Multiscale Local Polynomial Smoothing in a Lifted Pyramid for Non-Equispaced Data

This paper integrates Burt-Adelson's Laplacian pyramids with lifting schemes for the construction of slightly redundant decompositions. These decompositions implement multiscale smoothing on possibly non-equidistant point sets. Thanks to the slight redundancy and to the smoothing operations in the lifting scheme, the proposed construction unifies sparsity of the analysis, smoothness of the reconstruction and stability of the transforms. The decomposition is of linear computational complexity, with just a slightly larger constant than the fast lifted wavelet transform. This paper also discusses several alternatives in the design of non-stationary finite impulse response filters for a stable multiresolution smoothing system. These filters are adapted to each other and to the locations of the observations.