Fibonacci, Elias Gamma, and Elias Delta vs binary coding
Rice with k=2,3,4,5,8,16 vs binary In data compression, a universal code for integers is a prefix code that maps the positive integers onto binary codewords, with the additional property that whatever the true probability distribution on integers, as long as the distribution is monotonic (i.e.,  for all positive i), the expected lengths of the codewords are within a constant factor of the expected lengths that the optimal code for that probability distribution would have assigned. A universal code is asymptotically optimal if the ratio between actual and optimal expected lengths is bounded by a function of the information entropy of the code that, in addition to being bounded, approaches 1 as entropy approaches infinity. Image File history File links Download high resolution version (890x596, 7 KB) This image compares Fibonacci, Elias Gamma, and Elias Delta encoding schemes for bit usage. ...
Image File history File links Download high resolution version (890x596, 7 KB) This image compares Fibonacci, Elias Gamma, and Elias Delta encoding schemes for bit usage. ...
Image File history File links Download high resolution version (1033x704, 10 KB) This image compares the Rice Encoding Scheme using various parameters. ...
Image File history File links Download high resolution version (1033x704, 10 KB) This image compares the Rice Encoding Scheme using various parameters. ...
“Source coding†redirects here. ...
This article needs to be cleaned up to conform to a higher standard of quality. ...
In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ...
In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are...
In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are...
Claude Shannon In information theory, the Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable. ...
In general, most prefix codes for integers assign longer codewords to larger integers. Such a code can be used to efficiently communicate a message drawn from a set of possible messages, by simply ordering the set of messages by decreasing probability and then sending the index of the intended message. Universal codes are generally not used for precisely known probability distributions, and no universal code is known to be optimal for any distribution used in practice.
These are some universal codes for integers; an asterisk (*) indicates a code that can be trivially restated in lexicographical order, while a double dagger (‡) indicates a code that is asymptotically optimal: An asterisk (*) is a typographical symbol or glyph. ...
In mathematics, the lexicographic or lexicographical order, (also known as dictionary order, alphabetic order or lexicographic(al) product), is a natural order structure of the Cartesian product of two ordered sets. ...
A dagger (†, †, U+2020) is a typographical symbol or glyph. ...
- Elias gamma coding *
- Elias delta coding * ‡
- Elias omega coding * ‡
- Exp-Golomb coding *, which has Elias gamma coding as a special case. (Used in H.264/MPEG-4 AVC)
- Fibonacci coding ‡
- Levenstein (Левенштейн) coding * ‡, the original universal coding technique [1]
- Byte coding ‡, also known as comma coding, where a special bit pattern (with at least two bits) is used to mark the end of the code — for example, if an integer is encoded as a sequence of nibbles representing digits in base 15 instead of the more natural base 16, then the highest nibble value (i.e., a sequence of four ones in binary) can be used to indicate the end of the integer.
These are non-universal ones: Elias gamma code is a universal code encoding positive integers. ...
Elias delta code is a universal code encoding the positive integers. ...
Elias omega coding is a universal code encoding the positive integers. ...
An Exponential-Golomb code (or just Exp-Golomb code) of order is a type of universal code, parameterized by a whole number . ...
H.264 is a standard for video compression. ...
The Fibonacci code is a universal code which encodes positive integers into binary code words. ...
Levenstein coding is the original [1] universal code encoding the non-negative integers. ...
A nibble (or less commonly but more accurately, nybble) is the computing term for a four-bit aggregation, or half an octet (an octet being an 8-bit byte). ...
In mathematics, hexadecimal or simply hex is a numeral system with a radix or base of 16 usually written using the symbols 0–9 and A–F or a–f. ...
Their nonuniversality can be observed by noticing that, if any of these are used to code the Gauss-Kuzmin distribution or the Zeta distribution with parameter s=2, expected codeword length is infinite. For example, using unary coding on the Zeta distribution yields an expected length of Unary coding is an entropy encoding that represents a number n with n-1 ones followed by a zero. ...
Golomb coding is a form of entropy encoding invented by Solomon W. Golomb that is optimal for alphabets following geometric distributions, that is, when small values are vastly more common than large values. ...
