Application of reproducing and invariance properties of wavelet and Fourier-Wigner transforms

Leon H. Sibul

doi:10.1117/12.217598

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1 September 1995 Application of reproducing and invariance properties of wavelet and Fourier-Wigner transforms

Leon H. Sibul

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Proceedings Volume 2569, Wavelet Applications in Signal and Image Processing III; (1995) https://doi.org/10.1117/12.217598
Event: SPIE's 1995 International Symposium on Optical Science, Engineering, and Instrumentation, 1995, San Diego, CA, United States

Abstract

We use group representation theory to establish common theoretical foundation for wavelet, Fourier-Wigner, Gabor and short-time Fourier transforms as well as for the narrow and wideband ambiguity functions. These transforms are coefficients of unitary representations of either affine or Heisenberg groups. From this fact, many important properties of these transforms and ambiguity functions, including volume conservation and admissibility conditions, follow. In this paper we use a generalization of the Frobenius-Shur-Godement theorem (generalized resolution of identity) to derive the reproducing kernels associated with these transforms and ambiguity functions. This result has several new applications to the well- established reproducing kernel Hilbert space theory. First of all, it establishes the conditions for the general resolution of identity and identifies spaces on which transforms are invertible. These results can be used to solve inverse problems that arise in remote sensing and characterization of stochastic propagation and scattering channels. Since reproducing kernels are positive definite functions, they can be used as approximating functions, analogously to the radial bases functions, for neural network expansions, interpolation and optimization. Because auto-wavelet and auto-Fourier-Wigner transforms are reproducing kernels on a well defined space of functions, we have a powerful method for generating a rich set of 2n dimensional positive definite functions for multi-dimensional interpolation, approximation, and sampling.

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Leon H. Sibul "Application of reproducing and invariance properties of wavelet and Fourier-Wigner transforms", Proc. SPIE 2569, Wavelet Applications in Signal and Image Processing III, (1 September 1995); https://doi.org/10.1117/12.217598

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