Extensions of the Heisenberg group by dilations and frames
type: Item
instance of: scholarly article
title: Extensions of the Heisenberg group by dilations and frames
zbMATH Open document ID: 0840.43017
published in: Applied and Computational Harmonic Analysis
publication date: 1995-07-20T00:00:00Z
review text: Theory and application of wavelet analysis have recently become a very active field of research because they are standing at the intersection of the frontiers of computerized Fourier analysis, algorithmic computing, computer controlled non-algorithmic parallel computing, massively parallel photonic signal processing, digital signal analysis, filter bank design, remote sensing, and visualization. From the mathematical point of view, the aforementioned disciplines overlap significantly. Over the years, however, the notational conventions and methodologies of the disciplines unfortunately diverged to the point that engineers and scientists working in one of the areas such as data compression, feature extraction, image processing, pattern recognition, flow visualization, synthetic aperture radar (SAR) imagery, voice signal processing, magnetic resonance imaging (MRI), or filter bank design often find similar or even identical work in the other of these areas nearly unrecognizable. Harmonic analysis on the Heisenberg nilpotent Lie group suggests to regard square integrability of continuous irreducible unitary group representations as an essential part of the Stone-von Neumann theorem of quantum mechanics. Then the nonlocality phenomenon of quantum physics is a consequence of quantum holography [\textit{W. Schempp}, Geometric analysis: The double-slit interference experimental and magnetic resonance imaging. In: Cybernetics and Systems `96, Vol. I, R. Trappl, Editor, pp. 179-183, Austrian Society for Cybernetic Studies, Vienna 1996]. The symplectic structure of the coadjoint orbits allows for implementation of the Keplerian temporospatial strategy used in physical astronomy for the synchronous and stroboscopic recording of phase histories in frequency subbands, and the design of multichannel coherent wavelet perfect reconstruction analysis-synthesis filter banks of matched filter type [\textit{W. Schempp}, The structure-function problem of Fourier transform magnetic reson…
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 94A12
Mathematics Subject Classification ID: 42C15
zbMATH DE Number: 772200
zbMATH Keywords: wavelet analysis
zbMATH Keywords: Fourier analysis
zbMATH Keywords: algorithmic computing
zbMATH Keywords: signal processing
zbMATH Keywords: digital signal analysis
zbMATH Keywords: filter bank design
zbMATH Keywords: remote sensing
zbMATH Keywords: visualization
zbMATH Keywords: data compression
zbMATH Keywords: feature extraction
zbMATH Keywords: image processing
zbMATH Keywords: pattern recognition
zbMATH Keywords: flow visualization
zbMATH Keywords: synthetic aperture radar
zbMATH Keywords: voice signal processing
zbMATH Keywords: magnetic resonance imaging
zbMATH Keywords: Heisenberg nilpotent Lie group
zbMATH Keywords: unitary group representations
zbMATH Keywords: Stone-von Neumann theorem
zbMATH Keywords: quantum mechanics
zbMATH Keywords: quantum holography
author: Jeffrey A. Hogan
author: Joseph Lakey
reviewed by: Walter Schempp
MaRDI profile type: Publication
full work available at URL: acha.1995.1013
OpenAlex ID: W2043225182
DOI: 10.1006/ACHA.1995.1013
description: scientific article; zbMATH DE number 772200