General Reconstruction of Laplacian Pyramid and its Dual Frame Solutions

The Laplacian pyramid (LP) is a useful tool in image processing and computer vision. It provides an overcomplete signal representation, thus can be treated as a frame operator. In this paper, we first propose a lifting-based factorization for the LP decomposition. Based on such lifting factorization, a parameterization reconstruction algorithm is presented to characterize all synthesis banks yielding the perfect reconstruction property. Compared with other LP reconstruction algorithms in the literature, our proposed reconstruction contains much less number of parameters - the reduction is M times for an LP with the decimation factor of M. For the proposed reconstruction algorithm, noise analysis of the LP is carried out, and the closed-form solutions of the dual frame for any LP are also presented.