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Exponential-Golomb coding
Type of universal code, parameterized by a nonnegative integer

An exponential-Golomb code (or just Exp-Golomb code) is a type of universal code. To encode any nonnegative integer x using the exp-Golomb code:

  1. Write down x+1 in binary
  2. Count the bits written, subtract one, and write that number of starting zero bits preceding the previous bit string.

The first few values of the code are:

0 ⇒ 1 ⇒ 1 1 ⇒ 10 ⇒ 010 2 ⇒ 11 ⇒ 011 3 ⇒ 100 ⇒ 00100 4 ⇒ 101 ⇒ 00101 5 ⇒ 110 ⇒ 00110 6 ⇒ 111 ⇒ 00111 7 ⇒ 1000 ⇒ 0001000 8 ⇒ 1001 ⇒ 0001001 ...1

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.

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Extension to negative numbers

Exp-Golomb coding is used in the H.264/MPEG-4 AVC and H.265 High Efficiency Video Coding video compression standards, in which there is also a variation for the coding of signed numbers by assigning the value 0 to the binary codeword '0' and assigning subsequent codewords to input values of increasing magnitude (and alternating sign, if the field can contain a negative number):

0 ⇒ 0 ⇒ 1 ⇒ 1 1 ⇒ 1 ⇒ 10 ⇒ 010 −1 ⇒ 2 ⇒ 11 ⇒ 011 2 ⇒ 3 ⇒ 100 ⇒ 00100 −2 ⇒ 4 ⇒ 101 ⇒ 00101 3 ⇒ 5 ⇒ 110 ⇒ 00110 −3 ⇒ 6 ⇒ 111 ⇒ 00111 4 ⇒ 7 ⇒ 1000 ⇒ 0001000 −4 ⇒ 8 ⇒ 1001 ⇒ 0001001 ...3

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

Exp-Golomb coding is also used in the Dirac video codec.4

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  k. To encode a nonnegative integer x in an order-k exp-Golomb code:

  1. Encode ⌊x/2k⌋ using order-0 exp-Golomb code described above, then
  2. Encode x mod 2k in binary with k bits

An equivalent way of expressing this is:

  1. Encode x+2k−1 using the order-0 exp-Golomb code (i.e. encode x+2k using the Elias gamma code), then
  2. Delete k leading zero bits from the encoding result
Exp-Golomb-k coding examples
 x k=0k=1k=2k=3 x k=0k=1k=2k=3 x k=0k=1k=2k=3
011010010001000010110011000111001001020000010101000101100011000011100
10101110110011100011000011010111101001121000010110000101110011001011101
201101001101010120001101001110001000001010022000010111000110000011010011110
30010001011111011130001110001111001000101010123000011000000110010011011011111
40010101100100011001400011110001000000100100101102400001100100011010001110000100000
5001100111010011101150000100000001000100100110101112500001101000011011001110100100001
600111001000010101110160000100010001001000101000110002600001101100011100001111000100010
70001000001001010111111170000100100001001100101010110012700001110000011101001111100100011
800010010010100110001000018000010011000101000010110011010280000111010001111000010000000100100
900010100010110110101000119000010100000101010010111011011290000111100001111100010000100100101

See also

References

  1. Richardson, Iain (2010). The H.264 Advanced Video Compression Standard. Wiley. pp. 208, 221. ISBN 978-0-470-51692-8. 978-0-470-51692-8 ↩

  2. Rupp, Markus (2009). Video and Multimedia Transmissions over Cellular Networks: Analysis, Modelling and Optimization in Live 3G Mobile Networks. Wiley. p. 149. ISBN 9780470747766. 9780470747766 ↩

  3. Richardson, Iain (2010). The H.264 Advanced Video Compression Standard. Wiley. pp. 208, 221. ISBN 978-0-470-51692-8. 978-0-470-51692-8 ↩

  4. "Dirac Specification" (PDF). BBC. Archived from the original on 2015-05-03. Retrieved 9 March 2011. https://web.archive.org/web/20150503015104/http://diracvideo.org/download/specification/dirac-spec-latest.pdf ↩