Ryuichi Ashino, Osaka Kyoiku University
Image separation by wavelet analysis
Blind image separation is the separation of a set of images,
called source images, from a set of mixed images, called observed
images, without information about the source images nor the mixing
process. Besides methods based on independent component analysis,
several methods based on time-frequency analysis have been proposed.
In this talk, a new method using wavelet analysis is proposed.
Allan Greenleaf, University of Rochester
Basic Microlocal Analysis
Microlocal analysis allows one to precisely describe the singularities
of functions and distributions (generalized functions), and analyze
how operators transform these singularities. The locations of
singularities are expressed in terms of the cotangent bundle,
which includes both spatial location and momentum (phase space)
direction. Since many of the material parameters one is interested
in medicine have discontinuities across interfaces, microlocal
analysis lends itself to applications in medical imaging. This
two-lecture mini-course will present some of the basics of microlocal
analysis,
including a discussion of elliptic partial differential operators,
their fundamental solutions and hypoellipticity; the notions of
support, singular support and wave-front set; the calculus of
pseudodifferential operators;and construction of parametrices
for elliptic pseudodifferential operators.
Alexander Katsevich, University of Central Florida
Recent advances in tomographic image reconstruction from interior
x-ray data
Using the Gelfand-Graev formula, the interior problem of tomography
reduces to invertion of the finite Hilbert transform (FHT) from
incomplete data. In this talk we study several aspects of inverting
the FHT when the data are incompelte. Using the Cauchy transform
and an approach based on the Riemann-Hilbert problem we derive
a differential operator that commutes with the FHT. Our second
result is the characterization of the null-space of the FHT in
the case of incomplete data. Also we derive the asymptotics of
the singular values of the FHT in three different cases of incomplete
data.
Michael Lamoureux, University of Calgary
Gabor methods for imaging
The Gabor transform of a signal creates a time-frequency representation
of physical data which can be directly manipulated through multiplication
by symbols, analogous to the action of pseudodifferential operation
and their representation through symbols in the time-frequency
domain.
This two-lecture minicourse describes the basics of Gabor transforms
and Gabor multipliers, windowing considerations, representation
of linear operators including differential operators, and the
use of Gabor multipliers in nonstationary filtering, deconvolution,
and numerical wave propagation. Some particular imaging applications
are described.
Cheng Liu, Kinectrics Inc.
An Adaptive S Transform with Applications in Studying Brain Functions
Discovering how the brain functions has been proved valuable
for understanding the brain's behavioral control as well as guiding
treatment of mental diseases. In response to stimuli, the brain
generates a mix of brain waves that are dynamic and frequency-specified.
Thus, time-frequency analysis has been widely used in analyzing
brain signals. However, due to huge variation of the characteristics
of brain signals, analysis measures providing the signal-invariant
resolution cannot well reveal dynamic structure of various brain
signals.
We introduce an adaptive S transform (AST), a new multi-resolution
time-frequency representation whose resolution is adaptively adjusted
to its analyzed signal. The proposed representation is built on
the S transform with additional parameters to control its resolution.
Given any specific signal, we implement a numerical procedure
that automatically determines optimal parameters so that the resulting
representation has the signal energy highly concentrated at the
involved frequencies and time duration. It hence provides a time-frequency
analysis tool offering good resolution to describe behaviors of
various signals. We then use the AST to derive a number of measures
for analyzing brain time series recorded by electroencephalography
and magnetoencephalography. These measures include the AST-based
power spectrum for revealing the characteristics of functional
activity at a single brain area, and the AST-based coherence and
phase-locking statistic for investigating the functional connectivity
between multiple brain areas. Numerical simulations are presented
to demonstrate performances of the AST and the corresponding brain
time series measures. Finally, we apply the proposed AST-based
analysis tools to investigate functional activity of motor cortices
when subjects perform the multi-source interference task, a behavioral
experiment involving tasks at multiple levels of difficulty.
