A shift-invariant multiscale multidirection image decomposition

This paper presents a shift-invariant complex directional pyramid transform constructed by a dual-tree pyramidal directional filter banks (DFB). The double filter bank framework consists of a shift-invariant Laplacian pyramid and a dual-tree DFB. The two binary tree structure for the (primal and dual) DFBs employed in the structure are identical except for the filter bank employed at the second level of the dual DFB, where special conditions on the phase of the filters are required. It is proven analytically and experimentally that each pair of corresponding directional filters produced by the primal and dual filter banks are symmetric and anti-symmetric, which can be interpreted as the real and imaginary parts of a complex filter. Therefore, the two subband coefficients can be viewed as the real and imaginary parts of a complex-valued subband image. It is proven that there is no aliasing in the decimated complex-valued signal, which implies that the system is shift-invariant in the energy sense. In addition, the proposed shift-invariant, multiscale, multidirectional image decomposition has two unique characteristics that other shiftinvariant decompositions do not possess. First, the directional resolution of the image transform can be arbitrarily high. Secondly, the two-dimensional filter bank is implemented in a separable fashion, which makes the entire structure very computational efficient.