Chapter 3: Multimedia Data Compression

Chapter 3:Multimedia Data

Compression•Compression with loss and lossless

•Huffman coding

•Entropy coding

•Adaptive coding

•Dictionary-based coding(LZW)

1chapter3: Multimedia Compression

Data Compression

• Branch of information theory

– minimize amount of information to be transmitted

• Transform a sequence of characters into a new string of bits

– same information content

– length as short as possible

chapter3: Multimedia Compression

2

Why Compress

• Raw data are huge.

• Audio:CD quality music44.1kHz*16bit*2 channel=1.4Mbps

• Video:near-DVD quality true color animation640px*480px*30fps*24bit=220Mbps

• Impractical in storage and bandwidth


chapter3: Multimedia Compression 4

CompressionGraphic file formats can be regarded as being of three types.

• The first type stores graphics uncompressed.

– Windows BMP (.bmp) files are an example of this sort offormat.

• The second type is called "non-lossy“ or “lossless”compression formats.

– Most graphic formats use lossless compression - the GIFformats are among them.

• The third type of bitmapped graphic file formats is called"lossy" compression.

– the details are what prevent areas from being all the samecolor, and as such from responding well to compression.

– perhaps too subtle to be discernable by your eye

Lossless is not enough!

• The best lossless audio and imagecompression ratio is normally a half

• Lossy audio compression like mp3 or oggachieve 1/20 ratio while remainacceptable quality, and 1/5 ratio forperfect quality

• Lossy video compression reduce a film to1/300 size


Lossy Compression

• Massively reduce information we don’tnotice

• Highly content specific

• Psychology


Lossy Audio Compression

• Frequency domain

• Quantization– The importance varies in bands

– Higher frequency, larger quantum

• Psychoacoustics– Pitch resolution of ear is only 2Hz without

beating

– Threshold of hearing varies in bands

– Simultaneous and temporal masking effect


Lossy Image Compression

• Frequency domain

– Discrete Cosine Transform (in Jpeg)

– Discrete Wavelet Transform (in J2k)

• Quantization

– Reduce less important data

Transform QuantizationEntropy

Coding

Image

data

Output

data


Broad Classification• Entropy Coding (statistical)

– lossless; independent of data characteristics

– e.g. RLE( Run Length Encoding), Huffman, LZW, Arithmetic coding

• Source Coding– lossy; may consider semantics of the data

– depends on characteristics of the data

– e.g. DCT, DPCM, ADPCM, color model transform

• Hybrid Coding (used by most multimedia systems)– combine entropy with source encoding

– e.g., JPEG-2000, H.264, MPEG-2, MPEG-4, MPEG-7


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Huffman Coding• Huffman codes can be used to compress

information– Like WinZip – although WinZip doesn’t use the

Huffman algorithm

– JPEGs do use Huffman as part of their compressionprocess

• The basic idea is that instead of storing eachcharacter in a file as an 8-bit ASCII value, we willinstead store the more frequently occurringcharacters using fewer bits and less frequentlyoccurring characters using more bits– On average this should decrease the filesize (usually ½)


Huffman Coding

• As an example, lets take the string:

“duke blue devils”

• We first to a frequency count of the characters:• e:3, d:2, u:2, l:2, space:2, k:1, b:1, v:1, i:1, s:1

• Next we use a Greedy algorithm to build up a Huffman Tree

– We start with nodes for each character

e,3 d,2 u,2 l,2 sp,2 k,1 b,1 v,1 i,1 s,1


Huffman Coding

• We then pick the nodes with the smallestfrequency and combine them together toform a new node– The selection of these nodes is the Greedy part

• The two selected nodes are removed fromthe set, but replace by the combined node

