Arithmetic Coding
This material is based on K. Sayood, “Introduction to data compression,” San
Francisco, CA: Morgan Kaufmann Publishers, 1996. III Edition 2006.
Internet: mkp@mkp.com
web site: http://mkp.com
Useful for sources with small alphabets (ex-binary 0 or 1) and alphabets with
highly skewed (unequal) probabilities. (Ex-facsimile, text etc.).
The rate for Huffman Coding is within Pmax + 0.086 of entropy (Pmax =
probability of most frequently occurring symbol). For small alphabet size
and highly skewed probability Pmax can be very large.
Example: A = (a1, a2, a3)
Huffman Code
Letter
a1
a2
a3
Probability
.95
.02
.03
Code
0
11
10
Entropy = ∑3𝑖=1 𝑃𝑖 𝑙𝑜𝑔2 𝑃𝑖 = 0.335 bits/symbol
Redundancy = 1.05-.335 = 0.67 bit/symbol
Huffman code gives 1.05 bits/symbol = (.95) 1 + (.02) 2 + (.03) 2
Group symbols {a1, a2, a3} in blocks of two.
{a1a1, a1a2, a1a3, a2a1, ...., a3a3}
# of messages = 32 = 9
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Symbol Probability
Code
0
a1a1
.9025
a1a3
.0285 o
0
a3a1
.0285 o
1
a1a2
.0190
0
a2a1
.0190
0
a3a3
.0009
Root
o 1.0
1
o .0975
o
0
.0570
0
100
1
101
o .0405
1
110
o
.0215
1
1110
111100
0
o
1
a2a3
.0006
a3a2
.0006
0
.0015
o
.0025
111101
1
111110
0
o
.001
a2a2
.0004
1
111111
Huffman Tree
Average rate = 1.222bits/message (coding in blocks of 2 symbols)
Average rate = 0.611bit/symbol of original alphabet
Symbol
a1
a2
a3
Probability
0.95
0.02
0.03
P (a1) = 0.95, P(a2) = .02, P(a3) = .03
Average bit rate = .611 bits/symbol, entropy = .335 bits/symbol
Redundancy = 0.611-0.335 = 0.276
Group symbols (a1, a2, a3) in blocks of three,
Alphabet size = 27 = 33, (a1 a1 a1, a1 a1 a2,.....,a3 a3 a2, a3 a3 a3)
Average bit rate by Huffman Coding can be reduced but alphabet size grows
exponentially. Group symbols in blocks of four. Alphabet size = 3 4 = 81
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To assign a code for a particular sequence (groups of symbols) of length m,
Huffman Code requires developing codewords for all possible sequences of
length m. Arithmetic Coding assigns codes to particular sequences of length
m without having to generate codes for all sequences of length m.
Procedure:
I.
II.
Generate a unique identifier or tag for the sequence of length m to be
coded.
Assign a unique binary code to this tag.
Generating a tag:
Alphabet size = 3, A= (a1, a2, a3)
P(a1) = .7, P(a2) = .1, P(a3) = .2
Cumulative distribution function CDF
i
Fx (i) =  P(X = k)
k=1
0
a1
0
.49
.546
a1
a1
a1
a2
.7
.8
.49
.56
a2
.539
.546
a2
a3
.5558
.5572
a3
a3
Mid point of
tag interval
.5565
1.0
.7
Tag for input sequence a1 a2 a3 a2
.56
.56
a1a1 = 0.7*0.7 = 0.49
a1a2 = .7*0.1 = 0.07,
0.49+0.07 = 0.56
a1a3 = 0.7*0.2 = 0.14,
0.56+0.14 = 0.7
a1a2a1 = 0.07*0.7 = 0.049,
0.49+0.049 = 0.539
a1a2a2 = 0.07*0.1 = 0.007,
0.539+0.007 = 0.546,
a1a2a3 = 0.07*0.2 = 0.014,
0.546+0.014 = 0.56
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Tag for input sequence (a1, a2, a3.....)
Each new symbol restricts the tag to a subinterval that is disjoint from any
other subinterval. The tag interval for (a1, a2, a3, a1) is disjoint from the tag
interval for (a1, a2, a3, a2)
Any member of an interval can be used to identify a tag.
1.
2.
3.
