Journal of Fourier Analysis and Applications

, Volume 18, Issue 1, pp 146–181 | Cite as

On the Representation of Functions with Gaussian Wave Packets

Article

Abstract

We introduce Gaussian wave packets in pursuit of representations of functions, in which the representation is invariant under translation, modulation, scale, rotation and anisotropic dilation. Properties of both continuous and discrete representations are discussed. For the discrete (two-dimensional) case, we develop fast algorithms for the application of the analysis and synthesis operators. A main objective for using Gaussian wave packets is to obtain sparse approximations of functions. However, due to the many invariance properties, the representations will have a high degree of redundancy. Therefore, we also introduce sparse methods for highly redundant representations, that employ some of the analytic properties of Gaussian wave packet for gaining computational efficiency.

Keywords

Gaussian wave packets Sparse representations Fast algorithms Compression 

Mathematics Subject Classification

41A25 42B05 65D15 65R99 

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Fredrik Andersson
    • 1
  • Marcus Carlsson
    • 2
  • Luis Tenorio
    • 3
  1. 1.Centre for Mathematical SciencesLund UniversityLundSweden
  2. 2.Departamento de Matema’ticasUniversidad de Santiago de ChileSantiagoChile
  3. 3.Mathematical and Computer SciencesColorado School of MinesGoldenUSA

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