We presented a new algorithm for image compression that uses an adaptive joint space and frequency image expansion. The expansion is produced by building a graph that cascades both filter bank and segmentation operations. The nodes in the graph form a space and frequency library from which the best basis is selected. The library includes other expansions, such as wavelet, wavelet packet and double tree decompositions. The space and frequency graph offers the most rich expansion, and image compression performance increases when using the best basis selected from the space and frequency library.