We proposed an improvement of the double tree that treats the space and frequency operations symmetrically and results in a graph structured cascade [SC95]. Both the double tree and its dual are embedded in the space and frequency graph. The graph structured decomposition provides a more complete joint space and frequency expansion of the signal. Using the same number of nodes as the double tree library, the space and frequency graph greatly increases the number of accessible bases that represent the signal. The space and frequency graph will be described in more detail in the following sections.