• This continues until we have only 1 nodeleft in the set


Huffman Coding

e,3 d,2 u,2 l,2 sp,2 k,1 b,1 v,1 i,1 s,1


Huffman Coding

e,3 d,2 u,2 l,2 sp,2 k,1 b,1 v,1

i,1 s,1

2


Huffman Coding

e,3 d,2 u,2 l,2 sp,2 k,1

b,1 v,1 i,1 s,1

22


Huffman Coding

e,3 d,2 u,2 l,2 sp,2

k,1 i,1 s,1

2

b,1 v,1

2

3


Huffman Coding

e,3 d,2 u,2

l,2 sp,2 k,1 i,1 s,1

2

b,1 v,1

2

34


Huffman Coding

e,3

d,2 u,2 l,2 sp,2 k,1 i,1 s,1

2

b,1 v,1

2

344


Huffman Coding

e,3

d,2 u,2 l,2 sp,2

k,1i,1 s,1

2

b,1 v,1

2

3

44 5


Huffman Coding

e,3

d,2 u,2

l,2 sp,2

k,1i,1 s,1

2

b,1 v,1

2

3

4

4

57


Huffman Coding

e,3

d,2 u,2 l,2 sp,2

k,1i,1 s,1

2

b,1 v,1

2

3

44 5

7 9


Huffman Coding

e,3

d,2 u,2 l,2 sp,2

k,1i,1 s,1

2

b,1 v,1

2

3

44 5

7 9

16


Huffman Coding

• Now we assign codes to the tree byplacing a 0 on every left branch and a 1 onevery right branch

• A traversal of the tree from root to leafgive the Huffman code for that particularleaf character

• Note that no code is the prefix of anothercode


Huffman Coding

e,3

d,2 u,2 l,2 sp,2

k,1i,1 s,1

2

b,1 v,1

2

3

44 5

7 9

16

e 00

d 010

u 011

l 100

sp 101

i 1100

s 1101

k 1110

b 11110

v 11111


Huffman Coding

• These codes are then used to encode the string

• Thus, “duke blue devils” turns into:010 011 1110 00 101 11110 100 011 00 101 010 00 11111 1100 100 1101

• When grouped into 8-bit bytes:01001111 10001011 11101000 11001010 10001111 11100100 1101xxxx

• Thus it takes 7 bytes of space compared to 16characters * 1 byte/char = 16 bytesuncompressed


Huffman Coding• Uncompressing works by reading in the file bit

by bit– Start at the root of the tree– If a 0 is read, head left– If a 1 is read, head right– When a leaf is reached decode that character and start

over again at the root of the tree

• Thus, we need to save Huffman tableinformation as a header in the compressed file– Doesn’t add a significant amount of size to the file for

large files (which are the ones you want to compressanyway)

– Or we could use a fixed universal set ofcodes/freqencies


Entropy Coding Algorithms

(Content Dependent Coding)

• Run-length Encoding (RLE)

– Replaces sequence of the same consecutive bytes with number of occurrences

– Number of occurrences is indicated by a special flag (e.g., !)

– Example:

• abcccccccccdeffffggg (20 Bytes)

• abc!9def!4ggg (13 bytes)


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Variations of RLE (Zero-suppression

technique)

• Assumes that only one symbol appearsoften (blank)

• Replace blank sequence by M-byte and abyte with number of blanks in sequence

– Example: M3, M4, M14,…

• Some other definitions are possible

– Example:

• M4 = 8 blanks, M5 = 16 blanks, M4M5=24 blanks


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Adaptive Coding

Motivations:– The previous algorithms (Huffman) require the statistical

knowledge which is often not available (e.g., live audio, video).

– Even when it is available, it could be a heavy overhead.

– Higher-order models incur more overhead. For example, a 255entry probability table would be required for a 0-order model. Anorder-1 model would require 255 such probability tables. (Aorder-1 model will consider probabilities of occurrences of 2symbols)

The solution is to use adaptive algorithms. AdaptiveHuffman Coding is one such mechanism that we willstudy.

The idea of “adaptiveness” is however applicable to otheradaptive compression algorithms.


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Adaptive Coding

ENCODER

Initialize_model();

do {

c = getc( input );

encode( c, output );

update_model( c );

} while ( c != eof)

DECODER

Initialize_model();

while ( c = decode (input)) != eof) {

putc( c, output)

update_model( c );

}

r The key is that, both encoder and decoder use exactly the same initialize_model and update_model routines.


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The Sibling Property

The node numbers will be assigned in such a waythat:

1. A node with a higher weight will have a higher nodenumber

2. A parent node will always have a higher node numberthan its children.

In a nutshell, the sibling property requires that thenodes (internal and leaf) are arranged in order ofincreasing weights.

The update procedure swaps nodes in violation ofthe sibling property.– The identification of nodes in violation of the sibling

property is achieved by using the notion of a block.

– All nodes that have the same weight are said to belongto one block


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Flowchart of the update procedure

START

First appearance of symbol

Go to symbol external node

Node number max

in block?

Incrementnode weight

Switch node withhighest numbered

node in block

Is thisthe root

node?

Go toparent node

STOP

NYT gives birthTo new NYT and

external node

Increment weight of external node

and old NYT node; Adjust node

numbers

Go to oldNYT node

Yes

No

No

Yes

Yes

No

The Huffman tree is

initialized with a single

node, known as the Not-

Yet-Transmitted (NYT) or

escape code. This code

will be sent every time

that a new character,

which is not in the tree,

is encountered, followed

by the ASCII encoding of

the character. This allows

for the de-compressor

to distinguish between a

code and a new

character. Also, the

procedure creates a new

node for the character chapter3: Multimedia Compression

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Example

BW=2#1

CW=2#2

DW=2#3

W=2#4

W=4#5

W=6#6

EW=10#7

Root W=16#8

Counts:(number of occurrences)

B:2C:2D:2E:10

Example Huffman tree after some symbols have been processed in accordance

with the sibling property

NYT

NYT

#0

Initial Huffman Tree

#0


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Example

W=1#2

BW=2#3

CW=2#4

DW=2#5

W=2+1#6

W=4#7

W=6+1#8

EW=10#9

Root W=16+1

#10

Counts:(number of occurrences)

A:1B:2C:2D:2E:10

A Huffman tree after first appearance of symbol A

AW=1#1

NYT

#0


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Increment

BW=2#3

CW=2#4

DW=2#5

W=3+1#6

W=4#7

W=7+1#8

EW=10#9

Root W=17+1

#10

Counts:A:1+1

B:2C:2D:2E:10

An increment in the count for A propagates up to the root

W=1+1#2

AW=1+1

#1

NYT

#0


36

Swapping

BW=2#3

CW=2#4

DW=2#5

W=4#6

W=4#7

W=8#8

EW=10#9

Root W=18#10

Counts:A:2+1

B:2C:2D:2E:10

BW=2#3

CW=2#4

AW=2+1

#5

W=4#6

W=4+1#7

W=8+1#8

EW=10#9

Root W=18+1

#10

Counts:A:3B:2C:2D:2E:10

Swap nodes 1 and 5

Another increment in the count for A results in swap

W=2#2

AW=2#1

NYT

W=2#2

DW=2#1

NYT

#0

#0chapter3: Multimedia Compression

37

Swapping … contd.

BW=2#3

CW=2#4

AW=3+1

#5

W=4#6

W=5+1#7

W=9+1#8

EW=10#9

Root W=19+1

#10

Counts:A:3+1

B:2C:2D:2E:10

Another increment in the count for A propagates up

W=2#2

DW=2#1

NYT

#0


38

Swapping … contd.

BW=2#3

CW=2#4

AW=4#5

W=4#6

W=6#7

W=10#8

EW=10#9

Root W=20#10

Counts:A:4+1

B:2C:2D:2E:10

Swap nodes 5 and 6

Another increment in the count for A causes swap of sub-tree

W=2#2

DW=2#1

NYT

#0


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Swapping … contd.

CW=2#4

W=6#7

W=10#8

EW=10#9

Root W=20#10

Counts:A:4+1

B:2C:2D:2E:10

BW=2#3

W=4#5

AW=4+1

#6

Swap nodes 8 and 9

Further swapping needed to fix the tree

W=2#2

DW=2#1

NYT

#0


40

Swapping … contd.

CW=2#4

W=6#7

W=10+1#9

Root W=20+1

#10

Counts:A:5B:2C:2D:2E:10

BW=2#3

W=4#5

AW=5#6

EW=10#8

W=2#2

DW=2#1

NYT

#0


Lempel-Ziv-Welch (LZW) Compression Algorithm

Introduction to the LZW Algorithm

Example 1: Encoding using LZW

Example 2: Decoding using LZW

LZW: Concluding Notes


Introduction to LZW

As mentioned earlier, static coding schemes requiresome knowledge about the data before encoding takesplace.

Universal coding schemes, like LZW, do not requireadvance knowledge and can build such knowledge on-the-fly.

LZW is the foremost technique for general purpose datacompression due to its simplicity and versatility.

It is the basis of many PC utilities that claim to “doublethe capacity of your hard drive”

LZW compression uses a code table, with 4096 as acommon choice for the number of table entries.


Introduction to LZW (cont'd)

Codes 0-255 in the code table are always assigned torepresent single bytes from the input file.

When encoding begins the code table contains only thefirst 256 entries, with the remainder of the table beingblanks.

Compression is achieved by using codes 256 through4095 to represent sequences of bytes.

As the encoding continues, LZW identifies repeatedsequences in the data, and adds them to the code table.

Decoding is achieved by taking each code from thecompressed file, and translating it through the code tableto find what character or characters it represents.


LZW Encoding Algorithm

1 Initialize table with single character strings

2 P = first input character

3 WHILE not end of input stream

4 C = next input character

5 IF P + C is in the string table

6 P = P + C

7 ELSE

8 output the code for P

9 add P + C to the string table

10 P = C

11 END WHILE

12 output code for P


Example 1: Compression using LZW

Example 1: Use the LZW algorithm to compress the string

BABAABAAA


Example 1: LZW Compression Step 1

BABAABAAA P=A

C=empty

STRING TABLEENCODER OUTPUT

stringcodewordrepresentingoutput code

BA256B66



BABAABAAA P=B

C=empty



BA256B66

AB257A65



BABAABAAA P=A

C=empty



BA256B66

AB257A65

BAA258BA256



BABAABAAA P=A

C=empty



BA256B66

AB257A65

BAA258BA256

ABA259AB257



BABAABAAA P=A

C=A



BA256B66

AB257A65

BAA258BA256

ABA259AB257

AA260A65



BABAABAAA P=AA

C=empty



BA256B66

AB257A65

BAA258BA256

ABA259AB257

AA260A65

AA260


LZW Decompression

The LZW decompressor creates the same string tableduring decompression.

It starts with the first 256 table entries initialized tosingle characters.

The string table is updated for each character in theinput stream, except the first one.

Decoding achieved by reading codes and translatingthem through the code table being built.


LZW Decompression Algorithm

1 Initialize table with single character strings

2 OLD = first input code

3 output translation of OLD

4 WHILE not end of input stream

5 NEW = next input code

6 IF NEW is not in the string table

7 S = translation of OLD

8 S = S + C

9 ELSE

10 S = translation of NEW

11 output S

12 C = first character of S

13 OLD + C to the string table

14 OLD = NEW

15 END WHILE


Example 2: LZW Decompression 1

Example 2: Use LZW to decompress the output sequence of

Example 1:

<66><65><256><257><65><260>.


Example 2: LZW Decompression Step 1

<66><65><256><257><65><260> Old = 65 S = A

New = 66 C = A


stringcodewordstring

B

BA256A



<66><65><256><257><65><260> Old = 256 S = BA

New = 256 C = B



B

BA256A

AB257BA



<66><65><256><257><65><260> Old = 257 S = AB

New = 257 C = A



B

BA256A

AB257BA

BAA258AB



<66><65><256><257><65><260> Old = 65 S = A

New = 65 C = A



B

BA256A

AB257BA

BAA258AB

ABA259A



<66><65><256><257><65><260> Old = 260 S = AA

New = 260 C = A



B

BA256A

AB257BA

BAA258AB

ABA259A

AA260AA


LZW: Some Notes

This algorithm compresses repetitive sequences of datawell.

Since the codewords are 12 bits, any single encodedcharacter will expand the data size rather than reduce it.

In this example, 72 bits are represented with 72 bits ofdata. After a reasonable string table is built, compressionimproves dramatically.

Advantages of LZW over Huffman: LZW requires no prior information about the input data stream. LZW can compress the input stream in one single pass. Another advantage of LZW its simplicity, allowing fast

execution.


LZW: Limitations

What happens when the dictionary gets too large (i.e., when all the4096 locations have been used)?

Here are some options usually implemented:

Simply forget about adding any more entries and use the tableas is.

Throw the dictionary away when it reaches a certain size.

Throw the dictionary away when it is no longer effective atcompression.

Clear entries 256-4095 and start building the dictionary again.

Some clever schemes rebuild a string table from the last Ninput characters.


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Chapter 3: Multimedia Data Compression