Mathematical Algorithms for Medical Imaging
Org: Sima Noghanian (Manitoba) [PDF]
 JORGE ALPUCHE, Department of Physics & Astronomy, University of Manitoba,
Winnipeg, MB
The Application of the Filtered Back Projection (FBP)
Algorithm to Quantitative Scatter CT Reconstruction
[PDF] 
Introduction: Computed Tomography (CT) uses FBP to acquire Linear
Attenuation Coefficient (LAC) images at a specific energy using
transmitted photons. In typical CT studies a large proportion of
photons are scattered and can lead to a reduction in contrast and a
decrease in signal to noise ratio. However these photons carry
valuable material information and this work presents a technique which
uses a variant of the FBP algorithm to reconstruct images of Electron
Density (ED) from scattered photons.
Methods: In the absence of attenuation the total number of scattered
photons is given by the integral of ED along a narrow strip of
material (eq. (1)).
N_{s} (E_{0}) = N_{0} (E_{0})_{e} s(E_{0}) 
ó õ

r_{e} (x) dx \tag1 
 (1) 
where N_{s} (E_{0}) is the number of scattered photons resulting from
N_{0} incident photons of energy E_{0}, _{e} s(E_{0}) is the
probability of scattering for a photon of energy E_{0} and r_{e}(x) is the electron density at point x. In conventional CT the FBP
algorithm is used to reconstruct ray integrals of the LAC. Similarly,
our Scatter CT system reconstructs images of ED using FBP. Our system
was simulated both with and without attenuation and Attenuation
Correction Factors (ACFs), which were applied iteratively, were
developed to correct for the attenuation.
Results and Conclusions: Attenuation free simulations reconstructed
using the FBP yielded EDs with errors ranging from 0.5% to
2.1%. Simulations which included attenuation were
reconstructed using the correct ACFs and yielded EDs with errors
ranging from 2.7% to 1.1% after six iterations.
These results show that under appropriate attenuation corrections,
Scatter CT is capable of quantifying EDs assuming monoenergetic beam
conditions and an accurate rejection of multiple scatter.
 ALI ASHTARI, University of Manitoba
Resolution limits in nonlinear microwave image
reconstruction
[PDF] 
The resolution of microwave imaging methods based on linear radar
signal processing such as synthetic aperture radar (SAR) is well known
[1] and is directly dependent on the radar signal bandwidth. There
are also studies about the resolution of other inverse scattering
methods such as diffraction tomography, Born iterative method and
distorted Born iterative method [2]. However, to the best of our
knowledge, there has been no study on the resolution of optimization
based inverse scattering methods. In this presentation, we emphasize
on the differences between the optimization based methods and other
inverse scattering methods in terms of the resolution. Preliminary
observations on the possibility of having very high resolution images
using optimization based methods are shown. Also, different natures
of the SAR imaging and optimization based imaging are highlighted so
that the major difference in the dependability of the resolution on
bandwidth in these two methods is justified.
References
 [1]

M. Soumekh,
Synthetic Aperture Radar Signal Processing with MATLAB
Algorithms.
Wiley Interscience, New York, 1999.
 [2]

T. Cui, W. C. Chew, X. X. Yin and W. Hong,
Study of resolution and super resolution in electromagnetic
imaging for halfspace problems.
IEEE Trans. Antennas Propagation 52(2004), 13981411.
 HOMA FASHANDI, University of Manitoba
Frequency Based Portal Image Registration for Radiotherapy
Treatment
[PDF] 
In External Beam Radiotherapy (EBRT), one cancer treatment method,
external source of radiation is directed at the tumor from an external
source produced by linear accelerator (LINAC). EBRT consists of two
major parts; planning and treatment phases. In the planning phase,
the shape and location of the tumor is determined by a simulator and
in the treatment phase high energy beams irradiate the tumor.
Reducing patient positioning uncertainty for each fraction of
treatment process is crucial for both cancerous and healthy cells. To
reduce geometric error, increasing the frequency of treatment
verification with portal imaging would be an effective method. Portal
images are taken with linear accelerator devices with therapeutic
beams. Patient positioning problem could be considered as an image
registration problem between the images taken during the planning
phase and the ones taken during each treatment fraction (portal
images). Images taken by therapeutic beams have low resolution and
contrast. Limited contrast adaptive histogram equalization is used to
enhance their quality. The major focuses of our method are logpolar
and Fourier transforms properties.
Logpolar transformation converts scaling and rotation into
translation; so we can deal with any angle of rotation and large
scaling in the logpolar domain. To recover the parameters of the
affine transformation, we map frequency domain to the log polar space
and as an optimization method the normalized cross power spectrum of
mapped portal and reference image is calculated. The scale and
rotation will be recovered and phase correlation method will be used
for determining translation.
