Noise reduction in an adaptive multiscale local polynomial decomposition

This paper investigates the selection of coefficients in an adaptive multiscale local polynomial decomposition. The multiscale local polynomial (MLP) decomposition is a slightly overcomplete alternative for a critically downsampled fast wavelet transform. Thanks to the redundancy, the MLP transform reconciles numerically well conditioned analyses and syntheses with smooth reconstructions for data observed on irregular point sets on a real line. The MLP can also be seen as an extension of a Burt and Adelson´s Laplacian pyramid on irregular point sets, or as a new lifting scheme where the classical interpolating prediction is replaced by smoothing prediction, using statistical nonparametric estimation techniques. The MLP allows easy extension towards adaptive decompositions, but the adaptivity is incompatible with some of the design options for good numerical condition. This paper implements an adaptive decomposition based on techniques from statistical testing and investigates noise reduction within the adaptive scheme with special attention to the numerical condition of the decomposition.