# Non-Asymptotic Bounds and a General Formula for the Rate-Distortion Region of the Successive Refinement Problem.

In the successive refinement problem, a fixed-length sequence emitted from aninformation source is encoded into two codewords by two encoders in order togive two reconstructions of the sequence. One of two reconstructions isobtained by one of two codewords, and the other reconstruction is obtained byall two codewords. For this coding problem, we give non-asymptotic inner andouter bounds on pairs of numbers of codewords of two encoders such that eachprobability that a distortion exceeds a given distortion level is less than agiven probability level. We also give a general formula for the rate-distortionregion for general sources, where the rate-distortion region is the set of ratepairs of two encoders such that each maximum value of possible distortions isless than a given distortion level.

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