On Optimal Multi-resolution Scalar Quantization

• Authors: Xiaolin Wu;Sorina Dumitrescu
• Venue: DCC '02 Proceedings of the Data Compression Conference
• Year: 2002

Any scalar quantizer of 2h bins, where h is a positive integer, can be structured by a balanced binary quantizer tree T of h levels. Any pruned subtree T of T corresponds to an operational rate R(T) and distortion D(T) pair. Denote by Sn the set of all pruned subtrees of n leaf nodes, 1 n2h. We consider the problem of designing a 2h-bin scalar quantizer that minimizes the weighted average distortion D(T)W(N), where W(N) is a weighting function in the size of pruned subtrees (or the resolution of the underlying quantizer). We present an O(hN3) algorithm to solve the underlying optimization problem (N is the number of points of the histogram that represents the source probability mass function), and call the resulting quantizer optimal multi-resolution scalar quantizer in the sense that it minimizes a global distortion measure averaged over all quantization resolutions of T. Interestingly, a similar quantizer design problem studied by Brunk et al. [1] is a special case of our formulation, and can thus be solved exactly and efficiently using our algorithm.Furthermore, we present an algorithm to generate a sequence of 2h nested pruned subtrees of T, from the root of T to the entire tree T itself, which minimizes an expected distortion over a range of operational rates. The resulting nested pruned subtree sequence generates an optimized embedded (rate-distortion scalable) code stream with maximum granularity of 2h quantization stages, as opposed to existing successively refinable quantizers, such as the popular bit-plane coding scheme, which offer only h stages.