Home page of Edward R. Vrscay
Edward R. Vrscay
Department of Applied Mathematics
Faculty of Mathematics
University of Waterloo
Waterloo, Ontario, Canada N2L 3G1
Tel: (519) 888 4567 x 35455
Office: MC 6326
e-mail: ervrscay "at" uwaterloo.ca
- Mathematical imaging:
Nonlocal image processing: theory and applications.
- Fractal image coding (a particular example of nonlocal image processing) and
the self-similarity of images.
- The use of fractal-based coding methods in image processing: compression,
Image quality measures -- in particular, the ``structural similarity'' measure
(originally due to Prof. Z. Wang, my collaborator from E&CE, UW).
Novel spaces of image functions and their applications: Most recently: (i) measure-valued image mappings and (ii) function-valued image mappings. The latter are ideally suited for the representation of hyperspectral images and diffusion MRI images.
Intensity-based metrics for image functions, including those that
accommodate Weber's models of perception.
- "Diagnostically lossless" medical image compression.
- Fractal-based methods of analysis and approximation: Iterated function systems,
"generalized fractal transforms" over various metric spaces, inverse problems of
approximation using fractal-based methods (e.g., "collage method for contraction mappings").
- For a brief and quite readable introduction to the ideas behind
fractal image coding, please
consult A Hitchhiker's Guide to Fractal Image Coding
(Admittedly, it's an old document (1996), but people still find it helpful.)
Dynamical systems and their applications, e.g., iteration of rational mappings
in the complex plane, chaotic dynamics.
The ''Chrysler-Waterloo Project:''
Design of a new generation
of conformable high-pressure vessels for gaseous fuels in automotive applications
This work was
supported by a Natural Sciences and Engineering Research
Council Collaborative Research and Development (CRD) with Chrysler Canada Inc.
-- now Fiat Chrysler Automotive Canada (FCA Canada)--
as industrial sponsor during the years 2014-2018.
(The heading of this section
is the title of the CRD Grant.)
In collaboration with FCA, we were engaged in
developing a framework
for the design of compressed gaseous fuel vessels that will occupy
arbitrary geometries. Our goal was to develop
algorithms for fitting a network of tubes with a range of diameters into
an arbitrary three-dimensional region.
The following three faculty members were involved in this project:
This research was highly interdisciplinary in nature, involving
various aspects of optimization, fluid mechanics, solid mechanics,
software design and computing. Both theory and application played important
Edward R. Vrscay, Dept. of Applied Mathematics, UW
Sean Peterson, Dept. of Mechanical and Mechatronics Engineering, UW.
Franklin Mendivil, Dept. of Mathematics and Statistics, Acadia
University, Wolfville, NS.
Three M.Math. students, one M.Sc. student and one
Postdoctoral Research worked
on this project (see below).
I was originally approached by Chrysler Canada because of our
Waterloo Fractal Coding and Analysis Group website. Our project would eventually involve very little
fractal content, except for the important idea of branching.
Nevertheless, the entire exercise was
a very good opportunity for me, and others as well, to
``expand our horizons'' by learning new ideas and methods. For example,
one of the important components of our project and subsequent
algorithms was circle packing, which was used to pack tubes
Here are two slide presentations on our work
delivered at the AMMCS-CAIMS 2015 meeting, June 7-12, 2015,
held at Wilfrid Laurier University, Waterloo, Ontario, Canada.
Here is a set of notes on an early method of circle
packing for arbitrary polygonal regions which I developed
a little later in the project (and for which I
wrote some primitive code) in collaboration with my colleagues
Sean Peterson (UW) and Franklin Mendivil:
Mathematical physics, in particular quantum theory. At one time, this
represented a major research activity of mine. However, as time progressed
and my activities in mathematical imaging were expanding,
there was less and less time (and energy!) available to supervise
graduate students in this area. As a result, I decided in 2007 that I would not
take any new graduate students. It was a difficult decision for a number of reasons:
Here is a brief list of areas of quantum mechanics in which I have worked,
arranged chronologically from past to most recent:
I enjoyed very much the most recent work in the de Broglie-Bohm causal interpretation
of quantum mechanics with my students Caroline Colijn (Ph.D.) and Jeff Timko (M.Math.),
I was still receiving many requests from potential students to supervise them in the
area of foundations of quantum theory, especially "Bohmian mechanics",
I am still very much interested in the foundations (or lack thereof!) of quantum theory.
Quantum mechanical perturbation theory and summability of divergent perturbation expansions:
Continued fraction representations of divergent series
Coherent states in quantum mechanics
Classical limit of quantum mechanics, including classical limits of perturbation expansions
The de Broglie-Bohm causal interpretation of quantum mechanics
D. La Torre (Milan), F. Mendivil (Acadia), H. Kunze (Guelph): We
Waterloo Fractal Coding and Analysis Group .
