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Properties of Discrete Framelet Transforms

Published online by Cambridge University Press:  28 January 2013

Bin Han
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Abstract

As one of the major directions in applied and computational harmonic analysis, theclassical theory of wavelets and framelets has been extensively investigated in thefunction setting, in particular, in the function spaceL2(ℝd). A discrete wavelettransform is often regarded as a byproduct in wavelet analysis by decomposing andreconstructing functions in L2(ℝd)via nested subspaces of L2(ℝd) ina multiresolution analysis. However, since the input/output data and all filters in adiscrete wavelet transform are of discrete nature, to understand better the performance ofwavelets and framelets in applications, it is more natural and fundamental to directlystudy a discrete framelet/wavelet transform and its key properties. The main topic of thispaper is to study various properties of a discrete framelet transform purely in thediscrete/digital setting without involving the function spaceL2(ℝd). We shall develop acomprehensive theory of discrete framelets and wavelets using an algorithmic approach bydirectly studying a discrete framelet transform. The connections between our algorithmicapproach and the classical theory of wavelets and framelets in the function setting willbe addressed. Using tensor product of univariate complex-valued tight framelets, we shallalso present an example of directional tight framelets in this paper.

Keywords

discrete framelet transformperfect reconstructionsparsitystabilitydual framelet filter banksdiscrete affine systemslinear-phase momentsvanishing momentssum rules
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 1: Harmonic analysis , 2013 , pp. 18 - 47
Copyright
© EDP Sciences, 2013

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