 DANIEL FLORES, University of Manitoba
Wavefront Reconstruction Method for Microwave Imaging
[PDF] 
In recent years, radar technology has started to being used in a wide
range of subsurface imaging applications. Traditionally, linear scan
trajectories were used to acquire data in most of the subsurface radar
applications. However, novel applications, such as Breast Microwave
Imaging, require the use of non linear scan trajectories in order to
perform their data acquisition process. This paper proposes a novel
reconstruction algorithm for subsurface radar data acquired along
quasielliptical and circular trajectories. The spectrum of the
collected data is processed in order to locate the spatial origin of
the target reflections and remove the diffraction artifacts introduced
by the scan trajectory. The effects of the antenna mainlobe beamwidth
on the quality of the reconstructed images is discussed and
illustrated. The proposed algorithm was tested using simulated
examples and data collected from phantoms that mimic breast and cancer
tissue.
 MICKAEL GERMAIN, Université de Montréal, CRM, Montréal
Lie algebra data processing applied to medical images
[PDF] 
We propose to describe and develop a data processing technique that
uses a new type of discrete transforms based on the orbit functions of
compact Lie groups on the 3dimensional (3D) case. These discrete
orbitfunction transforms (DOFT) are, in the particular case of a
rectangular lattice of dimension n=2 and of the Lie group SU(2)×SU(2), reduced to the transform known as 2dimensional (2D)
Discrete Cosine Transform TypeI. However, the DOFTs are unique as
they are fast and allow processing of images defined on grids in the
form lattices of other symmetries. A crucial property of the DOFTs is
that it allows construction of continuous functions (trigonometric
polynomials) providing continuous extension of the (inverse) transform
from the discrete grid points to any point of the surface in between.
It was demonstrated that the continuous extensions of DOFTs (CEDOFT)
have superior analytic properties, such as convergence, localization,
and differentiability of the trigonometric series when compared to
other techniques. These properties (fast processing and continuous
extension) suggest that these tools are ideal for interpolation
processing. The proposal is aimed at using this process to
interpolate data extracted from a MRI system. Compared to standard
interpolation processes (tricubic, spline, etc.), we increase the
quality of the interpolation, and the time computation is faster.
 COLIN GILMORE, University of Manitoba, Winnipeg, Manitoba, Canada
An Overview of The Electromagnetic Inverse Problem with
Biomedical Applications
[PDF] 
Electromagnetic inversion and imaging as applied to the problems
associated with biomedical imaging are described and reviewed. The
mathematics of the electromagnetic inverse scattering problem will be
outlined, as well as the basic problems of nonlinearity,
illposedness and nonuniqueness. Different inversion methods, such
as linear and nonlinear optimization will be discussed.
In particular, we will consider a nonlinear inversion algorithm known
as the Contrast Source Inversion (CSI) method. The CSI method
formulates the inversion problem as an optimization problem which
allows for nonconstant velocities within the biological tissue and
take into account multiple scattering. While slightly more
mathematically complicated than linear inversion methods, these
algorithms offer the possibility of reconstructing the quantitative
material parameter values, such as permittivity and conductivity,
within the biological material.
Computational results for 2D breast models are presented and show that
the CSI algorithm provides a promising technique for biomedical
imaging.
 IAN JEFFREY, University of Manitoba
Adaptive Basis Functions Suitable for a WellConditioned
Formulation to the Inverse Electromagnetic Scattering Problem
under the BIM
[PDF] 
The work presented shows that through an adaptive set of basis
functions, the MoM solution to the linearized scalar inverse
electromagnetic scattering problem is capable of alleviating the
illconditioning of the resulting matrix equation. The selected basis
functions, wholedomain and harmonic, provide a perfectly conditioned
system of equations under the firstorder Born approximation when
appropriately selected field frequencies are chosen.
By analogy, we iteratively solve the full nonlinear problem using the
Born Iterative Method (BIM) by introducing variability in the basis
function expansion through a single phase parameter. By selecting the
parameter value that minimizes the condition number of the discrete
matrix operator, we demonstrate that it is possible to maintain a
wellconditioned, linearized inverse problem, at each iteration of the
BIM. The benefit of this approach is that it removes the requirement
for Tikhonov regularization (or equivalent regularization schemes)
usually needed to obtain physically meaningful solutions to the
discrete system at each iteration.
 AMIR MEGHDADI, University of Manitoba, Department of Electrical and
Computer Engineering
Temporal and Spatial Imaging of Brain Epileptic Activities
by Dynamical Characterization of EEG Signal
[PDF] 
EEG signals may be used for detecting abnormal brain activities
including epileptic seizures. Nonlinear time series analysis methods
employ a dynamical system approach in order to better characterize the
EEG signal. These methods assume a deterministic though very complex
nature for the time series of EEG signal.
