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Journal of the Optical Society of America A

Unitary transformation between Cartesian- and polar-pixellated screens

Luis Edgar Vicent, Kurt Bernardo Wolf

Published Jul 2, 2008• 17 citations
Publisher

Abstract

A unitary transformation between Cartesian and polar pixellations of finite two-dimensional images is obtained from the su(2) model for discrete and finite signals. This transformation analyzes the original image into its finite Cartesian "Laguerre-Kravchuk" modes (involving Wigner little-d functions) and synthesizes it back using a polar mode basis with the same set of mode coefficients. The polar basis is derived from the quantum angular momentum theory, and its modes are given by Clebsch-Gordan coefficients.

DOI
10.1364/josaa.25.001875

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A unitary transformation between Cartesian and polar pixellations of finite two-dimensional images is obtained from the su(2) model for discrete and finite signals. This transformation analyzes the original image into its finite Cartesian "Laguerre-Kravchuk" modes (involving Wigner little-d functions) and synthesizes it back using a polar mode basis with the same set of mode coefficients. The polar basis is derived from the quantum angular momentum theory, and its modes are given by Clebsch-Gordan coefficients.
Using the coefficients introduced by Bargmann and Moshinsky for the reduction of the su(333) algebra of Cartesian three-dimensional oscillator multiplet states into so(333) angular momentum submultiplets, we implement unitary rotations of three-dimensional Cartesian arrays that form finite pixellated "volume images." Transforming between the Cartesian and spherical bases, the subgroup of rotations in the latter is converted into rotations of the former, allowing for proper concatenation and inversion of these unitary transformations, which entail no loss of information.
In this essay we finitely quantize the Hamiltonian system of geometric optics to a finite system that is also Hamiltonian, but where signals are described by complex N-vectors, which are subject to unitary transformations that form the group U(N). This group can be decomposed into U(2)-paraxial and aberration transformations. Proper irreducible representation bases are thus provided by quantum angular momentum theory. For one-dimensional systems we have waveguide models. For two-dimensional systems we can have Cartesian or polar sensor arrays, where digital images are subject to unitary rotation, gyration or asymmetric Fourier transformations, as well as a unitary map between the two arrays.
The one-dimensional finite oscillator model satisfies that under commutator brackets, all operators close into the (dynamical) Lie algebra su(2); therefore the eigenvalues of position, momentum and energy operators are discrete and the eigenvalue of a compact u(l) central operator fixes the dimension of the su(2) representation. In this work we review the one-dimensional finite oscillator model with u(2) as the dynamical algebra where the wave functions, which involve Kravchuk polynomials, are discrete functions of angular momentum theory.
Linear canonical transforms (LCTs) are important in several areas of signal processing; in particular, they were extended to complex-valued parameters to describe optical systems. A special case of these complex LCTs is the Bargmann transform. Recently, Pei and Huang [J. Opt. Soc. Am. A34, 18 (2017)JOAOD60740-323210.1364/JOSAA.34.000018] presented a normalization of the Bargmann transform so that it becomes possible to delimit it near infinity. In this paper, we follow the Pei-Huang algorithm to introduce the discrete normalized Bargmann transform by the relationship between Bargmann and gyrator transforms in the SU(2) finite harmonic oscillator model, and we compare it with the discrete Bargmann transform based on coherent states of the SU(2) oscillator model. This transformation is invertible and unitary. We show that, as in the continuous analog, the discrete normalized Bargmann transform converts the Hermite-Kravchuk functions into Laguerre-Kravchuk functions. In addition, we demonstrate that the discrete su(1,1) repulsive oscillator functions self-reproduce under this discrete transform with little error. Finally, in the space spanned by the wave functions of the SU(2) harmonic oscillator, we find that the discrete normalized Bargmann transform commutes with the fractional Fourier-Kravchuk transform.
Based on the coherence property under the Fourier transform, we introduce two types of coherent states into the finite discrete oscillator model of s u (2) algebra, and show that they are stable under the fractional Fourier–Kravchuk transform. Using the set of discrete coherent states of this model as biorthonormal bases, we propose two discrete transforms. The first transform maps f ∈ C N functions to the unit circle, and the second transform maps the same functions to Bargmann space; both transforms have analytic inverses. These transformations establish a connection between Fourier and Bargmann expansions, respectively, with the basis of Kravchuk symmetric functions. They also provide a framework for the study of finite-dimensional systems in a periodic or complex continuous space. We give some examples of applications to qudit systems.