ABSTRACT
Just like B-spline wavelets in Chapter 4, the multiscale local polynomial transform (MLPT) in Chapter 11, lifts a well established uniscale smooth- ing method into a multiscale analysis. Since local polynomial smoothing works with kernel functions, rather than basis functions, the multiscale con- struction is different than that of B-spline wavelets. An important feature of the MLPT is the slight but necessary redundancy, realised in a variation of the lifting scheme known as the Laplacian pyramid. It is explained that the redundancy is needed to obtain smooth reconstructions. Fortunately, it offers the benefit of efficient variance reduction, which seems to out- perform even the nondecimated wavelet transform in that respect, while the redundancy in a MLPT is only by a constant factor of two as opposed to a logarithmic factor in the case of a nondecimated wavelet transform. Whereas in uniscale kernel based smoothing, the bandwidth is a smooth- ing parameter, its role in the MLPT is different and twofold. First, it defines the scale of each resolution level, which in the MLPT, unlike in the wavelet transform, is a nondyadic, even continuous and user controlled notion. Second, it defines adjacency, allowing the MLPT to proceed on scattered data without multiscale triangulations.
