Separable and nonseparable multiwavelets in multiple dimensions

Authors:
C. J. Tymczak;Anders M. N. Niklasson;Heinrich Röder
Affiliations:
Theoretical Division, Los Alamos National Laboratory, MS B221, Los Alamos, New Mexico 87545;Theoretical Division, Los Alamos National Laboratory, MS B221, Los Alamos, New Mexico 87545;Theoretical Division, Los Alamos National Laboratory, MS B221, Los Alamos, New Mexico 87545
Venue:
Journal of Computational Physics
Year:
2002

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Abstract

We report on a method of constructing multidimensional biorthogonal interpolating multiwavelets. The approach is based upon polynomial interpolation in Rd (C. de Boor and A. Ron, Math. Comput. 58, 198 (1997)) and an extension of the lifting scheme (J. Kovacević and W. Sweldens, IEEE Trans. Image Process. 9, No. 3, 480 (2000)). The constructed wavelets have compact support, are nearly isotropic, and retain partial scale invariance leading to a fast and efficient multidimensional wavelet transform. We demonstrate an implementation for these wavelets of variable polynomial order up to four dimensions. Finally, we show that these wavelets have a much sparser representation of discontinuous functions as compared to tensor product wavelets, which allows for a more compact and efficient representation.