The role of the coarsest resolution subband in the wavelet-based reconstruction of signals from gradients

Abstract

An efficient wavelet-based algorithm to reconstruct non-square/non-cubic signals from gradient data is proposed. This algorithm is motivated by applications such as image or video processing in the gradient domain. In some earlier approaches, the non-square/non-cubic gradients were extended to enable a square/cubic Haar wavelet decomposition and the coarsest resolution subband was derived from the mean value of the signal. In this paper, a non-square/non-cubic wavelet decomposition is obtained directly without extending the gradient data. The challenge comes from finding the coarsest resolution subband of the wavelet decomposition and an algorithm to compute this is proposed. The performance of the algorithm is evaluated in terms of accuracy and computation time, and is shown to outperform the considered earlier approaches in a number of cases. Further, a closer look on the role of the coarsest resolution subband coefficients reveals a trade-off between errors in reconstruction and visual quality which has interesting implications in image and video processing applications.