The phaselet transform-an integral redundancy nearly shift-invariant wavelet transform

This paper introduces an approximately shift invariant redundant dyadic wavelet transform the phaselet transform that includes the popular dual-tree complex wavelet transform of Kingsbury [1] as a special case. The main idea is to use a finite set of wavelets that are related to each other in a special way and hence called phaselets to achieve approximate shift-redundancy; bigger the set better the approximation. A sufficient condition on the associated scaling filters to achieve this is that they are fractional shifts of each other. Algorithms for the design of phaselets with a fixed number vanishing moments is presented building upon the work of Selesnick [2] for the design of wavelet pairs for Kingsbury’s dual-tree complex wavelet transform. Construction of 2-dimensional directional bases from tensor products of 1-d phaselets is also described. Phaselets as a new approach to redundant wavelet transforms and their construction are both novel and should be interesting to the reader independently of the approximate shift invariance property that this paper argues they possess.