Separable and Nonseparable Multiwavelets in Multiple Dimensions

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. Kovačević 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.