Improved bounds for the rate loss of multiresolution source codes
In this paper, we present new bounds for the rate loss of multiresolution source codes (MRSCs). Considering an M-resolution code, the rate loss at the ith resolution with distortion D-i is defined as L-i = R-i - R(D-i), where R-i is the rate achievable by the MRSC at stage i. This rate loss describes the performance degradation of the MRSC compared to the best single-resolution code with the same distortion. For two-resolution source codes, there are three scenarios of particular interest: i) when both resolutions are equally important; ii) when the rate loss at the first resolution is 0 (L-1 = 0); iii) when the rate loss at the second resolution is 0 (L-2 = 0). The work of Lastras and Berger gives constant upper bounds for the rate loss of an arbitrary memoryless source in scenarios i) and ii) and an asymptotic bound for scenario iii) as D-2 approaches 0. In this paper, we focus on the squared error distortion measure and a) prove that for scenario iii) L-1 < 1.1610 for all D-2 < D-1; b) tighten the Lastras-Berger bound for scenario ii) from L-2 less than or equal to 1 to L-2 < 0.7250; c) tighten the Lastras-Berger bound for scenario i) from L-i less than or equal to 1/2 to L-i < 0.3802, i is an element of {1, 2}; and d) generalize the bounds for scenarios ii) and iii) to M-resolution codes with M greater than or equal to 2. We also present upper bounds for the rate losses of additive MRSCs (AMRSCs). An AMRSC is a special MRSC where each resolution describes an incremental reproduction and the kth-resolution reconstruction equals the sum of the first k incremental reproductions. We obtain two bounds on the rate loss of AMRSCs: one primarily good for low-rate coding and another which depends on the source entropy.
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