EEG signal during and possibly just before seizure activity is shown
to be more deterministic. Detecting such determinism however is a
challenging task because the signals are usually affected by noise. A
new method for detecting determinism is proposed here which is robust
to measurement noise and provides a tool for characterization of
epileptic brain signals and locating the areas which are responsible
for seizure generation in the brain.
 PUYAN MOJABI, University of Manitoba, Electrical and Computer Engineering
The use of the Lcurve and NCP parameterchoice methods in
electromagnetic inverse scattering problems
[PDF] 
It is wellknown that the inverse scattering problem is inherently
illposed: the solution is not unique and does not depend continuously
on the data. For solving this illposed problem, we use Tikhonov
regularization, which can be formulated as a damped least squares
problem, in conjunction with a parameterchoice method for finding the
optimum regularization parameter. Finding the optimum regularization
parameter is very difficult and also computationally expensive because
the resulting solution can be very sensitive to the choice of the
regularization parameter. Many regularization parameterchoice
methods have been proposed in the literature: for example, generalized
discrepancy principle, generalized cross validation, the Lcurve and
Normalized Cumulative Periodogram (NCP) method. The Lcurve method
tries to balance the (semi) norm of the solution and the corresponding
residual by choosing the regularization parameter that puts one on the
corner of the Lcurve. The NCP method tries to use more available
information from the residual as opposed to just the norm of the
residual and it is based on the fact that there is similarity between
the SVD basis and Fourier basis.
Herein, the application of the Lcurve and NCP parameterchoice
methods to the Tikhonovregularized functional arising in the 2D/TM
inverse scattering problem which is formulated via an integral
equation and solving using the Born iterative method (BIM) is
investigated and adapted for this application.
 MALCOLM NG MOU KEHN, University of Manitoba
Accuracy Improvement of an Existing Permittivity Measurement
Technique for Dielectric Disk Samples
[PDF] 
A method for permittivity measurement is restudied. Though being
able to determine the dielectric constant of disk samples, it suffers
from frequency variation that can lead to errors as severe as 20% at
certain frequencies. A technique that improves the accuracy is
proposed here. It capitalizes on trends in the slopes of retrieved
permittivity vs. frequency graphs to sieve out the undesired
frequency dependence. These slope phenomena are characterized via
numerical simulations of the measurement structure, which comprises a
dielectric disk sample sandwiched between disjointed inner conductors
of a coaxial cavity. Readup data graphs for general usage are then
obtained. Repetitive generation of such graphs is thus not necessary.
With error levels of less than 1%, the accuracy of this improved
method is significantly higher than that obtained by just directly
applying the original technique alone. Being independent of the
required reference materials, the method is also shown to be stable.
In addition, an independently new technique for measuring the
permittivity of annular ring samples using quadratic curve fitting is
proposed. By measuring only three known materials (one of them may be
free space, thus reducing only to two required known solid
dielectrics), the permittivity of any unknown dielectric may
subsequently be determined with high accuracy over a wide frequency
range. Comparison results of accuracy between this new approach and
the improved method mentioned earlier will be presented.
 BARBARA PAWLAK, University of Manitoba
PET Image Reconstruction by Density Estimation
[PDF] 
PET (positron emission tomography) scans are still in the experimental
phase, as one of the newest breast cancer diagnostic techniques. It
is becoming the new standard in neurology, oncology and cardiology.
PET, like other nuclear medicine diagnostic and treatment technique
involves the use of radiation. Because of negative impact of
radioactivity to our body the radiation doses in PET should be small.
The existing computing algorithms for PET can be divided into two broad
categories: analytical and iterative methods. In the analytical
approach the relation between the picture and its projections is
expressed by a set of integral equations which are then solved
analytically. Iterative methods can be further divided into
deterministic and stochastic approaches. The ART (Algebraic
Reconstructed Technique) algorithm, developed and first used by
Gordon, et al., in the reconstruction of biological material in early
1970s, is an example of deterministic technique. The stochastic
approach, like EM (expectation maximization) algorithm, bases on the
assumption that radioactive emissions follow Poisson statistics. The
algorithm combines unique and properties of the Poisson process and
the maximum likelihood method of estimation.
The proposed kernel density estimation algorithm falls also into the
category of iterative methods. In this approach each coincidence
event is considered individually. The estimate location of the
annihilation event that caused each coincidence event bases on the
previously assigned location of events processed earlier. To
accomplish this, we construct a probability distribution along each
coincidence line. This is generated from previous annihilation points
by density estimation. It has been observed that density estimation
approach to PET can reconstruct an image of the existing tumor using
significantly lower data than the standard CT algorithms, like Fourier
backprojection. Therefore, it might be a very promising technique
allowing to reduce the radiation dose for patients.
