As illustrated in Figures 10(b), the space and frequency graph explores the trade-off in the joint resolution in space and spatial-frequency. The data at each node of the graph corresponds to a particular spatial position and spatial-frequency range. Fundamentally, the joint resolution is limited by the uncertainty principle, and is bounded by the relations 
and 
where 
gives resolution in space and 
gives resolution in spatial-frequency. In the space and frequency graph, the frequency resolution is doubled with each filter bank operation and spatial resolution is doubled with each segmentation. Therefore, to answer the first question, the space and frequency graph includes only a quaternary hierarchy of decomposition in space and spatial-frequency. The graph cannot produce any arbitrary resolution in space and spatial-frequency even within the bounds of the uncertainty principle. This manifests itself physically in that an arbitrary segment cannot be extracted from either the spatial plane or the spatial-frequency plane. In other words, the segmentations must be made on quad-tree boundaries and spatial-frequency subbands are extracted as quad-tree regions in the spatial-frequency plane.