Exponential-Golomb coding: Difference between revisions

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{{distinguish|Golomb coding}}

An '''exponential-Golomb code''' (or just '''Exp-Golomb code''') is a type of [[Universal code (data compression)|universal code]]. To encode any [[nonnegative integer]] ''x'' using the exp-Golomb code:

# Write down ''x''+1 in binary

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In the above examples, consider the case 3. For 3, x+1 = 3 + 1 = 4. 4 in binary is '100'. '100' has 3 bits, and 3-1 = 2. Hence add 2 zeros before '100', which is '00100'

Similarly, consider 8. '8 + 1' in binary is '1001'. '1001' has 4 bits, and 4-1 is 3. Hence add 3 zeros before 1001, which is '0001001'.

This is identical to the [[Elias gamma code]] of ''x''+1, allowing it to encode 0.<ref>{{cite book |last = Rupp |first = Markus |title = Video and Multimedia Transmissions over Cellular Networks: Analysis, Modelling and Optimization in Live 3G Mobile Networks |year = 2009 |publisher = Wiley |pages = 149 |isbn = 9780470747766 |url = https://books.google.com/books?id=H9hUBT-JvUoC&q=Exponential-Golomb+coding&pg=PA149 }}</ref>

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4 ⇒ 7 ⇒ 1000 ⇒ 0001000

−4 ⇒ 8 ⇒ 1001 ⇒ 0001001

...<ref name="richardson">{{cite book |last = Richardson |first = Iain |title = The H.264 Advanced Video Compression Standard |year = 2010 |publisher = Wiley |isbn = 978-0-470-51692-8 |pages = 208, 221 |url = https://books.google.com/books?id=LJoDiPnBzQ8C&q=Exponential-Golomb+coding&pg=PA221 }}</ref>

In other words, a non-positive integer ''x''≤0 is mapped to an even integer −2''x'', while a positive integer ''x''&gt;0 is mapped to an odd integer 2''x''−1.

Exp-Golomb coding is also used in the [[Dirac (video compression format)|Dirac video codec]].<ref>{{cite web |title = Dirac Specification |url = http://diracvideo.org/download/specification/dirac-spec-latest.pdf |publisher = BBC |access-date = 9 March 2011 |url-status = usurped |archive-url = https://web.archive.org/web/20150503015104/http://diracvideo.org/download/specification/dirac-spec-latest.pdf |archive-date = 2015-05-03 }}</ref>

==Generalization to order ''k''==

To encode larger numbers in fewer bits (at the expense of using more bits to encode smaller numbers), this can be generalized using a [[nonnegative integer]] parameter &nbsp;''k''. To encode a nonnegative integer ''x'' in an order-''k''&nbsp;exp-Golomb code:

# Encode ⌊''x''/2^''k''⌋ using order-0 exp-Golomb code described above, then

# Encode ''x'' mod 2^''k'' in binary with k bits

An equivalent way of expressing this is:

# Encode ''x''+2^''k''−1 using the order-0 exp-Golomb code (i.e. encode ''x''+2^''k'' using the Elias gamma code), then

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{{DEFAULTSORT:Exponential-Golomb Coding}}

[[Category:Entropy coding]]

[[Category:Numeral systems]]

[[Category:LosslessData compression algorithms]]