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Journal of the Optical Society of America A

Wigner functions from the two-dimensional wavelet group

S. Twareque Ali, Anna E. Krasowska, Romain Murenzi

Published Dec 1, 2000• 9 citations
Publisher

Abstract

Following a general procedure developed previously [Ann. Henri Poincaré 1, 685 (2000)], here we construct Wigner functions on a phase space related to the similitude group in two dimensions. Since the group space in this case is topologically homeomorphic to the phase space in question, the Wigner functions so constructed may also be considered as being functions on the group space itself. Previously the similitude group was used to construct wavelets for two-dimensional image analysis; we discuss here the connection between the wavelet transform and the Wigner function.

DOI
10.1364/josaa.17.002277

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Following a general procedure developed previously [Ann. Henri Poincaré 1, 685 (2000)], here we construct Wigner functions on a phase space related to the similitude group in two dimensions. Since the group space in this case is topologically homeomorphic to the phase space in question, the Wigner functions so constructed may also be considered as being functions on the group space itself. Previously the similitude group was used to construct wavelets for two-dimensional image analysis; we discuss here the connection between the wavelet transform and the Wigner function.
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The Wigner distribution function is one of the pillars of the phase space formulation of quantum mechanics. Its original formulation may be cast in terms of the unitary representations of the Weyl–Heisenberg group. Following the construction proposed by Wolf and coworkers in constructing the Wigner functions for general Lie groups using the irreducible unitary representations of the groups, we develop here the Wigner functions and Weyl operators on the Euclidean motion group of rank three. We give complete derivations and proofs of their important properties.
Although coherent states and wavelets have developed by now into full-fledged subdisciplines, with plenty of applications in physics, mathematics and engineering, both have deep roots in group theory,. The construction of coherent states was identified as a group-theoretical problem in the pioneering work of Perelomov and Gilmore, whereas wavelets are simply the coherent states associated to the affine group of the line (or the similitude group of the plane for the two-dimensional case). It was therefore appropriate to have both topics represented in the GROUP-24 Colloquium.
The present work concerns the phase space representation of special signals depending on two variables. These signals are of acoustic origin and correspond either to the passive observation of sources with a linear array or to the active study of evolutive sonar targets. The technique used to set up the phase space representation is founded on a constraint of covariance with respect to transformations coming from changes of reference systems. Due to the physical context these transformations correspond to changes of origin, scalings and Galilean boosts and make up the Weyl-Galileo group. A Wigner function affiliated with that group and depending on four variables is effectively obtained. It satisfies both localization and unitarity. The associated wavelet transform is recovered by smoothing.