Dual Framelets Transform on Manifolds and Graphs

Abstract

In this paper, the concept of dual framelets on manifolds and its characterization are introduced. The accuracy of the proposed dual framelets transform is determined by sparse representation on graphs. If any pair of the framelet system is associated with filter-bank transform, then compactly supported refinable functions can have vanishing moments at most one and framelet approximation is the order of at most two. An algorithm of decomposition and reconstruction for the dual framelets transform on graph is presented. A new method called dual framelets filter-bank transform (DFFT) is employed, which is faster than the existing method spectral graph wavelet transform (SGWT). The theoretical results along with algorithms for accurate and efficient computation of the DFFT on discrete data sets are provided. Subsequently, some numerical examples are provided to show the importance of DFFT over SGWT on graphs.