Existence and design of biorthogonal matrix-valued wavelets

In this paper, we study biorthogonal matrix-valued wavelets for analyzing matrix-valued signals based on matrix multiresolution analysis. Firstly, sufficient conditions for the existence of biorthogonal matrix-valued scaling function are established in terms of two-scale matrix symbols. Then we focus on the construction of biorthogonal matrix-valued wavelet. Two designs based on factorizations of biorthogonal two-scale matrix symbol are presented. In particular, explicit constructing formulations for biorthogonal matrix-valued wavelets are given. With these formulations, highpass filters {Gk}{Gk} and {G̃k} of biorthogonal matrix-valued wavelets can be given explicitly by lowpass filters {Hk}{Hk} and {H̃k} of their corresponding biorthogonal scaling functions. Finally, according to our designs, two examples of two-scale matrix filters are given.