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Edward R. Vrscay

Current Research Interests

Mathematical imaging, mathematics of image processing, fractal-based image coding

My interest in mathematical imaging began with postdoctoral work at Georgia Tech (1984-85) in the area of fractal geometry and dynamical systems.

Current interests: Image Resolution Enhancement, or "super-resolution".

We have also been investigating the use of fractal-based methods to perform other image processing tasks, for example, denoising.

Sample works:
  • G. S. Mayer and E. R. Vrscay, "Measuring information gain for frequency encoded super-resolution MRI", in Magnetic Resonance Imaging, to appear (2007).
  • G. S. Mayer and E. R. Vrscay, "Mathematical analysis of "phase ramping" for super-resolution magnetic resonance imaging", in Image Analysis and Recognition, ICIAR 2006}, Lecture Notes in Computer Science, Vol. 4141, pp. 82-93 (Springer-Verlag, 2006).
  • E. R. Vrscay (2006), "Continuous evolution of fractal transforms and nonlocal PDE imaging", in Image Analysis and Recognition, ICIAR 2006, Lecture Notes in Computer Science, Vol. 4141, pp. 446-457 (Springer-Verlag, 2006).
  • M. Ebrahimi and E. R. Vrscay (2006), "Fractal image coding as projections onto convex sets", in Image Analysis and Recognition, ICIAR 2006, Lecture Notes in Computer Science, Vol. 4141, pp. 493-506 (Springer-Verlag, 2006).
  • M. Ghazel, G. Freeman and E. R. Vrscay, "Fractal-wavelet image denoising revisited", IEEE Transactions on Image Processing, 15, No. 9, 2669-2675 (2006).
  • J. Bona and E. R. Vrscay, Continuous evolution of functions and measures toward fixed points of contraction mappings, in Fractals in Engineering: New Trends in Theory and Applications, edited by J. Levy-Vehel and E. Lutton (Springer-Verlag, London, 2005).
  • K.P. Wilkie and E. R. Vrscay, Mutual information-based methods to improve local region-of-interest image registration, in Image Analysis and Recognition, ICIAR 2005, Lecture Notes in Computer Science 3656, pp. 63-72 (Springer-Verlag, Berlin, 2005)
  • M. Ghazel, G. Freeman and E. R. Vrscay, Fractal image denoising, in IEEE Transactions on Image Processing, 12 (12), 1560-1578 (2003).
  • S. K. Alexander, P. Fieguth and E. R. Vrscay, "Hierarchical annealing for random image synthesis", Lecture Notes in Computer Science No. 2683, p. 194-210 (Springer Verlag, 2003). Proceedings of the EMMCVPR 2003 Conference, Lisbon, Portugal, May, 2003.
  • E. R. Vrscay, A Generalized Class of Fractal-Wavelet Transforms for Image Representation and Compression, Can. J. Elect. Comp. Eng. 23 (1,2), 69-83 (1998).

Dynamical Systems, Iterated Function Systems and their generalizations

Theory and application of IFS, including the formulation of "fractal transforms" over various spaces of interest and usefulness, e.g. measures, functions, distributions. More recently: Theory and application of the "GRID" (growth by random iteration of diffeomorphisms) model of U. Grenander, Brown University.

Sample works:

Inverse problems of approximation using IFS-type methods. Solving inverse problems using contraction mappings in other areas of mathematics.

Sample works:
  • H. E. Kunze, D. La Torre and E. R. Vrscay, "Random fixed point equations and inverse problems using "collage method" for contraction mappings", J. Math. Anal. Appl. 334, 1116-1129 (2007).
  • H. e. Kunze and E. R. Vrscay, Solving Inverse Problems for ODEs Using the Picard Contraction Mapping, Inverse Problems 15, 745-770 (1999).
  • H. e. Kunze, J. E. Hicken and E. R. Vrscay, "Inverse problems for ODEs using contraction maps and suboptimality of the 'collage method'", Inverse Problems 20, 977-991 (2004).
  • E. R. Vrscay, From Fractal Image Compression to Fractal-Based Methods in Mathematics, in Fractals in Multimedia, edited by M. F. Barnsley, D. Saupe and E. R. Vrscay (Springer-Verlag, New York, 2002).

For more information on the research programme outlined above, please consult our fractal imaging web page: Waterloo Fractal Compression Project.


Foundations of quantum mechanics. The deBroglie-Bohm interpretation of quantum mechanics

We have been investigating the properties of simple quantum systems according to the so-called de Broglie-Bohm ontological interpretation of quantum mechanics. Contrary to the standard Copenhagen interpretation of QM, "Bohmian mechanics" assigns a definite velocity to a quantum mechanical particle. This scheme was introduced by Bohm in his two seminal papers, Phys. Rev. 85, 166-179, 180-193 (1952). Bohm begins in the same way as E. Madelung did in his hydro dynamical picture of the Schrodinger equation, Z. Phys. (1929), namely, by employing the transformation psi = R exp(iS/h). He shows that S satisfies a Hamilton-Jacobi equation with an additional term, the so-called quantum potential. On the basis of this observation, Bohm then suggested that the velocity of the quantum mechanical particle could be defined in terms of an expression involving the wave function psi. (This expression turns out to be the well-known quantum mechanical current associated with psi.)

Bohm's students showed that this interpretation could account for trajectories in the double-slit experiment that agreed with observations. In fact, with the assumption that the probability density is given by the squared modulus of the quantum mechanical wave function, the statistical predictions of Bohmian mechanics agree with those of the Copenhagen interpretation. This has, in fact, been a standard criticism of the Bohmian picture in that it yields no new predictions. However, what Bohmian mechanics gives us is an ontological description -- a possible interpretation of quantum reality -- which cannot be given by the standard CI.

Recent papers:
  • C. Colijn and E. R. Vrscay, "Relaxation of spin-dependent Bohm trajectories in electronic transitions of hydrogen", Phys. Lett. A 327, 113-122 (2004).
  • C. Colijn and E. R. Vrscay, "Spin-dependent Bohm trajectories associated with an electronic transition in hydrogen", J. Phys. A: Math. Gen. 36, 4689-4702 (2003).
  • C. Colijn and E. R. Vrscay, "Spin-dependent Bohm trajectories for Pauli and Dirac eigenstates of hydrogen", Found. Phys. Lett. 16(4), 303-323 (2003).
  • C. Colijn and E. R. Vrscay, "Spin-dependent Bohm trajectories for Bohm eigenstates of hydrogen", Phys. Lett. A 300, 334-340 (2003). (See also Erratum to this paper: Phys. Lett. A 316, 424 (2003).)


Last Modified:  Wednesday 6 April 2011
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