On the redundancy of lossy source coding with abstract alphabets

The redundancy problem of lossy source coding with abstract source and reproduction alphabets is considered. For coding at a fixed rate level, it is shown that for any fixed rate R>0 and any memoryless abstract alphabet source P satisfying some mild conditions, there exists a sequence {C_n}_n=1^∞ of block codes at the rate R such that the distortion redundancy of C_n (defined as the difference between the performance of C_n and the distortion rate function d(P, R) of P) is upper-bounded by |(∂d(P,R))/(∂R)| ln n/2n+o(ln n/n). For coding at a fixed distortion level, it is demonstrated that for any d>0 and any memoryless abstract alphabet source P satisfying some mild conditions, there exists a sequence {C_n}_n=1^∞ of block codes at the fixed distortion d such that the rate redundancy of C_n (defined as the difference between the performance of C _n and the rate distortion function R(P,d) of P) is upper-bounded by (7ln n)/(6n)+o(ln n/n). These results strengthen the traditional Berger´s (1968, 1971) abstract alphabet source coding theorem, and extend the positive redundancy results of Zhang, Yang, and Wei (see ibid., vol.43, no.1, p.71-91, 1997, and ibid., vol.42, p.803-21, 1996) on lossy source coding with finite alphabets and the redundancy result of Wyner (see ibid., vol.43, p.1452-64, 1997) on block coding of memoryless Gaussian sources