Home page of Edward R. Vrscay
Edward R. Vrscay
Professor
Department of Applied Mathematics
Faculty of Mathematics
University of Waterloo
Waterloo, Ontario, Canada N2L 3G1
Tel: (519) 888 4567 x 35455
Office: MC 6326
email: ervrscay "at" uwaterloo.ca
 Mathematical imaging:

Nonlocal image processing: theory and applications.
 Fractal image coding (a particular example of nonlocal image processing) and
the selfsimilarity of images.
 The use of fractalbased coding methods in image processing: compression,
denoising, superresolution.

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) measurevalued image mappings and (ii) functionvalued image mappings. The latter are ideally suited for the representation of hyperspectral images and diffusion MRI images.

Intensitybased metrics for image functions, including those that
accommodate Weber's models of perception.
 "Diagnostically lossless" medical image compression.
 Fractalbased methods of analysis and approximation: Iterated function systems,
"generalized fractal transforms" over various metric spaces, inverse problems of
approximation using fractalbased 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 ''ChryslerWaterloo Project:''
Design of a new generation
of conformable highpressure 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 20142018.
(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 threedimensional region.
The following three faculty members were involved in this project:

Edward R. Vrscay, Dept. of Applied Mathematics, UW
(Principal Investigator)

Sean Peterson, Dept. of Mechanical and Mechatronics Engineering, UW.

Franklin Mendivil, Dept. of Mathematics and Statistics, Acadia
University, Wolfville, NS.
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
roles.
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
in regions.
Here are two slide presentations on our work
delivered at the AMMCSCAIMS 2015 meeting, June 712, 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:

I enjoyed very much the most recent work in the de BroglieBohm 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.
Here is a brief list of areas of quantum mechanics in which I have worked,
arranged chronologically from past to most recent:

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 BroglieBohm causal interpretation of quantum mechanics

D. La Torre (Milan), F. Mendivil (Acadia), H. Kunze (Guelph): We
comprise the
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 (cosupervision with H. Tizhoosh, UW)

M.Math., in progress

D. Li, A novel class of intensitybased 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 $R3$ (20152017)
(ChryslerWaterloo project,
cosupervision with S. Peterson, UW and F. Mendivil, Acadia/UW)

Ph.D., completed

D. Otero, "Functionvalued mappings and SSIMbased optimization in imaging" (2015) (cosupervision with O. Michailovich, UW)

I. KowalikUrbaniak, "The quest for 'diagnostically lossless' medical
image compression using objective image quality measures" (2015)
(cosupervision 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)
(cosupervision with Z. Wang, UW)

N. Portman, "The modelling of biological growth using a pattern theoretic approach"
(2009) (cosupervision with U. Grenander, Brown University)

G.S. Mayer, "Resolution enhancement in magnetic resonance imaging by frequency extrapolation"
(2008)

M. Ebrahimi Kahrizsangi, "Inverse problems and selfsimilarity in imaging" (2008)

S.K. Alexander, "Multiscale methods in image modelling and image processing" (2005)

C. Colijn, "The de BroglieBohm causal interpretation of quantum mechanics
and its application to some simple systems" (2003)

M.Math., completed

J. Liang, Design of an Automatic Facial Expression Detector (2018)

H. Wang, A Novel Diffusionbased Empirical Mode Decomposition Algorithm
for Signal and Image Analysis (2018)
(cosupervision with R. Mann, Cheriton School of Computer Science, UW)

E. Maki, "Iterated function systems with placedependent
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)
(ChryslerWaterloo project,
cosupervision with F. Mendivil, Acadia/UW)

W. Jiang, "Construction of optimal tubular networks in arbitrary regions
in $R3$ " (2015)
(ChryslerWaterloo project,
cosupervision with S. Peterson, UW)

I.T. Ho, "Improvements on circle packing algorithms in twodimensional crosssectional areas " (2015)
(ChryslerWaterloo project,
cosupervision with S. Peterson, UW)

J. Ladan, "An analysis of Stockwell transforms, with applications to
image processing" (2014)

D. Glew, "Selfsimilarity 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 twoelectron helium atom" (2007)

Y. Li, "Determining NMR relaxation times for porous media: Theory, measurement
and the inverse problem" (2007)

S.K. Alexander, "Two and threedimensional coding schemes for wavelet
and fractalwavelet 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 higherorder 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,
Physicsbased Section 008, Fall 2012

MATH 138, Honours Calculus II,
Physicsbased Section 005, Winter 2017

MATH 227, Honours Calculus III for
Physics, Fall 2010

MATH 228, Differential Equations for Physics and Chemistry,
Winter 2012

AMATH 231, Honours Calculus IV  Vector Calculus and Fourier Series,
Winter 2018

AMATH 351, Ordinary Differential Equations II,
Fall 2016

AMATH 353, Partial Differential Equations I,
Winter 2010

AMATH 391, From Fourier to Wavelets,
Fall 2017

PMATH 370, Chaos and Fractals,
Winter 2018

AMATH 731, Applied Functional Analysis,
Fall 2018