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The Projection Method for Multidimensional Framelet and WaveletAnalysis

Published online by Cambridge University Press:  17 July 2014

B. Han
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Abstract

The projection method is a simple way of constructing functions and filters byintegrating multidimensional functions and filters along parallel superplanes in the spacedomain. Equivalently expressed in the frequency domain, the projection method constructs anew function by simply taking a cross-section of the Fourier transform of amultidimensional function. The projection method is linked to several areas such as boxsplines in approximation theory and the projection-slice theorem in image processing. Inthis paper, we shall systematically study and discuss the projection method in the area ofmultidimensional framelet and wavelet analysis. We shall see that the projection methodnot only provides a painless way for constructing new wavelets and framelets but also is auseful analysis tool for studying various optimal properties of multidimensional refinablefunctions and filters. Using the projection method, we shall explicitly and easilyconstruct a tight framelet filter bank from every box spline filter having at least orderone sum rule. As we shall see in this paper, the projection method is particularlysuitable to be applied to frequency-based nonhomogeneous framelets and wavelets in anydimensions, and the periodization technique is a special case of the projection method forobtaining periodic wavelets and framelets from wavelets and framelets on Euclidean spaces.

Keywords

Projection methodwavelets and frameletstight framelets from box splinesdual framelet filter banksinterpolatory filtersorthonormal filters, frequency-based dual frameletsnonhomogeneous and homogeneous affine systemsFourier transform
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 5: Spectral problems , 2014 , pp. 83 - 110
Copyright
© EDP Sciences, 2014

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