The multiple description rate region for high resolution source coding

Consider encoding a memoryless source using two descriptions, the first at rate R₁ and distortion d₁, the second at rate R₂ and distortion d₂. Combining the two descriptions the source can be reconstructed with distortion d₀ . For a Gaussian source of variance σ², Ozarow (1980) found an explicit characterization of the region R*(σ² ; d₁,d₂,d₀)⊂R² of achievable rate pairs (R₁, R₂) with given mean squared distortions d₁, d₂, and d₀ . 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 ρ(x,xˆ)=w(x)²(x-xˆ)²+o((x-xˆ) ²), the region of admissible rate pairs is arbitrary well approximated in the limit of small distortions by the region R*(P_X 2^{2E{log m(X)}}; d₁,d₂,d₀) where R*(σ²; d₁,₂, d₀) denotes the multiple description rate region of a Gaussian source with variance σ² , and where P_X is the entropy-power of the source. Applications to companding quantization are also considered