Wavelet transforms for vector fields using omnidirectionally balanced multiwavelets

Abstract

Vector wavelet transforms for vector-valued fields can be implemented directly from multiwavelets; however, existing multiwavelets offer surprisingly poor performance for transforms in vector-valued signalprocessing applications. In this paper, the reason for this performance failure 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 scalarsignal 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 omnidirectionally balanced vector wavelet transform in which filter-coefficient matrices are constrained to be diagonal. Additionally, a family of symmetricantisymmetric multiwavelets is designed according to the omnidirectionalbalancing 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.

Matrix-Valued and Quaternion Wavelets

“…As in [16], [19], we will design the scaling filters {G k } by directly solving a set of design equations. One of the advantages of this approach is that we can use it to prove uniqueness and non-existence results.…”

“…This is the case in [8], [9], where only filters of length 3 are considered, in [12], [30] where ad-hoc constructions centered on the eigenvalues of the scaling filter transfer function are used, and in [13] where a symmetry condition is imposed. More general methods of designing MVWs include multichannel lifting schemes [2], [3], the spectral factorisation of interpolatory vector subdivision schemes [11], and directly solving the system of equations imposed by the design constraints [16], [19].…”

Wavelet-based coding of time-varying vector fields of ocean-surface winds

“…The simplest approach to creating a VWT is to use scalar-valued filters from a traditional DWT to create diagonal VWT matrix filters, e.g., diag , where is a scalarvalued DWT filter. In [5] and [6], we designed more general, nondiagonal VWT filters that also have good signal processing performance.…”

“…The reference frame is subtracted from the current frame to produce a residual frame, which is then subjected to frame-by-frame intraframe coding. For intraframe coding, we use the vector-valued successive-approximation runlength (VSARL) coder developed in [5] and [6], in which we first apply a full-frame VWT to the entire residual frame. Subsequently, vector quantization (VQ) using the successive-approximation VQ (SAVQ) of [7] is applied to the VWT coefficient vectors.…”

“…In our previous work [5], [6], we constructed an intraframe coder for static vector fields. In doing so, we designed a vector wavelet transform (VWT) to exploit the spatial correlation inherent in vector fields.…”