Notes on Conditions for Successive Refinement of Information

The source divisibility [V. N. Koshelev, "Hierarchical coding of discrete sources", Problemy peredachi informatsii 16 (3) (1980) 31–49], [V. N. Koshelev, "An evaluation of the average distortion for discrete scheme of sequential approximation", Problemy peredachi informatsii 17 (3) (1981) 20–30], alternatively, the successive refinement of information [W. H. R. Equitz and T. M. Cover, "Successive refinement of information", IEEE Trans. on Inform Theory IT–37 (2) (March 1991) 269–275] under error exponent [A. Kanlis and P. Narayan, "Error exponents for successive refinement by partitioning", IEEE Trans. Inform. Theory 42 (January 1996) 275–282] or reliability [E. A. Haroutunian and A. N. Harutyunyan, "Successive refinement of information with reliability criterion", in Proc. IEEE Int. Symp. Inform. Theory, Sorrento, Italy, June 2000, p. 205] constraint is discussed. The result by Tuncel and Rose [E. Tuncel, K. Rose, "Error exponents in scalable source coding", IEEE Trans. Inform. Theory 49 (1) (January 2003) 289–296] on conditions of successive refinability in distortion-reliability pair contradicting to our earlier work [E. A. Haroutunian and A. N. Harutyunyan, "Successive refinement of information with reliability criterion", in Proc. IEEE Int. Symp. Inform. Theory, Sorrento, Italy, June 2000, p. 205] is restated. The proof employs the rate-reliability-distortion region for hierarchical source coding [R. Sh. Maroutian, "Achievable rates for multiple descriptions with given exponent and distortion levels", Problemy peredachi informatsii, (in Russian) 26 (1) (1990) 83–89] (in particular resulting in the rate-distortion region of [V. N. Koshelev, "Hierarchical coding of discrete sources", Problemy peredachi informatsii 16 (3) (1980) 31–49] and [B. Rimoldi, "Successive refinement of information: Characterization of the achievable rates", IEEE Trans. on Inform Theory IT–40 (1) (January 1994) 253–259]) with a correcting representation. Specializing the concept of successive refinability in "purely" reliability sense (zero-distortion case), it is noted also that such a property always holds for any discrete memoryless source.