Redirected from Arithmetic encoding
Arithmetic coding in data compression is the process of encoding a stream of data into a large binary fraction. It can achieve nearoptimal entropy encoding.
Arithmetic coding actually refers to half of an Arithmetic Coding data compression system.
It has two parts:
At any point in time in coding the input stream of data,
The number of bits used to encode a character depends on how much of the range (0, 1) the character's probability is.
The larger the range, the less bits it takes to code the character. The smaller the range, the more bits it takes to code the character.
Typically, the model used to code the data changes based on the data input stream contents. This is known as adaptive coding[?].
The following text has been moved from Arithmetic encoding:
An arithmetic encoder takes a string of symbols as input and produces a rational number in the interval [0, 1) as output. As each symbol is processed, the encoder will restrict the output to a smaller interval.
Let N be the number of distinct symbols in the input; let x_{1}, x_{2} ... x_{N} represent the symbols, and let P_{1}, P_{2} ... P_{N} represent the probability of each symbol appearing. At each step in the process, the output is restricted to the current interval [y, y+R). Partition this interval into N disjoint subintervals:
Therefore the size of I_{i} is P_{i}R. If the next symbol is x_{i}, then restrict the output to the new interval I_{i}.
Note that at each stage, all the possible intervals are pairwise disjoint. Therefore a specific sequence of symbols produces exactly one unique output range, and the process can be reversed.
Since arithmetic encoders are typically implemented on binary computers, the actual output of the encoder is generally the shortest sequence of bits representing the fractional part of a rational number in the final interval.
Suppose our entire input string contains M symbols: then x_{i} appears exactly P_{i}M times in the input. Therefore, the size of the final interval will be
By Shannon's theorem, this is the total entropy of the original message. Therefore arithmetic encoding is nearoptimal entropy encoding.
However, IBM and other companies own patents in the United States and other countries on algorithms essential for implementing an arithmetic encoder. But are those patent holders willing to license the patents royaltyfree for use in opensource software?
An earlier version of the above article was posted on PlanetMath (http://planetmath.org/encyclopedia/ArithmeticEncoding). This article is open content
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