Orthogonal Wavelet Frames and Vector-valued Wavelet Transforms

Motivated by the notion of orthogonal frames, we describe sufficient conditions for the construction of orthogonal MRA wavelet frames in L(R) from a suitable scaling function. These constructions naturally lead to filter banks in `(Z) with similar orthogonality relations and, through these filter banks, the orthogonal wavelet frames give rise to a vector-valued discrete wavelet transform (VDWT). The novelty of these constructions lies in their potential for use with vector-valued data, where the VDWT seeks to exploit correlation between channels. Extensions to higher dimensions are natural and the constructions corresponding to the bidimensional case are presented along with preliminary results of numerical experiments in which the VDWT is applied to color image data.