arithmetic coding - unisi.itdiligmic/bdm2011... · gabriele monfardini - corso di basi di dati...

Arithmetic Coding

Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a. 2005-2006 2

How we can do better than Huffman? - I

As we have seen, the main drawback of Huffman scheme is that it has problems when there is a symbol with very high probabilityRemember static Huffman redundancy bound

where is the probability of the most likely simbol

1redundancy 0.086p≤ +

1p


How we can do better than Huffman? - II

The only way to overcome this limitation is to use, as symbols, “blocks” of several characters.In this way the per-symbol inefficiency is spread over the whole block

However, the use of blocks is difficult to implement as there must be a block for every possible combination of symbols, so block number increases exponentially with their length


How we can do better than Huffman? - III

Huffman Coding is optimal in its framework

static model

one symbol, one word

adaptive Huffman

blocking

arithmetic coding


The key idea

Arithmetic coding completely bypasses the idea of replacing an input symbol with a specific code.

Instead, it takes a stream of input symbols and replaces it with a single floating point number in

The longer and more complex the message, the more bits are needed to represents the output number

[0,1)


The key idea - II

The output of an arithmetic coding is, as usual, a stream of bitsHowever we can think that there is a prefix 0, and the stream represents a fractional binary number between 0 and 1

In order to explain the algorithm, numbers will be shown as decimal, but obviously they are always binary

01101010 0110 00. 101→


An example - I

String bccb from the alphabet {a,b,c}Zero-frequency problem solved initializing at 1 all character countersWhen the first b is to be coded all symbols have a 33% probability (why?)

The arithmetic coder maintains two numbers, low and high, which represent a subinterval [low,high) of the range [0,1)Initially low=0 and high=1


An example - II

The range between low and high is divided between the symbols of the alphabet, according to their probabilities

1high

c(P[c]=1/3)

0.6667

b(P[b]=1/3)0.3333

a(P[a]=1/3)0low

9

An example - IIIb

1high

a

c

P[a]=1/4

P[b]=2/4

P[c]=1/4

new probabilities

high = 0.6667

low 0

0.3333

0.6667

b

low = 0.3333


An example - IVc

high = 0.6667

new probabilitiesP[a]=1/5P[b]=2/5P[c]=2/5

low 0.3333

0.4167

0.6667highc

0.5834

a

b

(P[c]=1/4)

(P[b]=2/4)

(P[a]=1/4)

low = 0.5834


An example - Vc

high = 0.6667

new probabilitiesP[a]=1/6P[b]=2/6P[c]=3/6

low 0.5834

0.6001

0.6667high

c

0.6334

a

b

(P[c]=2/5)

(P[b]=2/5)

(P[a]=1/5)

low = 0.6334


An example - VI

Final interval[0.6390,0.6501)

we can send 0.64

low 0.63340.6390

high 0.6667

0.6501

a

b

c

bhigh = 0.6501

(P[c]=3/6)

(P[b]=2/6)

(P[a]=1/6)

low = 0.6390


An example - summary

Starting from the range between 0 and 1 we restrict ourself each time to the subinterval that codify the given symbol

At the end the whole sequence can be codified by any of the numbers in the final range (but mind the brackets...)

14

An example - summary1

0

0.3333

0.6667

a

b

c 0.66671/3

0.3333

1/3

1/3

0.4167

0.5834

1/4

1/4 a

b2/4

c

0. 5834

0. 6667

2/5

2/5

1/50.6001

0.6334

b

c

a

0. 6667

0.6334 a

b

c

0.6390

0.6501

3/6

2/6

1/6

[0.6390, 0.6501) 0.64


Another example - I

Consider encoding the name BILL GATESAgain, we need the frequency of all the characters in the text.

chr freq.space 0.1A 0.1B 0.1E 0.1G 0.1I 0.1L 0.2S 0.1T 0.1


Another example - II

character probability rangespace 0.1 [0.00, 0.10)A 0.1 [0.10, 0.20)B 0.1 [0.20, 0.30)E 0.1 [0.30, 0.40)G 0.1 [0.40, 0.50)I 0.1 [0.50, 0.60)L 0.2 [0.60, 0.80)S 0.1 [0.80, 0.90)T 0.1 [0.90, 1.00)


