Subband Image Coding with Hexagonal Quadrature Mirror Filters

Subband coding schemes, originally developed for the compression of one-dimensional signals, are becoming increasingly popular as a means of compressing images and image sequences. In particular, critically-sampled lterbank systems using Quadrature Mirror lters have been found to be quite eeective 1, 2, 3]. In one dimension, a pair of QMF lters is formed by modulating and shifting a single baseband lter, H(!). The formation of the highpass lter through shifts and modulations ensures that the system aliasing (produced when the subband signals are subsampled) will cancel when the signal is reconstructed. The remaining linear shift-invariant (LSI) system response may be written in terms of the baseband lter: 1 2 h jH (!)j 2 + jH (! +)j 2 i : (1) The advantages of these systems are many. A QMF system provides an orthogonal subband decomposition without the arbitrary block boundaries of the blocked DCT. Moreover , the ringing and computational ineeciency of a sharp subband (sinc lter) system are reduced through the use of relatively small lters with soft transitions (this is possible due to the cancellation of the overall system aliasing). The lter design process is simpliied by the relationships between the various subband lters. Since the system aliasing cancels, the only constraint on the baseband lter H(!) is that expression (1) have a value of unity for all !. Finally, these systems may be cascaded uniformly or non-uniformly (to form pyramid structures). Unlike traditional Gaussian or Laplacian pyramids, the representation is not overcomplete and may contain oriented subbands. We have been exploring the use of nonseparable QMF's with hexagonal symmetry. The basic framework is outlined in 4], and a more thorough description may be found in 5]. On a hexagonal sampling lattice system it is possible to construct a pyramid decomposition in which all of the subbands are selective for orientation { something that is not possible with the separable systems that are commonly used. An idealized illustration of the frequency subdivision performed by such a system is given in gure 1. The hexagonal QMF's are strictly analogous to the one-dimensional case, in that the subband lters are formed by shifting and modulating a baseband lter. The system aliasing cancels exactly, and the remaining LSI system response is in the form 1 4 3 X i=0 jH (! + ~ i)j 2 ; (2)