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Arithmetic Coding

This material is based on K. Sayood, “Introduction to data compression,” San Francisco, CA: Morgan Kaufmann Publishers, 1996. III Edition 2006. Internet: [email protected] 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

Entropy = ∑i=1

3

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) 2Group symbols {a1, a2, a3} in blocks of two.{a1a1, a1a2, a1a3, a2a1, ...., a3a3}# of messages = 32 = 9

P.S: Note that Morgan Kaufmann is bought by Elsevier. http:// www.books.elsevier.com, http://textbooks.elsevier.com

a1(0.95)

a3(0.03)

a2(0.02)

0

0 11 (0.05)

(1.0)Root

Huffman Tree

1

Letter Probability Codea1 .95 0a2 .02 11a3 .03 10

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Symbol Probability Code0 Root

a1a1 .9025 o 1.0 0 1

a1a3 .0285 o 0 o 0 o .0975 100 .0570

a3a1 .0285 o 1 1 101

a1a2 .0190 0 o .0405 110 1

a2a1 .0190 0 o 1110 .0215

a3a3 .0009 1 111100 0

o 0 o 1 .0015 .0025

a2a3 .0006 111101

a3a2 .0006 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 Probability a1 0.95 a2 0.02 a3 0.03

P (a1) = 0.95, P(a2) = .02, P(a3) = .03Average bit rate = .611 bits/symbol, entropy = .335 bits/symbol


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Redundancy = 0.611-0.335 = 0.276Group 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 = 34 = 81To 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. Generate a unique identifier or tag for the sequence of length m to be coded.

II. 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) = .2Cumulative distribution function CDF

iFx (i) = P(X = k)

k=1

0 0 .49 .546

a1 a1 a1 a1

a2

.7 .49 a2 .539 a2 .5558 Mid point of

.8 .56 .546 .5572 tag intervala3 a3 a3 .5565

1.0 .7 .56 .56

Tag for input sequence a1 a2 a3 a2

a1a1 = 0.7*0.7 = 0.49a1a2 = .7*0.1 = 0.07, 0.49+0.07 = 0.56a1a3 = 0.7*0.2 = 0.14, 0.56+0.14 = 0.7a1a2a1 = 0.07*0.7 = 0.049, 0.49+0.049 = 0.539


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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

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. Lower limit of the interval2. Midpoint of the interval3. 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-1TX (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, 60 01 .0833 1 1/721/6 1/362 .25 2 3/721/3 2/363 .4166 3 5/721/2 3/364 .5883 4 7/722/3 4/36

5 .75 5 9/72

5/6 5/36

6 .9166 6 11/72

1.0 1/6

4TX (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 = 5/72We 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 Xl(n) = lower limit of tag for Xmidpoint of the interval for tag


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TX (n) (X) = (u (n) + l (n) ) 2

Example: A = {a1, a2, a3} P(a1) = .8, P(a2) = .02, P(a3) = .18

(small alphabet size & highlyEncode a sequence 1 3 2 1 (a1 a3 a2 a1) skewed probability)

0 0 .656 .7712

a1 a1 a1 a1 .772352

.8 .64 .7712 .773504 a2 a2 a2 a2

.82 .656 .77408 a3 a3 a3 a3

1.0 .8 .77408 .8

FX (1) = .8 , FX (2) = .82 P(a1) = 0.8P(a2) = 0.02

FX (3) = 1, FX (k) = 0, k < 0 P(a3) = 0.18 FX (k) = 1, k> 3

Sequence 1, 3, 2, 1 Recursion relations

l(0) = 0, u(0) = 1 l(n) = l(n-1) + [u(n-1) - l(n-1)] FX (Xn-1 ) First element = 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


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u(2) = 0 + (.8- 0) FX (3) = .8

a1a1 = 0.8*0.8 = 0.64 a1a3 = 0.8*0.18 = 0.144a1a2 = 0.8*0.02 = 0.016 0.656+0.144 = 0.80.64+0.016 = 0.656III 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 = 0.1250

a1 .25 TX (1) Binary code for TX (x)= Represent TX(x)

.5 in binary and truncate to l (x),a2 .625 TX (2) l (x) = log2 1/p(x) + 1.75

a3 .8125 TX (3).875

a4 .9375 TX (4)


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X = smallest integer > (X)

Binary Code

P(x) Symbol Fx TX in binary log2 1/P(x) +1 Code

1/2 1 .5 .25 .010 2 011/4 2 .75 .625 .101 3 1011/8 3 .875 .8125 .1101 4 11011/8 4 1.0 .9375 .1111 4 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 By increasing ‘m’ bothHuffman and arithmetic

H (x) = Entropy (block m symbols together) coding can reach closeto 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


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while still preserving the information being transmitted. This is called synchronized rescaling. In general the tag is confined to lower [0, .5) or upper

half [.5, 1.0) of the [0, 1) interval. If this is not valid, the algorithm is modified. Mapping is

E1 : [0, .5) [0, 1), E1 (X) = 2XE2 : [.5, 1) [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. Advantageous for small alphabet size and highly skewed probabilities2. 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.

3. 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 standards1. JPEG: Extended DCT-based process and lossless process2. JBIG: 1024 to 4096 ‘ACS’. QM Coder. Also JBIG-2: Context based

AC.


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3. H.263 Optional mode: Syntax based ‘AC’ ‘SBAC’

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).

References:

1. T. C. Bell, J. G. Cleary, and I. H. Witten. Text Compression. Advanced Reference Series. Englewood Cliffs, NJ: Prentice Hall, 1990.2. G. G. Langdon, Jr. An Introduction to Arithmetic Coding, IBM Journal

of Research and Development, 28: 135-149, March 1984.3. J. J. Rissanen and G. G. Langdon. Universal Modeling and Coding.

IEEE Trans. on Information Theory, vol. IT-27(1),pp.12-22, Jan. 1981.4. G. G. Langdon and J. J. Rissanen. Compression of Black-White

Images with Arithmetic Coding. IEEE Trans. on Communications, vol. 29(6): pp.858- 867, June 1981.


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5. T. C. Bell, I. H. Witten, and J. G. Cleary. Modeling for Text Compression. ACM Computing Survey, vol. 21: pp. 557-591, Dec. 1989.

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