On the rate loss of multiple description source codes and additive successive refinement codes

The rate loss of a multi-resolution source code (MRSC) describes the difference between the rate needed to achieve distortion D_i in resolution i and the rate-distortion function R(D_i). We generalize the rate loss definition and bound the rate losses of multiple description source codes (MDSCs) and additive MRSCs (AMRSCs). For a 2-description MDSC (2DSC), the rate loss of description i with distortion D_i is defined as L_i=R_i-R(D_i), i=1, 2, where R_i is the rate of the ith description; the rate loss associated with decoding the two descriptions together to achieve central distortion D₀ is measured as L₀=R₁+R₂-R(D₀) or as L₁₂=L₁+L₂. We show that given an arbitrary source with variance σ², there exists a 2DSC with L₁≤0.5 and (a) L₀≤1 if D₀≤D₁+D₂-σ², (b) L₁₂≤1 if 1/D₀≤1/D₁+1/D₂-1/σ², (c) L₀≤L_G0+1.5 and L₁₂≤L_G12+1 otherwise, where L_G0 and L_G12 are the joint rate losses of a normal (0, σ²) source. An AMRSC is an MRSC with the kth-resolution reconstruction equal to the sum of the first k side reproductions of an MDSC. We obtain one bound on the rate loss of an AMRSC.