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
e-mail: ervrscay "at" uwaterloo.ca

• Ongoing research activities
• Recent industrial research and collaboration: The "Chrysler-Waterloo Project"
• Primary research collaborators
• Publications (recently updated)
• Some past conference talks and seminars
• Current and recent graduate students and undergraduate research assistants
• Lecture notes for courses taught recently (and not so recently)
• Other websites of possible interest

• 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, denoising, super-resolution.
  • 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 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.

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 AMMCS-CAIMS 2015 meeting, June 7-12, 2015, held at Wilfrid Laurier University, Waterloo, Ontario, Canada.

• Some novel circle-packing algorithms, presented by W. Jiang.
• The barbeque pool heater: An algorithm to construct tubular networks that occupy arbitrary regions in $R3$, presented by B. Kettlewell.

• "NEWPACK" - a modified circle-packing scheme,

• 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 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, Content-based image retrieval (CBIR) for digital histopathology (co-supervision with H. Tizhoosh, Systems Design Engineering, UW)
  • T. Costa (Dept. of Electrical and Computer Engineering, UW), Perceptual Quality Assessment of High Dynamic Range and Wide Colour Gamut Images and Video (co-supervision with V. Gaudet, E&CE Dept., UW)

• M.Math., in progress

• Postdoctoral Research Associate, completed

  • F. Ghasempour, Design and analysis of conformable tubular networks which occupy arbitrary regions in $R3$ (2015-2017) (Chrysler-Waterloo project, co-supervision with S. Peterson, UW and F. Mendivil, Acadia/UW)

• Ph.D., completed

  • 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" (2008)
  • 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)

• M.Math., completed

  • D. Li, A Novel Class of Intensity-based Metrics for Image Functions which Accommodate a Generalized Weber's Model of Perception (2020)
  • M. Miao, Monte Carlo Simulation of Diffusion Magnetic Resonance Imaging (2019)
  • 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) (Chrysler-Waterloo project, co-supervision with F. Mendivil, Acadia/UW)
  • W. Jiang, "Construction of optimal tubular networks in arbitrary regions in $R3$ " (2015) (Chrysler-Waterloo project, co-supervision with S. Peterson, UW)
  • I.-T. Ho, "Improvements on circle packing algorithms in two-dimensional cross-sectional areas " (2015) (Chrysler-Waterloo project, 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

  • H. Chen, Investigating the self-similarity of images (Physics 437 research project, Fall 2019, Winter 2020)
    • H. Chen, PHYS 437A Project, Fall 2019
      
  • 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)

• 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, Winter 2012
  
• AMATH 231, Honours Calculus IV - Vector Calculus and Fourier Series, Winter 2018
  
• AMATH 343, Discrete Models in Applied Mathematics, Fall 2019
  
• AMATH 351, Ordinary Differential Equations II, Fall 2016
  
• AMATH 353, Partial Differential Equations I, Winter 2010
  
• AMATH 391, From Fourier to Wavelets, Fall 2019
  
• PMATH 370, Chaos and Fractals, Winter 2018
  
• AMATH 731, Applied Functional Analysis, Fall 2018