We formulate the compression of images as the following optimization problems: (1) joint optimization over all stages, and (2) joint optimization over stages 1-2. In addition, we utilize a new and extensive library to provide the large number of potential bases. The elementary building blocks of the adaptive image expansion are (1) the two-channel quadrature mirror filter bank (QMF) filter bank, and (2) a disjoint and complete binary segmentation. Cascading permutations of these two units produces the expansion of the image in space and spatial-frequency. The building blocks of the signal expansion system are described in the next section.