<title>Lifting scheme: a new philosophy in biorthogonal wavelet constructions</title>

TLDR

Wavelets on the sphere illustrate the class of second‑generation wavelets that are not simple translates and dilates of a single function. The paper presents the basic idea behind the lifting scheme, a new construction of biorthogonal wavelets that does not rely on the Fourier transform. The authors introduce lifting directly from a wavelet‑transform viewpoint, postponing basis‑function considerations, and show it can generate second‑generation wavelets that need not be simple translates or dilates. Lifting yields a faster, fully in‑plane implementation of the wavelet transform.

Abstract

In this paper we present the basic idea behind the lifting scheme, a new construction of biorthogonal wavelets which does not use the Fourier transform. In contrast with earlier papers we introduce lifting purely from a wavelet transform point of view and only consider the wavelet basis functions in a later stage. We show how lifting leads to a faster, fully in- plane implementation of the wavelet transform. Moreover, it can be used in the construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one function. A typical example of the latter are wavelets on the sphere.