ABSTRACT
In these cases, Sweldens says, translations and dilations can't be used, so the Fourier transform is of no help. The basic idea behind lifting is to take a simple or even trivial wavelet transform and gradually improve ("lift") properties such as smoothness and vanishing moments. The important realization was that translation and dilation are not essential to building wavelets with the desired properties. "The notion that a basis function can be written as a finite linear combination of basis functions at a finer, more subdivided level, is maintained and forms the key behind the fast transform," he writes in a paper with Peter Schroder [Schroder, Sweldens]. "The main difference with the classical wavelets is that the filter coefficients of second generation wavelets are not the same through-
Schroder and Sweldens have shown how to use the lifting scheme to produce spherical wavelets, with pdtential in geophysics and geography, computer graphics (virtual reality), and astronomy. The Jet Propulsion Laboratory is considering using spherical wavelets to analyze solar winds.
