The shiftable complex directional pyramid - Part I: Theoretical aspects

This paper presents an over-complete multiscale decomposition by combining the Laplacian pyramid and the complex directional filter bank (DFB). The filter bank is constructed in such a way that each complex directional filter is analytical using the dual-tree structure of real fan filters. Necessary and sufficient conditions in order for the resulting multirate filter bank to be shift-invariant in energy sense (shiftablity) are derived in terms of the magnitude and phase responses of these filters. Their connection to 2-D Hilbert transform relationship is established. The proposed transform possesses several desirable properties including multiresolution, arbitrarily high directional resolution, low redundant ratio, and efficient implementation.