The space and frequency graph provides a very convenient structure for the joint decomposition of the image in space and spatial-frequency. It requires only the basic building blocks of a two-channel filter bank and binary segmentation. The graph is a natural mechanism for accessing and indexing the nodes in the joint expansion. But several interesting questions remain, such as, can a more complete joint decomposition be found and what is the physical significance to each node in the graph? To answer these questions we analyze more closely the expansion produced by the space and frequency graph.