Multiscale kernel smoothing using a lifting scheme

This paper discusses the idea of a lifting scheme for multiscale implementation of kernel estimation procedures used in statistical estimation. The resulting decomposition is related to the Burt-Adelson pyramid, but the design of the filters is well adapted to nonequispaced samples. The proposed decomposition has an oversampling rate of 2, where the oversampling can be seen as an alternative to primal lifting steps (update steps) as a tool for stabilising and anti-aliasing. We then propose an adaptive version of this multiscale kernel estimation with truncated kernels. Truncated kernels allow sharp representations of jumps. Illustrations show that our method is numerically well conditioned, suffers less from visual effects due to false detections, and allows indeed sharp transitions if equiped with an adaptive choice among truncated kernels. All variants of the proposed method have linear computational complexity.