The multiple description rate region for high resolution source coding

Consider encoding a memoryless source using two descriptions, the first at rate R/sub 1/ and distortion d/sub 1/, the second at rate R/sub 2/ and distortion d/sub 2/. Combining the two descriptions the source can be reconstructed with distortion d/sub 0/. For a Gaussian source of variance /spl sigma//sup 2/, Ozarow (1980) found an explicit characterization of the region R*(/spl sigma//sup 2/; d/sub 1/,d/sub 2/,d/sub 0/)/spl sub/R/sup 2/ of achievable rate pairs (R/sub 1/, R/sub 2/) with given mean squared distortions d/sub 1/, d/sub 2/, and d/sub 0/. This is the only case for which the multiple description rate-distortion region is completely known. We show that for a general real valued source X and a locally quadratic distortion measure of the form /spl rho/(x,x/spl circ/)=w(x)/sup 2/(x-x/spl circ/)/sup 2/+o((x-x/spl circ/)/sup 2/), the region of admissible rate pairs is arbitrary well approximated in the limit of small distortions by the region R*(P/sub X/2/sup 2E{log m(X)}/; d/sub 1/,d/sub 2/,d/sub 0/) where R*(/spl sigma//sup 2/; d/sub 1/,/sub 2/, d/sub 0/) denotes the multiple description rate region of a Gaussian source with variance /spl sigma//sup 2/, and where P/sub X/ is the entropy-power of the source. Applications to companding quantization are also considered.