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Copyright (c) Arturo San Emeterio Campos 1999. All rights reserved. Permission is granted to make verbatim copies of this document for private use only.
 

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Table of contents

Introduction
Arithmetic coding, is entropy coder widely used, the only problem is it's speed, but compression tends to be better than Huffman can achieve. This presents a basic arithmetic coding implementation, if you have never implemented an arithmetic coder, this is the article which suits your needs, otherwise look for better implementations.
 

Arithmetic coding
The idea behind arithmetic coding is to have a probability line, 0-1, and assign to every symbol a range in this line based on its probability, the higher the probability, the higher range which assigns to it. Once we have defined the ranges and the probability line, start to encode symbols, every symbol defines where the output floating point number lands. Let's say we have:
 

Symbol	Probability	Range
a	2	[0.0 , 0.5)
b	1	[0.5 , 0.75)
c	1	[0.7.5 , 1.0)

Note that the "[" means that the number is also included, so all the numbers from 0 to 5 belong to "a" but 5. And then we start to code the symbols and compute our output number. The algorithm to compute the output number is:

• Low = 0
• High = 1
• Loop. For all the symbols.
• Range = high - low
• High = low + range *  high_range of the symbol being coded
• Low = low + range * low_range of the symbol being coded

• Range, keeps track of where the next range should be.
• High and low, specify the output number.

• Loop. For all the symbols.
• Range = high_range of the symbol - low_range of the symbol
• Number = number - low_range of the symbol
• Number = number / range

Implementation
As you can see from the example is a must that the whole floating point number is passed to the decompressor, no rounding can be performed, but with today's fpu the higher precision which it ca offer is 80 bits, so we can't work with the whole number. So instead we'll need to redefine our range, instead of 0-1 it will be 0000h to FFFFh, which in fact is the same. And we'll also reduce the probabilities so we don't need the whole part, only 16 bits. Don't you believe that it's the same? let's have a look at some numbers:
 

0.000	0.250	0.500	0,750	1.000
0000h	4000h	8000h	C000h	FFFFh

If we take a number and divide it by the maximum (FFFFh) will clearly see it:

• 0000h: 0/65535 = 0,0
• 4000h: 16384/65535 =  0,25
• 8000h: 32768/65535 = 0,5
• C000h: 49152/65535 =  0,75
• FFFFh: 65535/65535 = 1,0

Underflow
Underflow occurs when both high and low get close to a number but theirs msb don't match: High = 0,300001  Low = 0,29997 if we ever have such numbers, and the continue getting closer and closer we'll not be able to output the msb, and then in a few itinerations our 16 bits will not be enough, what we have to do in this situation is to shift the second digit (in our implementation the second bit) and when finally both msb are equal also output the digits discarded.
 

Gathering the probabilities
In this example we'll use a simple statical order-0 model. You have an array initialized to 0, and you count there the occurrences of every byte. And then once we have them we have to adjust them, so they don't make us need more than 16 bits in the calculations, if we want to accomplish that, the total of our probabilities should be below 16,384. (2^14) To scale them we divide all the probabilities by a factor till all of them fit in 8 bits, however there's an easier (and faster) way of doing so, you get the maximum probability, divide it by 256, this is the factor that you'll use to scale the probabilities. Also when dividing, if the result ever is 0 or below put it to one, so the symbol is has a range. The next scaling deals with the maximum of 2^14, add to a value (initialized to 0) all the probabilities, and then check if it's above 2^14, if it is then divide them by a factor, (2 or 4) and the following assumptions will be true:

• All the probabilities are inside the range of 0-255. This helps saving the header with the probabilities.
• The addition of all of them don't get above 2^14, and thus we'll need only 16 bits for the computations.

Saving the probabilities
Our probabilities are one byte long, so we can save the whole array, as a maximum it can be 256 bytes, and it's only written once, so it will not hurt compression a lot. If you expect some symbols to not appear you could rle code it. If you expect some probabilities to have lower values than others, you can use a flag to say how many bits the next probability uses, and then code one with 4 or 8 bits, anyway you should tune the parameters.
 

