image compression

G52IIP, School of Computer Science, University of Nottingham

1

Image Compression

Fundamentals: Coding redundancy

The gray level histogram of an image can reveal a great deal of information about the image

That probability (frequency) of occurrence of gray level rk is p(rk),

1,...,2,0 Lkn

nrp kk


2

Image Compression


If the number of bits used to represent each value of rk is l(rk), the average number of bits required to represent each pixel is

To code an MxN image requires MNLavg bits

1

0

L

kkkavg rprlL


3

Image Compression


some pixel values more common than others


4

Image Compression



If m-bit natural binary code is used to represent the the gray levels, then

1

0

L

kkkavg rprlL

mrmprprlLL

kk

L

kkkavg

1

0

1

0


5

Image Compression



If m-bit natural binary code is used to represent the the gray levels, then

1

0

L

kkkavg rprlL

mrmprprlLL

kk

L

kkkavg

1

0

1

0


6

Image Compression

Fundamentals: Coding redundancy Variable length Coding: assign fewer bits to

the more probable gray levels than to less probable ones can achieve data compression


7

Image Compression

Fundamentals: Interpixel (spatial) redundancy

neighboring pixels have similar values

Binary imagesGray scale image (later)


8

Image Compression

Fundamentals: Interpixel (spatial) redundancy: Binary imagesRun-length coding

Mapping the pixels along each scan line into a sequence of pairs (g1, r1), (g2, r2), …,

Where gi is the ith gray level, ri is the run length of ith run


9

Image Compression

Example: Run-length coding

decode

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

123456

11

12

13

14

15

16

789

123456789

1 2 3 4 5 6 7 8 9 11 12 13 14 15 16

11

12

13

14

15

16

10

10 10

Row 1: (0, 16)Row 2: (0, 16)Row 3: (0, 7) (1, 2) (0, 7)Row 4: (0, 4), (1, 8) (0, 4)Row 5: (0, 3) (1, 2) (0, 6) (1, 3) (0, 2)Row 6: (0,2) (1, 2) (0,8) (1, 2) (0, 2)Row 7: (0, 2) (1,1) (0, 10) (1,1) (0, 2)Row 8: (1, 3) (0, 10) (1,3)Row 9: (1, 3) (0, 10) (1, 3)Row 10: (0,2) (1, 1) (0,10) (1, 1) (0, 2)Row 11: (0, 2) (1, 2) (0, 8) (1, 2) (0, 2)Row 12: (0, 3) (1, 2) (0, 6) (1, 3) (0, 2)Row 13: (0, 4) (1,8) (0, 4)Row 14: (0, 7) (1, 2) (0, 7)Row 15: (0, 16)Row 16: (0, 16)

encode


10

Image Compression

Fundamentals: Psychovisual redundancy

some color differences are imperceptible


11

Image Compression

Fidelity criteria

Root mean square error (erms) and signal to noise ratio (SNR):

Let f(x,y) be the input image, f’(x,y) be reconstructed input image from compressed bit stream, then

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12

Image Compression

Fidelity criteria

erms and SNR are convenient objective measures

Most decompressed images are view by human beings

Subjective evaluation of compressed image quality by human observers are often more appropriate


13

Image Compression

Fidelity criteria


14

Image Compression

Image compression models


15

Image Compression

Exploiting Coding Redundancy

These methods, from information theory, are not limited to images, but apply to any digital information. So we speak of “symbols” instead of “pixel values” and “sources” instead of “images”.

The idea: instead of natural binary code, where each symbol is encoded with a fixed-length code word, exploit nonuniform probabilities of symbols (nonuniform histogram) and use a variable-length code.


16

Image Compression


Entropy

is a measure of the information content of a source.

If source is an independent random variable then you can’t compress to fewer than H bits per symbol.

Assign the more frequent symbols short bit strings and the less frequent symbols longer bit strings. Best compression when redundancy is high (entropy is low, histogram is highly skewed).

1

02 )(log)(

L

kkk rprpH


17

Image Compression


Two common methods

Huffman coding and, LZW coding


18

Image Compression


Huffman Coding

Codebook is precomputed and static. Compute probabilities of each symbol by histogramming

source. Process probabilities to precompute codebook: code(i). Encode source symbol-by-symbol: symbol(i) -> code(i). The need to preprocess the source before encoding begins

is a disadvantage of Huffman coding


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Image Compression


Huffman Coding


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Image Compression


Huffman Coding


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Image Compression


Huffman Coding

Average length of the code is 2.2.bits/symbol The entropy of the source is 2.14 bits/symbol


22

Image Compression


Huffman Coding


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Image Compression

Exploiting Spatial/Interpixel Redundancy

Predictive Coding Image pixels are highly correlated (dependent) Predict the image pixels to be coded from those

already coded


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Image Compression


Predictive Coding Differential Pulse-Code Modulation (DPCM) Simplest form: code the difference between pixels

Original pixels:82, 83, 86, 88, 56, 55, 56, 60, 58,

55, 50, ……

DPCM:82, 1, 3, 2, -32, -1, 1, 4, -2, -3, -

5, ……


25

Image Compression


Predictive Coding Key features: Invertible, and lower entropy (why?)