FLAC, an acronym for Free Lossless Audio Codec, is a popular file format for audio data compression. ...
An audio codec is a computer program that compresses/decompresses digital audio data according to a given audio file format or streaming audio format. ...
Golomb coding is a form of entropy encoding invented by Solomon W. Golomb that is optimal for alphabets following geometric distributions, that is, when small values are vastly more common than large values. ...
In mathematics, the Gauss-Kuzmin distribution gives the probability distribution of the occurrence of a given integer in the continued fraction expansion of an arbitrary real number. ...
In probability theory and statistics, the zeta distribution is a discrete probability distribution. ...
On the other hand, using the universal Elias gamma coding for the Gauss-Kuzmin distribution results in an expected codeword length (about 3.51 bits) near entropy (about 3.43 bits)[2].
A universal code should not be confused with universal source coding, in which the data compression method need not be a fixed prefix code and the ratio between actual and optimal expected lengths must approach one. However, note that an asymptotically optimal universal code can be used on independent identically-distributed sources, by using increasingly large blocks, as a method of universal source coding. In probability theory, a sequence or other collection of random variables is independent and identically distributed (i. ...
The Railway Block Code is a system of bells used to send simple messages about train operations from one signalbox to another. ...
213.230.129.24 21:36, 30 September 2007 (UTC)Gerson== Relationship to practical compression == Huffman coding and arithmetic encoding (when they can be used) give at least as good, and often better compression than any universal code. In computer science and information theory, Huffman coding is an entropy encoding algorithm used for lossless data compression. ...
Arithmetic coding is a method for lossless data compression. ...
However, universal codes are useful when Huffman coding cannot be used — for example, when one does not know the exact probability of each message, but only knows the rankings of their probabilities.
Universal codes are also useful when Huffman codes are inconvenient. For example, when the transmitter but not the receiver knows the probabilities of the messages, Huffman coding requires an overhead of transmitting those probabilities to the receiver. Using a universal code does not have that overhead.
Each universal code, like each other self-delimiting (prefix) binary code, has its own "implied probability distribution" given by p(i)=2-l(i) where l(i) is the length of the ith codeword and p(i) is the corresponding symbol's probability. If the actual message probabilities are q(i) and Kullback–Leibler divergence DKL(q||p) is minimized by the code with l(i), then the optimal Huffman code for that set of messages will be equivalent to that code. Likewise, how close a code is to optimal can be measured by this divergence. Since universal codes are simpler and faster to encode and decode than Huffman codes (which is, in turn, simpler and faster than arithmetic encoding), the universal code would be preferable in cases where DKL(q||p) is sufficiently small. [3] In probability theory and information theory, the Kullback–Leibler divergence (or information divergence, or information gain, or relative entropy) is a natural distance measure from a true probability distribution P to an arbitrary probability distribution Q. Typically P represents data, observations, or a precise calculated probability distribution. ...
Arithmetic coding is a method for lossless data compression. ...
For any geometric distribution (an exponential distribution on integers), a Golomb code is optimal. With universal codes, the implicit distribution is approximately a power law such as 1 / n2. For the Fibonacci code, the implicit distribution is approximately 1 / nq, with In probability theory and statistics, the geometric distribution is either of two discrete probability distributions: the probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set { 1, 2, 3, ...}, or the probability distribution of the number Y = X − 1 of failures before...
See Also: Watt In physics, a power law relationship between two scalar quantities x and y is any such that the relationship can be written as where a (the constant of proportionality) and k (the exponent of the power law) are constants. ...
The Fibonacci code is a universal code which encodes positive integers into binary code words. ...
 where  is the golden ratio. For the ternary comma code (i.e., encoding in base 3, represented with 2 bits per symbol), the implicit distribution is a power law with  . These distributions thus have near-optimal codes with their respective power laws. Not to be confused with Golden mean (philosophy), the felicitous middle between two extremes, Golden numbers, an indicator of years in astronomy and calendar studies, or the Golden Rule. ...
The figures below compare Elias Gamma, Elias Delta, Fibonacci, and various Rice codings for bits used to encode integer values. A baseline, "direct binary", for binary encoding where the size is already known is also included.
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