Abdol-Reza Mansouri, Queen's University
Geometric Approaches in Image Diffusion
Image diffusion partial differential equations have been applied
to medical and non-medical images in applications such as denoising,
sharpening, and interpolating missing data, with great success.
In this talk, we will review two recent geometrical approaches
-- Beltrami and Hypoelliptic -- that have been proposed for deriving
image diffusion equations, and we will present theoretical and
experimental results on a class of geometrically inspired diffusion
equations that we have recently proposed. The diffusion equations
we obtain are derived by changing the Riemannian metric on the
space of images from L^2 to Sobolev, and lead to flows which could
not be obtained under the standard L^2 metric (Joint work with
J. Calder (Michigan) and A. Yezzi (Georgia Tech)).
Ross Mitchell, The Mayo Clinic - Arizona
Microlocal Analysis of Medical Images: Applications in Cancer
Imagenomics
Our expanding knowledge of the genetic basis and molecular mechanisms
of cancer is beginning to revolutionize the practice of oncology.
In fact, personalized medicine, using molecular biomarkers to
classify tumors and direct treatment decisions profiles, is becoming
the new standard of care. Unfortunately, genomic testing is invasive,
costly, and time consuming. In addition, different regions of
the same tumor, or the primary and metastatic tumors, can have
widely variable genetic signatures. Therefore, a single biopsy
in only one area of the tumor may provide an incomplete assessment.
Global assessment of tumors for genomic analysis would be preferred
but is limited due to high cost and practical limitations in obtaining
multiple biopsies.
Noninvasive global tumor assessment, however, is possible with
imaging such as Computed Tomography (CT), Magnetic Resonance (MR)
or Positron Emission Tomography (PET). The emerging field of "imagenomics"
is focused on identifying imaging traits that correlate with genetics.
If successfully validated, and proven to have suitable sensitivity
and specificity, the use of imagenomics tests could complement
conventional surgical biopsies. For example, this could be important
in the context of large heterogeneous lesions, multiple lesions,
surgically inaccessible lesions, and settings where disease progression
needs to be monitored frequently over time.
The Medical Imaging Informatics research group at the Mayo Clinic
is developing and applying microlocal analyses of medical images
to improve cancer imagenomics. This presentation will discuss
some of our recent efforts and results in this area.
Juri Rappoport, Russian Academy of Sciences
Lebedev transforms and some ophthalmic imaging applications
Lebedev transforms and some ophthalmic imaging applications are
discussed. The properties of Kontorovitch-Lebedev integral transforms,
Lebedev-Skalskaya integral tranforms and modified Bessel functions
are elaborated. The approximation and computation of the kernels
of transforms are given. The effective applications for some mixed
boundary value problems are described. The quantitative and qualitative
changes in the transplant endothelium after keratoplasty are analysed.
Luigi Riba, Universita di Torino and York University
Continuous Inversion Formulas for Multi-Dimensional Stockwell
Transforms
Stockwell transforms as hybrids of Gabor transforms and wavelet
transforms have been studied extensively. Starting from the well
known results on the two-diensional Stockwell transform we introduce
in this paper multi-dimensional Stockwell transforms that include
multi-dimensional Gabor transforms as special cases. Furthermore
we give continuous inversion formulas for multi-dimensional Stockwell
transforms.
Ervin Sejdic, University of Pittsburgh
Introduction to Time-Frequency Analysis and Its Biomedical Applications
Time-frequency analysis is of great interest when time or frequency
based techniques provide insufficient information about signals.
Time-frequency representations depict variations of the spectral
characteristics of signals as a function of time, which is ideally
suited for nonstationary signals, especially, non-stationary biomedical
signals. Many biomedical signals (e.g. heart sounds, swallowing
accelerometry signals) are multicomponent, one-dimensional signals.
The time-frequency analysis of these signals provides a two-dimensional
representation of signals' components, which is appropriate for
a diagnostic analysis. The resolution, that is, the quality of
a representation, depends on a specific time-frequency distribution.
