Tree-structured filter banks for M-block cyclic graphs

In this paper, we study the design of graph wavelet filter banks over M-block cyclic graphs. These graphs are natural directed extensions of bipartite graphs and their special structure is particularly suitable for the design of M-channel filter banks. Obtaining polynomial filter designs in this case that satisfy perfect reconstruction conditions is challenging since the Fourier domain of these graphs encompasses the entire complex-unit disc unlike just the complex unit-circle in the classical domain. Therefore, in this work, we consider a simpler setting where M is a power of 2 and propose a perfect reconstruction tree-structured biorthogonal filter bank solution comprised of a hierarchical 2-channel design. This approach significantly simplifies the design process by requiring the design of only one 2-channel filter bank for a directed bipartite graph, and repeating it across the hierarchy.