Arithmetic Coding 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 redundancy p1 0.086 where simbol p1 is the probability of the most likely Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a. 2005-2006 2 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 Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a. 2005-2006 3 How we can do better than Huffman? - III Huffman Coding is optimal in its framework static model adaptive Huffman one symbol, one word blocking arithmetic coding Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a. 2005-2006 4 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 [0,1) The longer and more complex the message, the more bits are needed to represents the output number Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a. 2005-2006 5 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 01101010 0.01101010 In order to explain the algorithm, numbers will be shown as decimal, but obviously they are always binary Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a. 2005-2006 6 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 Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a. 2005-2006 7 An example - II The range between low and high is divided between the symbols of the alphabet, according to their probabilities high 1 c (P[c]=1/3) 0.6667 b (P[b]=1/3) 0.3333 (P[a]=1/3) a low 0 Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a. 2005-2006 8 An example - III b high 1 high = 0.6667 c 0.6667 b 0.3333 a low 0 low = 0.3333 P[a]=1/4 new probabilities P[b]=2/4 P[c]=1/4 9 An example - IV c high 0.6667 (P[c]=1/4) 0.5834 0.4167 low c new probabilities P[a]=1/5 P[b]=2/5 P[c]=2/5 b (P[b]=2/4) (P[a]=1/4) high = 0.6667 0.3333 a low = 0.5834 Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a. 2005-2006 10 An example - V c high 0.6667 high = 0.6667 c (P[c]=2/5) new probabilities P[a]=1/6 P[b]=2/6 P[c]=3/6 0.6334 b (P[b]=2/5) (P[a]=1/5) low 0.6001 0.5834 a low = 0.6334 Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a. 2005-2006 11 An example - VI b high 0.6667 (P[c]=3/6) c high = 0.6501 0.6501 b (P[b]=2/6) (P[a]=1/6) low 0.6390 0.6334 a Final interval [0.6390,0.6501) we can send 0.64 low = 0.6390 Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a. 2005-2006 12 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...) Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a. 2005-2006 13 An example - summary 1 c 1/3 2/4 b 2/5 1/4 a 2/5 0.6001 0.4167 0.3333 0 c 0. 6667 0. 6667 c 1/5 a 0. 5834 3/6 0.6501 0.6334 b 1/3 1/4 0.5834 0.6667 1/3 0.6667 2/6 b a c b 0.6390 0.6334 a 1/6 0.3333 [0.6390, 0.6501) 0.64 14 Another example - I Consider encoding the name BILL GATES Again, we need the frequency of all the characters in the text. chr space A B E G I L S T freq. 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.1 0.1 Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a. 2005-2006 15 Another example - II character space A B E G I L S T probability 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.1 0.1 range [0.00, 0.10) [0.10, 0.20) [0.20, 0.30) [0.30, 0.40) [0.40, 0.50) [0.50, 0.60) [0.60, 0.80) [0.80, 0.90) [0.90, 1.00) Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a. 2005-2006 16 Another example - III chr B I L L Space G A T E S low 0.0 0.2 0.25 0.256 0.2572 0.25720 0.257216 0.2572164 0.25721676 0.257216772 0.2572167752 high 1.0 0.3 0.26 0.258 0.2576 0.25724 0.257220 0.2572168 0.2572168 0.257216776 0.2572167756 The final low value, 0.2572167752 will uniquely encode the name BILL GATES Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a. 2005-2006 17 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) Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a. 2005-2006 18 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 Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a. 2005-2006 19 Decoding - III 0.64 in [0.3333, 0.6667) 0.64 in [0.5834, 0.6667) b c... and so on... Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a. 2005-2006 20 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 Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a. 2005-2006 21 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 log Pr[ s ] bits to the output, that is equal to symbol probability content (uncertainty)!! Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a. 2005-2006 22 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 Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a. 2005-2006 23 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 Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a. 2005-2006 24 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 Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a. 2005-2006 25 The encoding step To code symbol s, where symbols are numbered from 1 to n and symbol i has probability Pr[i] low_bound = i 1 Pr[i] s high_bound = i 1 Pr[i] range = high - low low = low + range * low_bound high = low + range * high_bound s 1 Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a. 2005-2006 26 The decoding step The symbols are numbered from 1 to n and value is the arithmetic code to be processed Find s such that (value low) Pr[ i ] (high low) i 1 s 1 s Pr[i] i 1 Return symbol s Perform the same range-narrowing step of the encoding step Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a. 2005-2006 27 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 Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a. 2005-2006 28 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 29 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 Gabriele Monfardini - Corso di Basi di Dati Multimediali a.a. 2005-2006 30

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# Arithmetic Coding