Therefore, this talk provides an introduction to time-frequency
analysis through an overview of recent contributions.
Hau-Tieng Wu, Princeton University
Instantaneous frequency, shape functions, Synchrosqueezing transform
and some applications
Although one can formulate an intuitive notion of instantaneous
frequency, generalizing "frequency" as we understand
it in e.g. the Fourier transform, a rigorous mathematical definition
is lacking. In this talk, we consider a class of functions composed
of waveforms that repeat nearly periodically, and for which the
instantaneous frequency can be given a rigorous meaning. In other
words, we consider the problem of the following form: given a
function $$f(t)=\sum_{k=1}^K A_k(t)s_k(\phi_k(t)),\mbox{ with
} A_k(t),\phi'_k(t)>0 ~ \forall t,$$ and $s_k$ are $2\pi$ periodic,
compute $s_k(t)$, $A_k(t)$ and $\phi'_k(t)$ or describe their
properties from $f$. We introduce the Synchrosqueezing transforms,
which can be used to determine the instantaneous frequency of
functions in this class, even if the waveform is not harmonic.
The properties of the Synchrosqueezing transform, like robustness
to many kinds of noises, the ability to detect the dynamics of
the system will also be discussed. Finally we provide examples
in sleep depth detection, ventilator weaning and seasonality detection.
Hau-Tieng Wu, Princeton University
Vector Diffusion Maps, Connection Laplacian and Their Applications
We introduce vector diffusion maps (VDM), a new mathematical
framework for organizing and analyzing massive high dimensional
data sets, images and shapes. VDM is a mathematical and algorithmic
generalization of diffusion maps and other non-linear dimensionality
reduction methods, such as LLE, ISOMAP and Laplacian eigenmaps.
While existing methods are either directly or indirectly related
to the heat kernel for functions over the data, VDM is based on
the heat kernel for 1-forms and vector fields. VDM provides tools
for organizing complex data sets, embedding them in a low dimensional
space, and interpolating and regressing vector fields over the
data. In particular, it equips the data with a metric, which we
refer to as the {\em vector diffusion distance}. In the manifold
learning setup, where the data set is distributed on (or near)
a low dimensional manifold $\text{M}^d$ embedded in $\mathbb{R}^p$,
we prove the relation between VDM and the connection-Laplacian
operator for 1-forms over the manifold. The algorithm is directly
applied to the cryo-EM problem and the result will be demonstrated.
We will also discuss its application in determining the orient
ability of a given manifold.
Hau-Tieng Wu, Princeton University
Two-Dimensional tomography from noisy projections taken at unknown
random directions
Computerized Tomography (CT) is a standard method for obtaining
internal structure of objects from their projection images. While
CT reconstruction requires the knowledge of the imaging directions,
there are some situations in which the imaging directions are
unknown, for example, when imaging a moving object. It is therefore
desirable to design a reconstruction method from projection images
taken at unknown directions. Another difficulty arises from the
fact that the projections are often contaminated by noise, practically
limiting all current methods, including the recently proposed
diffusion map approach. In this paper, we introduce two denoising
steps that allow reconstructions at much lower signal-to-noise
ratios (SNR) when combined with the diffusion map framework. In
the first denoising step we use principal component analysis (PCA)
together with classical Wiener filtering to derive an asymptotically
optimal linear filter. In the second step, we denoise the graph
of similarities between the filtered projections using a network
analysis measure such as the Jaccard index. Using this combination
of PCA, Wiener filtering, graph denoising and diffusion map, we
are able to reconstruct the 2-D Shepp-Logan phantom from simulative
noisy projections at SNRs well below their currently reported
threshold values. We also report the results of a numerical experiment
corresponding to an abdominal CT. Although the focus of this paper
is the 2-D CT reconstruction problem, we believe that the combination
of PCA, Wiener filtering, graph denoising and diffusion maps is
potentially useful in other signal processing and image analysis
applications.
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