Another example - IIIchr low high

0.0 1.0B 0.2 0.3I 0.25 0.26L 0.256 0.258L 0.2572 0.2576Space 0.25720 0.25724G 0.257216 0.257220A 0.2572164 0.2572168T 0.25721676 0.2572168E 0.257216772 0.257216776S 0.2572167752 0.2572167756

The final low value, 0.2572167752 will uniquely encode the name BILL GATES


Decoding - I

Suppose we have to decode 0.64The decoder needs symbol probabilities, as it simulates what the encoder must have been doingIt starts with low=0 and high=1 and divides the interval exactly in the same manner as the encoder (a in [0, 1/3), b in [1/3, 2/3), c in [2/3, 1)


Decoding - II

The trasmitted number falls in the interval corresponding to b, so b must have been the first symbol encodedThen the decoder evaluates the new values for low (0.3333) and for high (0.6667), updates symbol probabilities and divides the range from low to high according to these new probabilitiesDecoding proceeds until the full string has been reconstructed


Decoding - III

0.64 in [0.3333, 0.6667) b0.64 in [0.5834, 0.6667) c...

and so on...


Why does it works?

More bits are necessary to express a number in a smaller intervalHigh-probability events do not decrease very much interval range, while low probability events result a much smaller next intervalThe number of digits needed is proportional to the negative logarithm of the size of the interval


Why does it works?

The size of the final interval is the product of the probabilities of the symbols coded, so the logarithm of this product is the sum of the logarithm of each termSo a symbol s with probability Pr[s]contributes

bits to the output, that is equal to symbol probability content (uncertainty)!!

log Pr[ ]s−


Why does it works?

For this reason arithmetic coding is nearly optimum as number of output bits, and it is capable to code very high probability events in just a fraction of bitIn practice, the algorithm is not exactly optimal because of the use of limited precision arithmetic, and because trasmission requires to send a whole number of bits


A trick - I

As the algorithm was described until now, the whole output is available only when encoding are finishedIn practice, it is possible to output bits during the encoding, which avoids the need for higher and higher arithmetic precision in the encodingThe trick is to observe that when low and highare close they could share a common prefix


A trick - II

This prefix will remain forever in the two values, so we can transmit it and remove from low and highFor example, during the encoding of “bccb”, it has happened that after the encoding of the third character the range is low=0.6334, high=0.6667We can remove the common prefix, sending 6 to the output and transforming low and high into 0.334 and 0,667


The encoding step

To code symbol s, where symbols are numbered from 1 to n and symbol i has probability Pr[i]

low_bound =high_bound =range = high - lowlow = low + range * low_boundhigh = low + range * high_bound

1

1Pr[ ]s

ii−

=∑1Pr[ ]s

ii

=∑


The decoding step

The symbols are numbered from 1 to n and value is the arithmetic code to be processed

Find s such that

Return symbol sPerform the same range-narrowing step of the encoding step

1

1 1

( )Pr[ ] Pr[ ]( )

s s

i i

value lowi ihigh low

−

= =

−≤ ≤

−∑ ∑


Implementing arithmetic coding

As mentioned early, arithmetic coding uses binary fractional number with unlimited arithmetic precisionWorking with finite precision (16 or 32 bits) causes compression be a little worser than entropy boundIt is possible also to build coders based on integer arithmetic, with another little degradation of compression

29

Arithmetic coding vs. Huffman coding

In tipical English text, the space character is the most common, with a probability of about 18%, so Huffman redundancy is quite small.Moreover this is an upper boundOn the contrary, in black and white images, arithmetic coding is much better than Huffman coding, unless a blocking technique is used

A A. coding requires less memory, as symbol representation is calculated on the fly

A A. coding is more suitable for high performance models, where there are confident predictions


Arithmetic coding vs. Huffman coding

H H. decoding is generally faster than a. decoding

H In a. coding it is not easy to start decoding in the middle of the stream, while in H. coding we can use “starting points”In large collections of text and images, Huffman coding is likely to be used for the text, and arithmeting coding for the images

arithmetic coding - unisi.itdiligmic/bdm2011... · gabriele monfardini - corso di basi di dati...

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