Lower limit of the interval
Midpoint of the interval
Upper and lower limits of the interval
Use midpoint of the interval to identify a tag. Let the alphabet be
A = (a1, a2, ...,am)
i-1
TX (ai) =  P (X=k) + (1/2) P(X=i)
k=1
= FX (i-1) + (1/2) P(X = i)
Example Roll of a die (1, 2, ......,6)
P(X = K) = 1/6, K = 1, 2, .......,6
Assign a tag to a particular sequence Xi
TX (m) (Xi) =  P(y) + (1/2) P(Xi)
y<xi
y<x means y precedes x in the ordering, m = length of sequence.
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Roll of a die 1, 2, 3, 4, 5, 6
0
1
1/6
2
1/3
3
1/2
4
2/3
5
0
.0833
1
.25
2
.4166
3
.5883
4
.75
5
5/6
6
1.0
1/72
1/36
3/72
2/36
5/72
3/36
7/72
4/36
9/72
5/36
.9166
6
11/72
1/6
4
TX (5) =  P(X = k) + (1/2) P(X = 5) = .75 = 0.67 + (.16/2)
k=1
TX (2) (13) = P(X = 11) + P(X = 12) + (1/2) P(X = 13) = 1/36 + 1/36 + (1/2)
1/36 = 5/72
We have to compute the probability of every sequence that is less than the
sequence for which the tag is generated. This is prohibitive. However, the
upper and lower limits of a tag interval can be computed recursively.
For a sequence X = (X1, X2.......Xn)
l(n) = l(n-1) + [u(n-1) - l(n-1)] FX (Xn-1 )
u(n) = l(n-1) + [u(n-1) - l(n-1)] FX (Xn)
u(n) = upper limit of tag for X
l(n) = lower limit of tag for X
midpoint of the interval for tag
TX (n) (X) = (u(n) + l(n))
2
Example: A = {a1, a2, a3}
P(a1) = .8,
P(a2) = .02, P(a3) = .18
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(small alphabet size & highly
skewed probability)
Encode a sequence 1 3 2 1 (a1 a3 a2 a1)
0
0
a1
a1
.8
.656
a1
.64
a2
.656
a3
a2
.82
a3
1.0
.7712
a1
.7712
a2
.772352
.773504
a2
.77408
a3
a3
.8
.77408
.8
FX (1) = .8 , FX (2) = .82
FX (3) = 1, FX (k) = 0, k < 0
FX (k) = 1, k> 3
Sequence 1, 3, 2, 1
l(0) = 0, u(0) = 1
First element = 1
P(a1) = 0.8
P(a2) = 0.02
P(a3) = 0.18
Recursion relations
l(n) = l(n-1) + [u(n-1) - l(n-1)] FX (Xn-1 )
u(n) = l(n-1) + [u(n-1) - l(n-1)] FX (Xn)
l(1) = 0 + (1-0) 0 = 0
u(1) = 0 + (1-0).8 = .8
II Element 3
l(2) = 0 + (.8 - 0) FX (2) = .656
u(2) = 0 + (.8- 0) FX (3) = .8
a1a1 = 0.8*0.8 = 0.64
a1a2 = 0.8*0.02 = 0.016
0.64+0.016 = 0.656
a1a3 = 0.8*0.18 = 0.144
0.656+0.144 = 0.8
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III Element 2
l(3) = .656 + (.8 - .656) .8 = .7712
u(3) = .656 + (.8 - .656) .82 = .77408
Last element 1
l(4) = .7712 + (.77408 - .7712) FX (0) = .7712 + 0.0 = .7712
u(4) = .7712 + (.77408 - .7712) FX (1) = .773504 = .7712 + .002888 x.8
Tag
TX(4) (1321) = .7712 + .773504
2
= .772352
Generating a binary code
Alphabet A = (a1, a2, a3, a4)
P(a1) = 1/2, P(a2) = 1/4, P(a3) = P(a4) = 1/8
= 0.5
= 0.25
= 0.125
0
a1
.25 TX (1)
.5
a2
.75
.625 TX (2)
a3
.875
.8125 TX (3)
a4
.9375 TX (4)
Binary code for TX (x)
= Represent TX(x)
in binary and truncate to l (x),
l (x) = log2 1/p(x) + 1
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X = smallest integer > (X)
Binary Code
P(x) Symbol
1/2
1/4
1/8
1/8
1
2
3
4
Fx
.5
.75
.875
1.0
TX
in binary
log2 1/P(x) +1
Code
2
3
4
4
01
101
1101
1111
.25 .010
.625 .101
.8125 .1101
.9375 .1111
This is a prefix code i.e., no codeword is a prefix of another codeword.
