Arithmetic coding

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 probability

Remember 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 bits

However 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 counters When 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

low

high

0

1

0.3333

0.6667

a

b

c(P[c]=1/3)

(P[b]=1/3)

(P[a]=1/3)

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An example - III

low

high

0

1

0.3333

0.6667

a

b

c

b

low = 0.3333

high = 0.6667

P[a]=1/4

P[b]=2/4

P[c]=1/4

new probabilities


An example - IV

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

low

high

0.3333

0.6667

0.4167

0.5834

a

b

c

c

low = 0.5834

high = 0.6667

(P[c]=1/4)

(P[b]=2/4)

(P[a]=1/4)


An example - V

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

low

high

0.5834

0.6667

0.6001

0.6334

a

b

c

c

low = 0.6334

high = 0.6667

(P[c]=2/5)

(P[b]=2/5)

(P[a]=1/5)


An example - VI

Final interval[0.6390,0.6501)

we can send 0.64

low

high

0.6334

0.6667

0.6390

0.6501

a

b

c

low = 0.6390

high = 0.6501b

(P[c]=3/6)

(P[b]=2/6)

(P[a]=1/6)


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...)

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An example - summary

0

1

0.3333

0.6667

a

b

c0.6667

0.3333

1/3

1/3

1/3

0.4167

0.5834

1/4

2/4

1/4 a

b

c

0. 5834

0. 6667

2/5

2/5

1/50.6001

0.6334

a

b

c

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 0.64 The decoder needs symbol probabilities, as it

simulates what the encoder must have been doing

It 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 encoded

Then 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 probabilities

Decoding proceeds until the full string has been reconstructed


Decoding - III

0.64 in [0.3333, 0.6667) b 0.64 in [0.5834, 0.6667) c...

and so on...


Why does it works?

More bits are necessary to express a number in a smaller interval

High-probability events do not decrease very much interval range, while low probability events result a much smaller next interval

The 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 term

So 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 bit

In 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 finished

In practice, it is possible to output bits during the encoding, which avoids the need for higher and higher arithmetic precision in the encoding

The trick is to observe that when low and high are 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 high

For example, during the encoding of “bccb”, it has happened that after the encoding of the third character the range is low=0.6334, high=0.6667

We 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 - low low = low + range * low_bound high = 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 s Perform the same range-narrowing step of the encoding step

1

1 1

( )Pr[ ] Pr[ ]

( )

s s

i i

value lowi i

high low


Implementing arithmetic coding

As mentioned early, arithmetic coding uses binary fractional number with unlimited arithmetic precision

Working with finite precision (16 or 32 bits) causes compression be a little worser than entropy bound

It is possible also to build coders based on integer arithmetic, with another little degradation of compression

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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 bound

On 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

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