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Nonlinear multiresolution signal decomposition schemes--part II: morphological wavelets.

H M Heijmans1, J Goutsias

  • 1Centre for Mathematics and Computer Science (CWI), Amsterdam. henkh@cwi.nl

IEEE Transactions on Image Processing : a Publication of the IEEE Signal Processing Society
|February 12, 2008
PubMed
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This study introduces a unified framework for wavelet decompositions, including novel nonlinear morphological wavelets. The lifting scheme provides a flexible method for constructing these advanced wavelet transforms.

Area of Science:

  • Signal Processing
  • Mathematical Analysis
  • Image Processing

Background:

  • The standard wavelet transform is a linear tool, limiting its application in certain complex signal analysis scenarios.
  • Nonlinear extensions of wavelet transforms offer enhanced capabilities for analyzing intricate data patterns.
  • The lifting scheme has significantly advanced the development of nonlinear wavelet transforms.

Purpose of the Study:

  • To present an axiomatic framework for diverse linear and nonlinear wavelet decompositions.
  • To introduce novel wavelets derived from mathematical morphology.
  • To explore the application of the lifting scheme in constructing nonlinear wavelets.

Main Methods:

  • Development of an axiomatic framework for wavelet decompositions.

Related Experiment Videos

  • Introduction of morphological wavelets, including the morphological Haar wavelet.
  • Application of the lifting scheme for nonlinear wavelet construction, exemplified by the max-lifting scheme.

    • Main Results:

    • A comprehensive framework encompassing various linear and nonlinear wavelet decompositions.
    • Novel one- and two-dimensional morphological wavelets.
    • Demonstration of the max-lifting scheme's ability to preserve local signal maxima across scales.

    Conclusions:

    • The lifting scheme offers a general and flexible approach for creating nonlinear (morphological) wavelets.
    • The proposed framework unifies existing and new wavelet decomposition methods.
    • Morphological wavelets, particularly those derived via max-lifting, present promising new tools for signal analysis.