Average length (bits/symbol) for coding groups of symbols of length m is
H (x) < lA < H(x) + 2/m
H (x) = Entropy (block m symbols together)
By increasing ‘m’ both
Huffman and arithmetic
coding can reach close
to entropy.
H (x) < lH < H(x) + 1/m
lA = Average bit length for arithmetic coding
lH = Average bit length for Huffman coding
Alphabet size K, # of possible sequences of length m is Km.
Codebook size = Km
Ex: K = 4 (a1, a2, a3, a4), m = 3, (codebook size = 43 = 64)
(a1a1a1, a1a1a2, a1a2a3,......a4a4a3, a4a4a4)
m = 4, Codebook size = 44 = 256, (a1a1a1a1, ....., a4a4a4a4)
Huffman Coding requires building the codes for the entire codebook. For
arithmetic coding, obtain the tag corresponding to a given sequence.
Synchronized rescaling.
As n gets larger (# of symbols in the group) l(n) and u(n) (lower and upper
limits of the tag interval get closer and closer). This is avoided by rescaling
while still preserving the information being transmitted. This is called
synchronized rescaling. In general the tag is confined to lower [0, .5) or upper
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half [.5, 1.0) of the [0, 1) interval. If this is not valid, the algorithm is
modified. Mapping is
E1 : [0, .5)
E2 : [.5, 1)
[0, 1), E1 (X) = 2X
[0, 1), E2 (X) = 2 (X-.5)
Incremental Encoding
Generating and transmitting portions of the code as the sequence is being
observed rather than wait until the end of the sequence.
Advantages of arithmetic coding
1.
2.
3.
Advantageous for small alphabet size and highly skewed probabilities
Easy to implement a system with multiple arithmetic codes. Once the
computational machinery to implement one arithmetic code is
developed, all that is needed to set up multiple arithmetic codes is the
availability of more probability tables.
Easy to adapt arithmetic codes to changing input statistics. No need to
generate a tree (Huffman Code), a priori. Modeling and coding
procedures can be separated.
QM Coder:
This is a modification of an adaptive binary arithmetic coder called the
Q coder. QM coder tracks the lower end of the tag interval l(n) and the size of
the interval A(n).
A(n) = u(n) - l(n)
where u(n) is the upper end of the tag interval.
Applications of arithmetic coding in standards
1.
JPEG: Extended DCT-based process and lossless process
2.
JBIG: 1024 to 4096 ‘ACS’. QM Coder. Also JBIG-2: Context based
AC.
3.
H.263 Optional mode: Syntax based ‘AC’ ‘SBAC’
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4.
MPEG-4 Context based arithmetic coding in shape coding ( MPEG-4
VM 8.0, July 1997)
5.
MPEG-4 Still frame image coding - wavelet based. Lowest subband is
DPCM coded. Prediction errors are coded using adaptive arithmetic
coder. Zerotree wavelet coding and quantized values are coded using
adaptive arithmetic coding.
6.
MPEG-2 AAC (Also MPEG-4 T/F audio coder) Scale factors are
coded based on bit-sliced arithmetic coding. AAC: advanced audio
coding, T/F: time/frequency
7.
JPEG-LS part 2 Lossless and nearly lossless compression of
continuous tone still images. [51]
8.
JPEG2000: context dependent binary arithmetic coding. Uses up to
9 contexts
9.
H.264/MPEG-4 Part 10: Context based adaptive binary arithmetic
coding (CABAC) [63].
JPEG: Joint Photographic Experts Group [20]
JBIG: Joint Binary Image Group [19]
H. 263 (Video Coding < 64 Kbps) [23]
MPEG: Moving Picture Experts Group
Adaptive arithmetic coding
Probabilities of source symbols are dynamically estimated based on the
changing symbol statistics observed in the message to be encoded, i.e.,
estimate probabilities on the fly (dynamic modeling).
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