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Wigner functions from the two-dimensional wavelet group.

S T Ali1, A E Krasowska, R Murenzi

  • 1Department of Mathematics and Statistics, Concordia University, Montréal, Québec, Canada. stali@neumann.concordia.ca

Journal of the Optical Society of America. A, Optics, Image Science, and Vision
|January 5, 2001
PubMed
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This study constructs Wigner functions on a 2D similitude group phase space, connecting them to wavelet transforms for image analysis. These functions offer new insights into quantum mechanics and signal processing applications.

Area of Science:

  • Theoretical physics
  • Quantum mechanics
  • Mathematical physics

Background:

  • Wigner functions are essential tools in quantum mechanics for phase space representation.
  • The similitude group in two dimensions has prior applications in wavelet-based image analysis.

Purpose of the Study:

  • To construct Wigner functions on a phase space associated with the two-dimensional similitude group.
  • To explore the relationship between these Wigner functions and the wavelet transform.

Main Methods:

  • Utilizing a general procedure previously established in Ann. Henri Poincaré 1, 685 (2000).
  • Constructing Wigner functions on the phase space of the two-dimensional similitude group.
  • Analyzing the topological homeomorphism between the group space and the phase space.

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Main Results:

  • Successfully constructed Wigner functions on the specified phase space.
  • Demonstrated that these Wigner functions can be viewed as functions on the group space itself.
  • Established a connection between the wavelet transform and the constructed Wigner functions.

Conclusions:

  • The Wigner functions constructed offer a novel perspective on phase space representation within the context of the similitude group.
  • The identified link between wavelet transforms and Wigner functions opens avenues for advanced two-dimensional image analysis and quantum signal processing.