Properties of Laplacian Pyramids for Extension and Denoising

This paper analyzes the Laplacian pyramids (LP) algorithm for extending a function sampled on a discrete set of points to outside values. This method was introduced in the context of machine learning by Rabin and Coifman in [20], and is modeled after the classical Laplacian pyramids algorithm of Burt and Adelson [10], which is a standard technique in image processing. The LP extension algorithm has been considered in a variety of applications [14, 11, 24, 1, 18, 12, 2], and several variants have been proposed [15, 21, 22]. The LP algorithm constructs a multiscale decomposition of the estimated function, consisting of averaged differences at successive levels. At each level, the residuals from the previous approximation are averaged and extended to the entire domain.