Non-Separable Extensions of Quadrature Mirror Filters to Multiple Dimensions

Quadrature Mirror Filter (QMF) banks have been used in a variety of one-dimensional signal processing applications, and have been applied separably in two dimensions. As with most one-dimensional filters, separable extension to multiple dimensions produces a transform in which the orientation selectivity of some of the high-pass filters is poor. We describe generalized non-separable extensions of QMF banks to two and three dimensions, in which the orientation specificity of the high-pass filters is greatly improved. In particular, we discuss extensions to two dimensions with hexagonal symmetry, and three dimensional spatio-temporal extensions with rhombic-dodecahedral symmetry. Although these filters are conceived and designed on non-standard sampling lattices, they may be applied to rectangularly sampled images. As in one dimension, these transformations may be hierarchically cascaded to form a multi-scale “pyramid” representation. We design a set of example filters and apply them to the problems of image compression, progressive transmission, orientation analysis, and motion analysis.