Existence and design of biorthogonal matrix-valued wavelets

Biorthogonal matrix-valued wavelets have been employed to analyse matrix-valued signals based on matrix multiresolution analysis.The sufficient condition for existence of a biorthogonal matrix-valued scaling function has been established in terms of the corresponding two-scale matrix symbols.Two designs based on factorization of biorthogonal two-scale matrix symbols are presented.In particular,explicit constructing formulations for biorthogonal matrix-valued wavelets are given.With these formulations,highpass filters of biorthogonal matrix-valued wavelets can be given explicitly by lowpass filters.Examples of two-scale matrix filter banks are given.