biology,
engineering, and
physics, and the past ten years have witnessed an exponential
increase in
the literature associated such systems. A shared conceptual
framework is
becoming apparent among challenges as seemingly different as the
following:
the search by mathematicians for exact high-order trigonometric
identities,
the search by engineers for stable control systems, the search by
biologists for stable protein structures, and the search by condensed
matter physicists for ground states.
Recent work has shown that separated product-sum representations
provide a
powerful and broadly applicable tool for analyzing complex
systems. Beylkin
and Mohlenkamp provide a good introduction to these
representations in their
recent preprint "Algorithms for Numerical Analysis in High
Dimensions"
(*). This talk will review some of the basic ideas of separated
product-sum
representations, and discuss how our UW Quantum System Engineering
(QSE)
Group is applying these ideas in polynomial-time simulations of
large-scale
quantum spin systems.
Our QSE Group has found that Beylkin and Mohlenkamp's methods can be
readily extended to dynamical systems by a two-fold trick: (1)
introduce
noise, and (2) convert the noise to an equivalent measurement
processes.
The second step exploits the same unitary invariance of operator-sum
representations that plays a central role in quantum computing
theory. The
resulting quantum trajectories are readily projected onto low-
dimensional
manifolds of Beylkin-Mohlenkamp type, where they can be integrated
using polynomial-time numerical algorithms.
The practical consequence is that a broad class of problems in
quantum
physics and engineering that were previously thought to be in the
(intractable) complexity class EXP can now be solved by algorithms
that are
in the (much simpler) complexity class NP. The lecture will close
with an
informal survey of physics problems that might be addressed by
these methods.
(*) http://amath.colorado.edu/activities/preprints/archive/519.pdf
Joint work with Gilbert Walter.
In Computerized Tomography (CT) an image must be recovered from data given
by the Radon transform of the image. This data is usually in the form of
sampled values of the transform. In our work a method of recovering the
image is based on the sampling properties of the prolate spheroidal
wavelets which are superior to other wavelets. It avoids integration and
allows the precomputation of certain coefficients. The approximation based
on this method
is shown to converge to the true image under mild hypotheses.
Another interesting application of wavelets is in functional Magnetic
Resonance Imaging (fMRI). To estimate the total intensity of the image
over the region of interest, a new method based on multi-dimensional
prolate spheroidal wave functions (PSWFs) was proposed in a series of
papers beginning with the work of Shepp and Zhang. We try to determine how
good the proposed approximations are and how they can be improved.
In optical tomography, conventionally the diffusion approximation to the
radiative transport equation (RTE) with a constant refractive index is
used to image highly scattering or turbid media. Recently we derived the
relevant RTE and its spherical harmonics approximation with a spatially
varying refractive index. We found that the model with spatially varying
refractive index for photon transport is substantially different than the
spatially constant model. We formulate the optical tomography inverse
problem based on the diffusion approximation to image a highly scattering
medium with a spatially varying refractive index. We have simulated the
forward and the inverse problem using the finite element method and have
reconstructed the spatially varying refractive index parameter in our
model for the inverse problem. Our simulations indicate that the
refractive index based optical tomography shows promise for the
reconstruction of the refractive index parameter.
Raman spectral imaging has been widely used for extracting chemical
information from biological specimens. One of the challenging
problems is to cluster the chemical groups from the vast amount of
hyperdimensional spectral imaging data so that functionally similar
groups can be identified. Furthermore, the poor signal to noise
ratio makes the problem more difficult. In this work, we introduce a
novel approach that combines a differential wavelet based noise
removal approach with a fuzzy clustering algorithm for the pixel-wise
classification of Raman image. The preprocessing of the spectral data
is facilitated by decomposing them in a special family of differential
wavelet domain, where the discrimination of true spectral features
and noises can be easily performed using a multi-scale pointwise product
criterion. The performance of the proposed approach is evaluated by
the improvement over the subsequent clustering of a dentin/adhesive
interface specimen under different noise levels. In comparison with
conventional denoising algorithms, the proposed approach demonstrates
the super performance. This is a joint work with Wang Yong and
Paulette Spencer of the School of Dentistry at the University of
Missouri-Kansas City.
Audio Video Recording only.
A mathematical framework based on band-limited functions has been
developed for modeling and analyzing images in two dimensions. The
foundation of this framework is a class of basis functions that are
locally compact in both frequency and image domains. Images
represented in such bases are visually smooth with neither ringing nor
blocky artifacts which frequently company processed images, and at the
same time preserve the original sharpness. Preliminary results in
image denoising will be presented.
We generalize the iterative regularization method, recently devoloped by
the authors, to a time-continuous inverse scale-space formulation.
Convergence and restoration properties, including a precise discrepancy
principle, still hold. The inverse flow is computed directly for
one-dimensional signals, yielding very high quality restorations. For
arbitrary dimensions, we introduce a simple relaxation technique using
two evolution equations, which allows for a fast and effective
implementation. This is a joint work with Martin Burger, Stanley Osher and
Guy Gilboa.
Elastography is an innovative new medical imaging technique that provides high
resolution/contrast images of elastic stiffness identifying abnormalities not seen
by standard ultrasound. Since the elastic stiffness increases signicantly (up
to 10 times) in cancerous tissue, elastography shows tumor as a bright spot in
the reconstructed image. Our data is the time dependent (10,000 frames/sec)
interior displacements (0.3mm grid spacing) initiated by a short-time pulse.
While standard inverse problems utilizing only boundary data suffer from the
inherent ill-posedness, our inverse problem for elastography doesn't because it
utilizes interior information.
For the isotropic tissue model, a series of uniqueness results for our inverse
problem are presented, and a fast stable algorithm to reconstruct the shear
stiffness based on arrival time is explained. For the anisotropic tissue model,
we assume an incompressible transversely isotropic model. It is important to
consider anisotropic tissue models, since some tumors exhibit anisotropy and
the structure of fiber orientation has a strong correlation with the malignancy
of tumor. In this model, two shear stiffness and the fiber orientation are recon-
structed by four measurements of SH-polarized shear waves, which are initiated
by line sources in the interior of human body based on supersonic remote pal-
pation interior excitation.