 ABAS SABOUNI, University of Manitoba, Winnipeg, Manitoba, Canada, R3T 5V6
Parallel FDTD/GA for Microwave Image Reconstruction
[PDF] 
In the past few years microwave imaging has received significant
interest due to its potential to detect breast tumors at an early
stage. Microwave imaging is the process by which radiofrequency
electromagnetic waves are used to generate an image of a body to enable
physicians to diagnose disease. To create images from microwave
measurements, it is necessary to construct an electromagnetic field,
which is able to transmit microwaves and measure the scattered waves at
one or more sampling points. "Tomography" is one of the methods used
in microwave imaging. In this method, to solve an inverse problem, a
forward solver and an optimization tool are needed. The numerical
Finite Difference Time Domain (FDTD) method is a powerful tool used as
forward solver for solving Maxwell's equations to compute the
scattered electric field at the observation points. The Genetic
Algorithm (GA) is a popular evolutionary global optimization method
that performs very well for problems with a high number of parameters
and high degrees of nonlinearity.
This talk presents an effective method of microwave imaging using FDTD
as forward solver and GA for optimization. Since both FDTD and GA are
computationally intensive parallel computations of the GA and FDTD
codes are proposed. Using MessagePassing Interface (MPI) libraries,
we are able to reach highquality images with a reasonable runtime.
The parallelization for the GA is based on master/slave protocol and
for FDTD based on the distributed heartbeat algorithm. To the best of
our knowledge, this implementation of hybrid method of parallel GA and
FDTD represent a novelty in the framework of the finding early stage
breast cancer application.
With contributions from Prof. Sima Noghanian and Prof. Stephen
Pistorius (both University of Manitoba).
 PARIMALA THULASIRAMAN, Univ. of Manitoba, Department of Computer Science, Winnipeg,
MB, R3T 2N2
Design, development and implementation of a parallel
algorithm for computed tomography using algebraic
reconstruction technique
[PDF] 
In this work, we examine the design and implementation of a parallel
algorithm for reconstructing images from projections using the
algebraic reconstruction technique (ART). This technique for
reconstructing pictures from projections is useful for applications
such as Computed Tomography (CT or CAT). The algorithm requires fewer
views, and hence less radiation, to produce an image of comparable or
better quality. However, the approach is not widely used because of
its computationally intensive nature in comparison with rival
technologies. A faster ART algorithm could reduce the amount of
radiation needed for CT imaging by producing a better image with fewer
projections.
A reconstruction from projections version of the ART algorithm for two
dimensions was first designed for a distributed memory machine and
implemented in parallel using the Message Passing Interface (MPI). The
results produced on the distributed memory machine did not produce
faster reconstructions due to prohibitively long and variant
communication latency. The algorithm was then redesigned for a
multithreaded, shared memory machine and implemented in OpenMP. This
version produced positive results, showing a clear computational
advantage for multiple processors and measured efficiency ranging from
6095%. Consistent with the literature, image quality proved to be
significantly better compared to the industry standard Filtered
Backprojection algorithm especially when reconstructing from fewer
projection angles.
 NIRANJAN VENUGOPAL, University of Manitoba
A 3D ModelBased Outer Volume Suppression Technique for MR
Spectroscopic Imaging of the Prostate
[PDF] 
Purpose: To adapt a new MR Spectroscopy (MRS) technique employing
noncuboidal voxels, called conformal voxel MRS (CVMRS), for use in
prostate spectroscopic imaging in order to reduce contamination of
spectra by lipid signal surrounding the prostate.
Method and Materials: CVMRS uses twenty or more spatial saturation
(SS) pulses, placed around the prostate, to reduce the lipid signal
affecting the spectra within the prostate. Use of the new CVMRS
technique reduced the lipid signal contamination by 84% as compared
to standard cuboidal voxel MRS. To further reduce the lipid
contamination, the routinely used 90 degree flip angle used for each
SS pulse was modified to take into account the regrowth of lipid
signal with its short T1 relaxation time.
Results: Resulting spectra from the optimized approach actually showed
an increase in lipid contamination by 10%. We tracked the problem
down to overlapping SS pulses. Using a simulated 3D model, we found
that 68% of the volume we were trying to saturate experienced
multiple overlapping SS pulses. Regions of the volume experiencing an
even number of SS pulses were found to increase the lipid
contamination signal by 88% to 200%. Conversely, regions
experiencing an odd number of SS pulses had a reduction in lipid
contamination of 55%.
Conclusion: Changing the ordering of the SS pulses, such that the
overlapping pulses occur later in the train of 20 SS pulses reduces
this problem. In summary, we have developed an improved outer volume
saturation technique which reduces lipid contamination problems in
prostate MRS.
