Phaselets of framelets

Phaselets are a set of dyadic wavelets that are related in a particular way such that the associated redundant wavelet transform is nearly shift-invariant. Framelets are a set of functions that generalize the notion of a single dyadic wavelet in the sense that dyadic dilates and translates of these functions form a frame in L²(IR). This paper generalizes the notion of phaselets to framelets. Sets of framelets that only differ in their Fourier transform phase are constructed such that the resulting redundant wavelet transform is approximately shift invariant. Explicit constructions of phaselets are given for frames with two and three framelet generators. The results in this paper generalize the construction of Hilbert transform pairs of framelets.