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Multiresolution Analysis Adapted to Irregularly Spaced Data

Abstract

This paper investigates the mathematical background of multiresolution analysis in the specific context where the signal is represented by irregularly sampled data at known locations. The study is related to the construction of nested piecewise polynomial multiresolution spaces represented by their corresponding orthonormal bases. Using simple spline basis orthonormalization procedures involves the construction of a large family of orthonormal spline scaling bases defined on consecutive bounded intervals. However, if no more additional conditions than those coming from multiresolution are imposed on each bounded interval, the orthonormal basis is represented by a set of discontinuous scaling functions. The spline wavelet basis also has the same problem. Moreover, the dimension of the corresponding wavelet basis increases with the spline degree. An appropriate orthonormalization procedure of the basic spline space basis, whatever the degree of the spline, allows us to (i) provide continuous scaling and wavelet functions, (ii) reduce the number of wavelets to only one, and (iii) reduce the complexity of the filter bank. Examples of the multiresolution implementations illustrate that the main important features of the traditional multiresolution are also satisfied.

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Correspondence to Anissa Mokraoui.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Mokraoui, A., Duhamel, P. Multiresolution Analysis Adapted to Irregularly Spaced Data. EURASIP J. Adv. Signal Process. 2009, 293952 (2009). https://doi.org/10.1155/2009/293952

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Keywords

  • Filter Bank
  • Scaling Function
  • Wavelet Function
  • Wavelet Basis
  • Multiresolution Analysis