Adaptive Orthogonal Matrix-valued Wavelets and Compression of Vector-valued Signals

Wavelet transforms using matrix-valued wavelets (MVWs) can process the components of vector-valued signals jointly, and thus offer potential advantages over scalar wavelets. For every matrix-valued scaling filter, there are infinitely many matrix-valued wavelet filters corresponding to rotated bases. We show how the arbitrary orthogonal factor in the choice of wavelet filter can be selected adaptively with a modified SIMPLIMAX algorithm. The 3×3 orthogonal matrix-valued scaling filters of length 6 with 3 vanishing moments have one intrinsic free scalar parameter in addition to three scalar rotation parameters. Tests suggest that even when optimising over these parameters, no significant improvement is obtained when compared to the naive scalar-based filter. We have found however in an image compression test that, for the naive scaling filter, adaptive basis rotation can decrease the RMSE by over 20%.