Multimedia Systems - SHARIF UNIVERSITY OF …ce.sharif.edu/courses/89-90/2/ce342-1/resources/root...Page 1 Multimedia Systems, Entropy Coding Data Compression Motivation Data storage

Multimedia Systems

Entropy Coding

Mahdi Amiri

February 2011

Sharif University of Technology

Course Presentation

Page 1 Multimedia Systems, Entropy Coding

Data CompressionMotivation

Data storage and transmission cost money

Use fewest number of bits to represent information source

Pro:

Less memory, less transmission time

Cons:

Extra processing required

Distortion (if using lossy compression )

Data has to be decompressed to be represented, this

may cause delay


Data CompressionLossless and Lossy

Lossless

Exact reconstruction is possible

Applied to general data

Lower compression rates

Examples: Run-length, Huffman, Lempel-Ziv

Lossy

Higher compression rates

Applied to audio, image and video

Examples: CELP, JPEG, MPEG-2


Lossless CompressionRun-length encoding

BBBBHHDDXXXXKKKKWWZZZZ 4B2H2D4X4K2W4Z

Image of a rectangle

0, 40

0, 40

0,10 1,20 0,10

0,10 1,1 0,18 1,1 0,10

0,10 1,1 0,18 1,1 0,10

0,10 1,1 0,18 1,1 0,10

0,10 1,20 0,10

0,40


Lossless CompressionFixed Length Coding (FLC)

A simple example

►♣♣♠☻►♣☼►☻The message to code:

5 different symbols at least 3 bits

Message length: 10 symbols

Total bits required to code: 10*3 = 30 bits

Codeword table


Lossless CompressionVariable Length Coding (VLC)

Intuition: Those symbols that are more frequent should have smaller codes, yet since

their length is not the same, there must be a way of distinguishing each code

►♣♣♠☻►♣☼►☻The message to code:

Total bits required to code: 3*2 +3*2+2*2+3+3 = 24 bits

Codeword tableTo identify end of a codeword as

soon as it arrives, no codeword can

be a prefix of another codeword

How to find the optimal codeword table?


Lossless CompressionVLC, Example Application

Morse code

nonprefix code

Needs separator symbol

for unique decodability


Lossless CompressionHuffman Coding Algorithm

Step 1: Take the two least probable symbols in the alphabet

(longest codewords, equal length, differing in last digit)

Step2: Combine these two symbols into a single symbol, and repeat.

P(n): Probability of

symbol number n

Here there is 9 symbols.

e.g. symbols can be

alphabet letters ‘a’, ‘b’, ‘c’,

‘d’, ‘e’, ‘f’, ‘g’, ‘h’, ‘i’


Lossless CompressionHuffman Coding Algorithm

David A. Huffman

1925-1999

Paper: "A Method for the Construction of

Minimum-Redundancy Codes“, 1952

Results in "prefix-free codes“

Most efficient

No other mapping will produce a smaller average output size

If the actual symbol frequencies agree with those used to create the code

Cons:

Have to run through the entire data in advance to find frequencies

„Minimum-Redundancy‟ is not favorable for error correction techniques (bits

are not predictable if e.g. one is missing)

Does not support block of symbols: Huffman is designed to code single

characters only. Therefore at least one bit is required per character, e.g. a word of

8 characters requires at least an 8 bit code


Entropy CodingEntropy, Definition

The entropy, H, of a discrete random variable X is a measure of the

amount of uncertainty associated with the value of X.

Measure of information content (in bits)

A quantitative measure of the disorder of a system

It is impossible to compress the data such that the average

number of bits per symbol is less than the Shannon entropy

of the source(in noiseless channel)

The Intuition Behind the Formula

2

1log

x X

H X P xP x

Claude E. Shannon

1916-2001 1

amount of uncertatinty P x HP x

2

1bringing it to the world of bits log , information content of H I x x

P x

weighted average number of bits required to encode each possible value and P x

X Information Source

P(x) Probability that symbol x in X will occur

Information Theory

Point of View


Lossless CompressionLempel-Ziv (LZ77)

Algorithm for compression of character sequences

Assumption: Sequences of characters are repeated

Idea: Replace a character sequence by a reference to an earlier occurrence1. Define a: search buffer = (portion) of recently encoded data

look-ahead buffer = not yet encoded data

2. Find the longest match between

the first characters of the look ahead buffer

and an arbitrary character sequence in the search buffer

3. Produces output <offset, length, next_character>

offset + length = reference to earlier occurrence

next_character = the first character following the match in the look ahead buffer


Lossless CompressionLempel-Ziv-Welch (LZW)

Drops the search buffer and keeps an explicit dictionary

Produces only output <index>

Used by unix "compress", "GIF", "V24.bis", "TIFF”

Example: wabbapwabbapwabbapwabbapwoopwoopwoo

Progress clip at 12th entry

Encoder output sequence so far: 5 2 3 3 2 1


Lossless CompressionLempel-Ziv-Welch (LZW)

Example: wabbapwabbapwabbapwabbapwoopwoopwoo

Progress clip at the end of above example

Encoder output sequence: 5 2 3 3 2 1

6 8 10 12 9 11 7 16 5 4 4 11 21 23 4


Lossless CompressionArithmetic Coding

Encodes the block of symbols into a single number, a

fraction n where (0.0 ≤ n < 1.0).

Step 1: Divide interval [0,1) into subintervals based on probability of

the symbols in the current context Dividing Model.

Step 2: Divide interval corresponds to the current symbol into sub-

intervals based on dividing model of step 1.

Step 3: Repeat Step 2 for all symbols in the block of symbols.

Step 4: Encode the block of symbols with a single number in the

final resulting range. Use the corresponding binary number in this

range with the smallest number of bits.

See the encoding and decoding examples in the following slides


Lossless CompressionArithmetic Coding, Encoding

Example: SQUEEZE

Using FLC: 3 bits per symbol 7*3 = 21 bits

P(‘E’) = 3/7 Prob. ‘S’ ‘Q’ ‘U’ ‘Z’: 1/7

We can encode the word SQUEEZE with a

single number in [0.64769-0.64772) range.

The binary number in this range with the

smallest number of bits is 0.101001011101,

which corresponds to 0.647705 decimal. The '0.'

prefix does not have to be transmitted because

every arithmetic coded message starts with this

prefix. So we only need to transmit the sequence

101001011101, which is only 12 bits.

Dividing Model


Lossless CompressionArithmetic Coding, Decoding

Input Probabilities: P(„A‟)=60%, P(„B‟)=20%, P(„C‟)=10%, P(„<space>‟)=10%

Decoding the input value of 0.538

Dividing model from input probabilities

60% 20% 10% 10%

The fraction 0.538 (the circular point) falls into the

sub-interval [0, 0.6) the first decoded symbol is 'A'

The subregion containing the point is successively

subdivided in the same way as diviging model.

Since .538 is within the interval [0.48, 0.54), the

second symbol of the message must have been 'C'.

Since .538 falls within the interval [0.534, 0.54), the

Third symbol of the message must have been '<space>'.

The internal protocol in this example indicates <space> as the

termination symbol, so we consider this is the end of decoding process


Lossless CompressionArithmetic Coding

Pros

Typically has a better compression ratio than

Huffman coding.

Cons

High computational complexity.

Patent situation had a crucial influence to decisions

about the implementation of an arithmetic coding

(Many now are expired).


Thank You

1. http://ce.sharif.edu/~m_amiri/

2. http://www.dml.ir/

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Entropy Coding

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