Fixed-slope universal lossy data compression

Corresponding to any lossless codeword length function l, three universal lossy data compression schemes are presented: one is with a fixed rate, another is with a fixed distortion, and a third is with a fixed slope. The former two universal lossy data compression schemes are the generalization of Yang-Kieffer´s (see ibid., vol.42, no.1, p.239-45, 1995) results to the general case of any lossless codeword length function l, whereas the third is new. In the case of fixed-slope λ>0, our universal lossy data compression scheme works as follows: for any source sequence xⁿ of length n, the encoder first searches for a reproduction sequence yⁿ of length n which minimizes a cost function n^-1l(yⁿ)+λρ_n(xⁿ, yⁿ) over all reproduction sequences of length n, and then encodes xⁿ into the binary codeword of length l(yⁿ) associated with yⁿ via the lossless codeword length function l, where ρ_n(xⁿ, yⁿ) is the distortion per sample between xⁿ and yⁿ. Under some mild assumptions on the lossless codeword length function l, it is shown that when this fixed-slope data compression scheme is applied to encode a stationary, ergodic source, the resulting encoding rate per sample and the distortion per sample converge with probability one to R_λ and D_λ, respectively, where (D_λ, R_λ) is the point on the rate distortion curve at which the slope of the rate distortion function is -λ. This result holds particularly for the arithmetic codeword length function and Lempel-Ziv codeword length function. The main advantage of this fixed-slope universal lossy data compression scheme over the fixed-rate (fixed-distortion) universal lossy data compression scheme lies in the fact that it converts the encoding problem to a search problem through a trellis and then permits one to use some sequential search algorithms to implement it. Simulation results show that this fixed-slope universal lossy data compression scheme, combined with a suitable search algorithm, is promising