  In the past decade, several multi-resolution representation theories for
graph signals have been proposed. Bipartite filter-banks stand out as the most
natural extension of time domain filter-banks, in part because perfect
reconstruction, orthogonality and bi-orthogonality conditions in the graph
spectral domain resemble those for traditional filter-banks. Therefore, many of
the well known orthogonal and bi-orthogonal designs can be easily adapted for
graph signals. A major limitation is that this framework can only be applied to
the normalized Laplacian of bipartite graphs. In this paper we extend this
theory to arbitrary graphs and positive semi-definite variation operators. Our
approach is based on a different definition of the graph Fourier transform
(GFT), where orthogonality is defined with the respect to the Q inner product.
We construct GFTs satisfying a spectral folding property, which allows us to
easily construct orthogonal and bi-orthogonal perfect reconstruction
filter-banks. We illustrate signal representation and computational efficiency
of our filter-banks on 3D point clouds with hundreds of thousands of points.
