As mentioned above, the building blocks for the joint space and frequency decomposition consist of expansions using the two-channel filter bank and binary segmentation. Since we are developing the compression system for images, which are 2-D signals, we note here that we augment these basic building blocks for image expansion. For spatial-frequency decomposition, we utilize a 2-D signal, 4-channel filter bank by cascading two 2-channel filter banks and by transposing filter channel outputs. For segmentation, we utilize a quad-tree spatial segmentation by cascading binary segmentations and by transposing the outputs of the segmentation operators. In other words, the 2-D filtering and segmentation used here are separable and composed of independent operations on the rows and columns of the image. In many of the descriptions and illustrations that follow, a 1-D signal is shown for simplicity. However, the results apply directly to the 2-D image case when using the augmented building blocks. We now present high-level diagrams that describe the joint signal expansion. The notation in Figure 6 is used to illustrate the joint expansions in space and spatial-frequency. As shown in Figure 6, each straight-line in the high-level notation corresponds to a tree.
Figure 6: High-level notation for expansions (a) tree-structured filter bank, (b) hierarchical segmentation.