Wavelet transforms for vector fields using omnidirectionally balanced multiwavelets

Abstract—Vector wavelet transforms for vector-valued fields can be im-plemented directly from multiwavelets; however, existing multiwavelets of-fer surprisingly poor performance for transforms in vector-valued signal-processing applications. In this paper, the reason for this performance fail-ure is identified, and a remedy is proposed. A multiwavelet design criterion, omnidirectional balancing, is introduced to extend to vector transforms the balancing philosophy previously proposed for multiwavelet-based scalar-signal expansion. It is shown that the straightforward implementation of a vector wavelet transform, namely the application of a scalar transform to each vector component independently, is a special case of an omnidirec-tionally balanced vector wavelet transform in which filter-coefficient ma-trices are constrained to be diagonal. Additionally, a family of symmetric-antisymmetric multiwavelets is designed according to the omnidirectional-balancing criterion. In empirical results for a vector-field compression system, it is observed that the performance of vector wavelet transforms derived from these omnidirectionally-balanced symmetric-antisymmetric multiwavelets is far superior to that of transforms implemented via other multiwavelets and can exceed that of diagonal transforms derived from popular scalar wavelets