We have been interested in various aspects
of fractal analysis including: iterated function systems, fractal image coding,
generalized fractal transforms and the inverse problem of approximation using fixed
points of contraction mappings. Here is a photo of our book, Fractal Based Methods in Analysis (Springer Verlag 2012).
You can read about it at the Springer website for the book.
Z. Wang, Department of Electrical and Computer Engineering, UW.
H. Tizhoosh, Department of Systems Design Engineering, UW.
D. Koff, Chair, Department of Radiology, McMaster University.
W. Wallace, Agfa HealthCare, Waterloo, Ontario.
O. Michailovich, Department of Electrical and Computer Engineering, UW.
Ph.D., in progress
A. Cheeseman (co-supervision with H. Tizhoosh, UW)
M.Math., in progress
D. Li, A novel class of intensity-based metrics for image functions
M. Miao, Mathematical modelling of diffusion magnetic resonance imaging
Postdoctoral Research Associate, completed
F. Ghasempour, Design and analysis of conformable tubular networks
which occupy arbitrary regions in (2015-2017)
co-supervision with S. Peterson, UW and F. Mendivil, Acadia/UW)
D. Otero, "Function-valued mappings and SSIM-based optimization in imaging" (2015) (co-supervision with O. Michailovich, UW)
I. Kowalik-Urbaniak, "The quest for 'diagnostically lossless' medical
image compression using objective image quality measures" (2015)
(co-supervision with Z. Wang, UW)
J. Vass, "On the Geometry of IFS Fractals and its Applications" (2014)
D. Brunet, "A study of the structural similarity image quality measure
with applications to image processing" (2012)
(co-supervision with Z. Wang, UW)
N. Portman, "The modelling of biological growth using a pattern theoretic approach"
(2009) (co-supervision with U. Grenander, Brown University)
G.S. Mayer, "Resolution enhancement in magnetic resonance imaging by frequency extrapolation"
M. Ebrahimi Kahrizsangi, "Inverse problems and self-similarity in imaging" (2008)
S.K. Alexander, "Multiscale methods in image modelling and image processing" (2005)
C. Colijn, "The de Broglie-Bohm causal interpretation of quantum mechanics
and its application to some simple systems" (2003)
J. Liang, Design of an Automatic Facial Expression Detector (2018)
H. Wang, A Novel Diffusion-based Empirical Mode Decomposition Algorithm
for Signal and Image Analysis (2018)
(co-supervision with R. Mann, Cheriton School of Computer Science, UW)
E. Maki, "Iterated function systems with place-dependent
probabilities and the inverse
problem of measure approximation using moments" (2017)
T. Qiao, "Design of tubular network systems using circle packing and discrete optimization" (2016)
co-supervision with F. Mendivil, Acadia/UW)
W. Jiang, "Construction of optimal tubular networks in arbitrary regions
in " (2015)
co-supervision with S. Peterson, UW)
I.-T. Ho, "Improvements on circle packing algorithms in two-dimensional cross-sectional areas " (2015)
co-supervision with S. Peterson, UW)
J. Ladan, "An analysis of Stockwell transforms, with applications to
image processing" (2014)
D. Glew, "Self-similarity of images, nonlocal image processing and image quality metrics" (2011)
C. Antonio Sanchez, "Dynamic magnetic resonance elastography: Improved direct methods
of shear modulus estimation" (2009)
J. Timko, "Bohmian trajectories of the two-electron helium atom" (2007)
Y. Li, "Determining NMR relaxation times for porous media: Theory, measurement
and the inverse problem" (2007)
S.K. Alexander, "Two- and three-dimensional coding schemes for wavelet
and fractal-wavelet image compression (2001)
Undergraduate RA, in progress
Undergraduate RA, completed
D. Li, A novel class of metrics for image functions designed
to accommodate Weber's model of perception
(Physics 437 research project, Fall 2017, Winter 2018)
A. Cheeseman, Methods of predicting the severity
of degradation of image blocks by JPEG and JPEG2000 compression methods
(Physics 437 research project, Fall 2014, Winter 2015)
P. Bendevis, Construction and analysis of a family of higher-order structural similarity
rational functions (URA, Fall 2013, Winter 2014)
A. Akulov, Indexing images by means of their fractal codes (NSERC USRA, Spring 2011)
Here are my lecture or supplementary notes for some
courses taught recently.
The "Spirit of Calculus"
MATH 137, Honours Calculus I,
Physics-based Section 008, Fall 2012
MATH 138, Honours Calculus II,
Physics-based Section 005, Winter 2017
MATH 227, Honours Calculus III for
Physics, Fall 2010
MATH 228, Differential Equations for Physics and Chemistry,
AMATH 231, Honours Calculus IV - Vector Calculus and Fourier Series,
AMATH 351, Ordinary Differential Equations II,
AMATH 353, Partial Differential Equations I,
AMATH 391, From Fourier to Wavelets,
PMATH 370, Chaos and Fractals,
AMATH 731, Applied Functional Analysis,