Golomb coding: Difference between revisions

Revision as of 03:05, 12 July 2021

Revision as of 03:55, 12 July 2021

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[[File:GolombCodeRedundancy.svg|thumb|upright 1.5|This image shows the redundancy, in bits, of the Golomb code, when {{mvar|M}} is chosen optimally, for {{math| 1 − ''p''(0) ≥ 0.45}}]]

Both {{mvar|q}} and {{mvar|r}} will be encoded using variable numbers of bits –: {{mvar|q}} by a unary code, and {{mvar|r}} by {{mvar|b}} bits for Rice code, or a choice between {{mvar|b}} and {{mvar|b}}+1 bits for Golomb code (i.e. {{mvar|M}} is not a power of 2). Let <math>b = \lfloor\log_2(M)\rfloor</math>. If <math>r < 2^{b+1} - M</math>, then use {{mvar|b}} bits to encode {{mvar|r}}; otherwise, use {{mvar|b}}+1 bits to encode {{mvar|r}}. Clearly, <math>b=\log_2(M)</math> if {{mvar|M}} is a power of 2 and we can encode all values of {{mvar|r}} with {{mvar|b}} bits.

The integer {{mvar|x}} treated by Golomb was the run length of a [[Bernoulli process]], which has a [[geometric distribution]] starting at 0. The best choice of parameter {{mvar|M}} is a function of the corresponding Bernoulli process, which is parameterized by <math>p = P(X=0)</math> the probability of success in a given [[Bernoulli trial]]. {{mvar|M}} is either the median of the distribution or the median +/− 1. It can be determined by these inequalities: