Signal Compression and JPEG

-1-

Overview

Sampling Fine Quantization

Transform Quantizer VLC encoding

A/D CONVERTER

COMPRESSION/SOURCE CODING

CHANNEL CODING

Error protect.Encipher

PCM encoded or “raw” signal(“wav”, “bmp”, …)

Compressed bit stream

(mp3, jpg, …)

Channel bit stream

Analog capturing

device(camera,

microphone)

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Reduce the #Amplitudes

24 bit (16777200 different colors)

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8 bit (256 different colors) CF 3


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8 bit (256 different gray values) CF 3


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JPEG CF 15

But there are More Cleaver Ways

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Why can Signals be Compressed?Because infinite accuracy of signal amplitudes is (perceptually) irrelevant

24 bit (16777200 different colors) 8 bit (256 different colors)Compression factor 3

Rate-distortion theory, scalar/vector quantization

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Why can Signals be Compressed?Because signal amplitudes are statistically redundant

Information theory, Huffman coding

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Why can Signals be Compressed?Because signal amplitudes are mutually dependent

0 500 1000 1500 2000 2500 3000 3500 4000 450050

100

150

200

250

“S”

0 1000 2000 3000 4000 5000100

150

200

“O”

Rate-distortion theory, transform coding

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Why can Signals be Compressed?Example of signal with no dependencies between successive amplitudes (Gaussian uncorrelated noise)

Indeed “noise” compresses badly

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System Overview



A/D CONVERTER


CHANNEL CODING



Compressed bit stream(mp3, jpg, …)

Channel bit stream

Analog capturingdevice(camera, microphone)

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Quantization

p(x)

Q(x)

Quantizer level more probable

Even Uniform Odd Uniform

Even Non-uniform Odd Non-uniform

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Information and Entropy

More ProbableLess information

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Huffman Code

y4 0.51 0.51 (0) 0y3 0.20 0.29 (0) 0.49 (1) 11y2 0.14 0.15 (0) 0.20 (1) 101y5 0.08 0.08 (0) 0.14 (1) 1000y6 0.04 0.04 (0) 0.07 (1) 10010y1 0.02 0.02 (0) 0.03 (1) 100110y7 0.005 (0) 0.01 (1) 1001110y0 0.005 (1) 1001111

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Run Length CodingRepresenting “0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 1 1 0 0 ...”“Runs” : 8 (“zeros”) 2 (“ones”) 6 (“zeros”) 5 (“ones”) ……The run lengths are also encoded (e.g. with Huffman coding)Efficient transforms (like DCT) used in compression produce

A lot of “zero” valuesAnd a “few” (significant) non-zero values

Typical symbol sequences to be coded “5 1 0 0 0 0 0 0 0 3 0 0 6 0 0 0 0 1 0 0 0 0 …..”

will be done by {zero-run, non-zero symbol/0} pairsHere: “{0,5},{0,1},{7,3},{2,6},{4,1},…..”The pairs will now be assigned a Huffman codeThis is used in JPEG

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General Compression System



A/D CONVERTER


CHANNEL CODING




Channel bit stream


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Correlation in Signals - IMeaningful signals are often highly predictable:

(Linear) Predictability has something to do with the autocorrelation function

0 500 1000 1500 2000 2500 3000 3500 4000 450050

100

150

200

2504

500 1000 1500 2000 2500 3000 3500 4000 4500100

0

100

Δx(n)=x(n)-x(n-1)

n

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Principle of Differential PCM

- Q VLC

Previoussignal values (“past”)

x(n) Δx(n) 001010010Δx*(n)

Predict x(n)

)(ˆ nx

x(n-1), x(n-2), x(n-3), ….

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Works Really Good - I

Variance = 3240

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Works Really Good - II

Variance = 315

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DPCM

Predicted signal

Signal to be encoded Prediction difference/error

Reconstructed signal

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What Linear Predictor to Use?Examples:

PCM

Simple differences

Average last two samples

General linear predictor

$( )x n = 0

$( ) ~( )x n x n= −1

$( ) ~( ) ~( )x n h x n h x n= − + −1 21 2

$( ) ~( )x n h x n kkk

N

= −=∑

1

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DPCM on Images

Same principle as 1-DDefinition of “Past”and “Future” in Images:

Predictions:horizontal (scan line)vertical (column)2-dimensional

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General Compression System



A/D CONVERTER


CHANNEL CODING




Channel bit stream


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Transform Coding

CorrelatingTransform

DecorrelatingTransform Q Q-1Vectorize

InverseVectorize

x(n) x θ $θ $x $( )x n

Which transform used?

T T-1channel

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Removing Correlation - I

0 200 400 600 800 1000100

50

0

50

100

40 20 0 20 40

40

20

20

40

xx

1

2

⎛

⎝⎜

⎞

⎠⎟

x1

x2

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Removing Correlation - II

0 200 400 600 800 1000100

50

0

50

100

xx

1

2

⎛

⎝⎜

⎞

⎠⎟

40 20 0 20 40

40

20

20

40

x1

x2

σ22

σ12

θ1

θ2

Rotate the coordinates

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Removing Correlation - III

xx

1

2

⎛

⎝⎜

⎞

⎠⎟

x1

x2

0 100 200 300 400 500 600 700 800 90050

0

50

40 20 0 20 40

40

20

20

40

θ1

θ2

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Decomposition

x t t t t= = + + +=∑ θ θ θ θk kk

N

N N1

1 1 2 2 L

t1

t8

0.707

0.25

-0.1250.00.00.063

-0.2-0.025

x

θ(Example)

3.1651.718

2.266

4.516

1.020

0.638

3.377

-0.702

θ

θ

k k n nn

N

n k n kk

N

t x k N

x t n N

= =

= =

=

=

∑

∑

,

,

, , ,

, , ,

1

1

1 2

1 2

K

K

θθ

θ=

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟=

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟=

1 11 1

1

1

M

L

M O M

L

M

N

N

N N N N

t t

t t

x

x

, ,

, ,

Tx

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Discrete Cosine Transform

t1

t8

t2

t3

t4

t5

t6

t7

N=8

Tt =

0.354

0.354

0.354

0.354

0.354

0.354

0.354

0.354

0.49

0.416

0.278

0.098

0.098

0.278

0.416

0.49

0.462

0.191

0.191

0.462

0.462

0.191

0.191

0.462

0.416

0.098

0.49

0.278

0.278

0.49

0.098

0.416

0.354

0.354

0.354

0.354

0.354

0.354

0.354

0.354

0.278

0.49

0.098

0.416

0.416

0.098

0.49

0.278

0.191

0.462

0.462

0.191

0.191

0.462

0.462

0.191

0.098

0.278

0.416

0.49

0.49

0.416

0.278

0.098

(All picture compression standards implement this matrix in a clever way)

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Image Transforms - I

6 0

4 6

1 1

1 1

1 -1

1 -1

1 1

-1 -1

1 -1

-1 14 1 -1 2=

t11 t12 t21 t22

θ11 θ12 θ21 θ22

xm n,

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Basis Images for 2-D DCT

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Discrete Cosine Transform (DCT) (1)

8 pixels

8 rows

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Cosine patterns/DCT basis functions

64 pixels

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Cosine patterns/DCT basis functions

64 pixels

89 8 8 1 0 -1 1 0

2 3 0 0 0 0 0 0

0 1 0 1 0 0 0 0

0 0 0 1 0 0 0 0

2 0 0 1 0 1 0 0

0 0 -1 1 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 0

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!

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2-D DCT of Picture - I

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2-D DCT of Image - II

A DCT coefficients isa weight of particularDCT basis function.

Low Low frequenciesfrequencies

High High frequenciesfrequencies

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“Grouped” DCT Coefficients

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JPEG Compression System

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DCT Weight Matrix and QuantizationQuantization:

Recommended JPEG normalization matrix

$ [ ], ,,

',

θ θθ

k l k lk l

k lQ

Q N= =

⎛

⎝⎜⎜

⎞

⎠⎟⎟round

N k l, =

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

16 11 10 16 24 40 51 6112 12 14 19 26 58 60 5514 13 16 24 40 57 69 5614 17 22 29 51 87 80 6218 22 37 56 68 109 103 7724 35 55 64 81 104 113 9249 64 78 87 103 121 120 10172 92 95 98 112 100 103 99

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User Controllable Quality

User has control over a “quality parameter” that runs from 100 (“perfect”) to 0 (“extremely poor”)

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6Q'

Q

Increasing quality

Par

amet

er u

sed

to s

cale

th

e no

rmal

izat

ion

mat

rix

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Entropy coding

Huffman codedRunlength coding then huffman orarithmetic coding

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Example

f k l, =

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

139 144 149 153 155 155 155 155144 151 153 156 159 156 156 156150 155 160 163 158 156 156 156159 161 162 160 160 159 159 159159 160 161 162 162 155 155 155161 161 161 161 160 157 157 157162 162 161 163 162 157 157 157162 162 161 161 163 158 158 158

θk l, =

− − − −− − − − − −− − − − −− −− − −

− −− − −− − − −

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

1260 1 12 5 2 2 3 123 17 6 3 3 0 0 111 9 2 2 0 1 1 07 2 0 1 1 0 0 01 1 1 2 0 1 1 1

2 0 2 0 1 1 1 11 0 0 1 0 2 1 13 2 4 2 2 1 1 0

DCT transform isexactly defined inJPEG standardaverage * 8

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Example

$,θk l =

−− −− −

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

79 0 1 0 0 0 0 02 1 0 0 0 0 0 01 1 0 0 0 0 0 0

0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0

DC: Difference with quantized DC coefficient of previous block is Huffman encoded AC: Zig-zag scan coefficients, and convert to (zero run-length, amplitude) combinations:

(79) 0 -2 -1 -1 -1 0 0 -1 EOB{1,-2}{0,-1}{0,-1}{0,-1} {2,-1} EOB

Quantizationusing Q’ = 1

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VLC Coding of AC CoefficientsThe (zero run-length, amplitudes) are put into categories

The (zero run-length, categories) are Huffman encodedThe sign and offset into a category are FLC encoded (required #bits = category number)

Category AC Coefficient Range1 -1,12 -3,-2,2,33 -7,...,-4,4,...,74 -15,...,-8,8,...,155 -31,...,-16,16,...,316 -63,...,-32,32,...,637 -127,...,-64,64,...,1278 -255,...,-128,128,...,2559 -511,...,-256,256,...,51110 -1023,...,-512,512,...,1023

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JPEG AC Huffman TableZ ero R un C ategory C ode leng th C odeword

0 1 2 000 2 2 010 3 3 1000 4 4 10110 5 5 110100 6 6 1110000 7 7 1111000. . . .. . .1 1 4 11001 2 6 1110011 3 7 11110011 4 9 111110110. . . .. . . .2 1 5 110112 2 8 11111000. . . .. . . .3 1 6 1110103 2 9 111110111. . . .. . . .4 1 6 1110115 1 7 11110106 1 7 11110117 1 8 111110018 1 8 111110109 1 9 111111000

10 1 9 11111100111 1 9 111111010. . . .. . . .

E O B 4 1010

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Example - III

The series {1,-2}{0,-1}{0,-1}{0,-1} {2,-1} EOBnow becomes

111001 01/00 0/00 0/00 0/11011 0/1010

Bit rate for AC coefficients in this DCT block 27 bits/64 pixels = 0.42 bit/pixel

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Ordering of Coefficient

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Sequential Encoding

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AcknowledgementProf. Inald Lagendijk, Delft University of TechnologyAlso try the software VcDemo

http://ict.ewi.tudelft.nl/index.php?Itemid=124

Signal Compression and JPEG

Technology

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