Assign ranges
For every symbol we have to define its high value and low value, they define the range, doing this is rather simple, we use its probability:
 

Symbol	Probability	Low	High	0-1 (x/4)
a	2	0	2	[ 0.0 , 0.5 )
b	1	2	3	[ 0.5 , 0.75 )
c	1	3	4	[ 0.75 , 1 )

What we'll use is high and low, and when computing the number we'll perform the division, to make it fit between 0 and 1. Anyway if you have a look at high and low you'll notice that the low value of the current symbol is equal to the high value of the last symbol, we can use it to use half the memory, we only have to care about setting the -1 symbol with a high value of 0:
 

Symbol	Probability	High
-1	0	0
a	2	2
b	1	3
c	1	4

And thus when reading the high value of a symbol we read it in its position, and for the low value we read the entry "position-1", I think you don't need pseudo code for doing such routine, you just have to assign to the high value the current probability of the symbol + the last high value, and set it up with the symbol "-1" with a high probability of 0. I.e.: When reading the range of the symbol "b" we read its high value at the current position (of the symbol in the table) "3" and for the low value, the previous: "2". And because our probabilities take one byte, the whole table will only take 256 bytes.
 

Pseudo code
And this is the pseudo code for the initialization:

• Get probabilities and scale them
• Save probabilities in the output file
• High = FFFFh (16 bits)
• Low = 0000h (16 bits)
• Underflow_bits = 0 (16 bits should be enough)

• High and low, they define where the output number falls.
• Underflow_bits, the bits which could have produced underflow and thus they were shifted.

• Range = ( high - low ) + 1
• High = low + ( ( range * high_values [ symbol ] ) / scale ) - 1
• Low = low + ( range * high_values [ symbol - 1 ] ) / scale
• Loop. (will exit when no more bits can be outputted or shifted)
• Msb of high = msb of low?
• Yes
• Output msb of low
• Loop. While underflow_bits > 0  Let's output underflow bits pending for output
• Output  Not ( msb of low )
• go to shift
• No
• Second msb of low = 1  and  Second msb of high = 0 ?  Check for underflow
• Yes
• Underflow_bits += 1  Here we shift to avoid underflow
• Low = low & 3FFFh
• High = high | 4000h
• go to shift
• No
• The routine for encoding a symbol ends here.

• Shift low to the left one time.  Now we have to put in low and high new bits
• Shift high to the left one time, and or the lsb with the value 1
• Repeat to the first loop.

• Note that the formulae before the loop should be done with 32 bit precision. (dword, long)
• Msb of high means the following, with a 16 bits number like that:  abbb bbbb bbbb bbbb, a is the msb bit of it.
• Not ( msb of low ) is "Bitwise complement operator" in C used in the following way: ~low in asm is just "not ax". First you perform not on low and then you output its msb bit.
• "&" means "bitwise and".
• "|" means "bitwise inclusive or". (or)
• Range must be 32 bits long, because the formula need this precision.
• Scale is the addition of all the probabilities.

Decoding
The first thing to do when decoding is read the probabilities, because the encode did the scaling you just have to read them and to do the ranges. The process will be the following: see in what symbol our number falls, extract the code of this symbol from the code. Before starting we have to init "code" this value will hold the bits from the input, init it to the first 16 bits in the input. And this is how it's done:

• Range = ( high - low ) + 1  See where the number lands
• Temp = ( ( code - low ) + 1 ) * scale)  - 1 ) / range )
• See what symbols corresponds to temp.
• Range = ( high - low ) + 1 Extract the symbol code
• High = low + ( ( range * high_values [ symbol ] ) / scale ) - 1
• Low = low + ( range * high_values [ symbol - 1 ] ) / scale  Note that those formulae are the same that the encoder uses
• Loop.
• Msb of high = msb of low?
• Yes
• Go to shift
• No
• Second msb of low = 1  and  Second msb of high = 0 ?
• Yes
• Code = code ^ 4000h
• Low = low & 3FFFh
• High = high | 4000h
• go to shift
• No
• The routine for decoding a symbol ends here.

Shift low to the left one time.  Now we have to put in low, high and code new bits

Shift high to the left one time, and or the lsb with the value 1

Shift code to the left one time, and or it the next bit in the input

Repeat to the first loop.

Closing words
First of all thanks to Mark Nelson for some help with it. This is the first version of this article, I hope to mend possible mistakes, which you should report if you find. Also any idea is welcome. There are faster implementations, but this was only an introduction, once you have a good encoder you only need a good model to have good compression, so research a little bit. If you want a faster arithmetic coder, look the range coder.
 

Contacting the author
You can reach me via email at: arturo@arturocampos.com  Also don't forget to visit my home page http://www.arturocampos.com where you can find more and better info. See you in the next article!
 
 

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