H I(k) HD(k)

high entropy image reduced entropy imageK-10 K-11-K

image histogram (high entropy)

DPCM histogram (low entropy)


26

Image Compression


Higher Order (Pattern) Prediction Use both 1D and 2D patterns for prediction

1D Causal:

1D Non-causal:

2D Causal:

2D Non-Causal:


27

Image Compression


2D Causal:


28

Image Compression

Quantization

Quantization: Widely Used in Lossy Compression

Represent certain image components with fewer bits (compression)

With unavoidable distortions (lossy)

Quantizer Design Find the best tradeoff between

maximal compression minimal distortion


29

Image Compression

Quantization

Scalar quantization

24 408 ... 248

21 3 4

Uniform scalar quantization:

Non-uniform scalar

quantization:


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Image Compression

Quantization

Scalar quantization


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Image Compression

Quantization

Vector quantization and palletized images (gif format)


32

Image Compression

Palletized color image (gif) A true colour image – 24bits/pixel, R – 8 bits, G – 8 bits, B –

8 bits

A gif image - 8bits/pixel

1677216 possible colours



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Image Compression

Palletized color image (gif) A true colour image – 24bits/pixel, R – 8 bits, G – 8 bits, B –

8 bits

A gif image - 8bits/pixel



Exploits psychovisua

l redundancy


34

Image Compression

Palletized color image (gif)

r0 g0 b0

r1 g1 b1

r255 g255 b255

(r, g, b)

Colour Table

For each pixel in the original image

Find the closest colour in the Colour Table

Record the index of that colour (for storage or transmission)

To reconstruct the image, place the indexed colour from the Colour Table at the corresponding spatial location


35

Image Compression

Palletized color image (gif)

r0 g0 b0

r1 g1 b1

r255 g255 b255

(r, g, b)

Colour Table

For each pixel in the original image

Find the closest colour in the Colour Table

Record the index of that colour (for storage or transmission)

To reconstruct the image, place the indexed colour from the Colour Table at the corresponding spatial location

How to choose the colours in the table/pallet?


36

Image Compression

Construct the pallet (vector quantization, k-means algorithm)

(r, g, b)

R

G

B

A pixel corresponding toA point in the 3 dimensional R, G, B space


37

Image Compression


R

G

B

Map all pixels into the R, G, B space, “clouds” of pixels are formed


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Image Compression


R

G

BGroup pixels that are close to each other, and replace them by one single colour


39

Image Compression

R

G

Br0 g0 b0

r1 g1 b1

r255 g255 b255

Colour Table

Representative colours placed in the colour table


40

Image Compression

Discrete Cosine Transform (DCT) 2D-DCT

Inverse 2D-DCT

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41

Image Compression

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2D-DCTimage block

DCT block

low frequenc

y

high frequenc

ylow

frequency

high frequency

DC component


42

JPEG Compression

Image

RG

B

YVU colorcoordinate

ChrominanceDownsampling(4:2:2 or 4:2:0)

8 X 8 FDCT

Quantizer

QuantizationTable

zigzag

DifferentialCoding

HuffmanEncoding

HuffmanEncoding

Bit-stream

DecodedImage

RG

B

ChrominanceUpsampling

(4:2:2 or 4:2:0)

8 X 8IDCT

YVU colorcoordinate

Dequantizer

QuantizationTable

De-zigzag

De-DC coding

HuffmanDecoding

HuffmanDecoding

Bit-stream


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JPEG Compression

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DCT

scalar quantization

zig-zag scan

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JPEG Compression

),(

),(),('

vuQ

vuXRoundvuX

X(u,v): original DCT coefficientX’(u,v): DCT coefficient after quantizationQ(u,v): quantization value

• The Baseline System – Quantization


45

JPEG Compression

Why quantization?

to achieve further compression by representing DCT coefficients with no greater precision than is necessary to achieve the desired image quality

Generally, the “high frequency coefficients” has larger quantization values

Quantization makes most coefficients to be zero, it makes the compression system efficient, but it’s the main source that make the system “lossy”


46

JPEG Compression

Quantization is the step where we actually throw away data.Luminance and Chrominance Quantization Table Smaller numbers in the upper left direction larger numbers in the lower right directionThe performance is close to the optimal condition

( , )( , )

( , )Quantization

F u vF u v round

Q u v

( , ) ( , ) ( , )deQ QuantizationF u v F u v Q u v

Quantization

Dequantization


47

JPEG Compression

( , )( , )

( , )Quantization

F u vF u v round

Q u v

( , ) ( , ) ( , )deQ QuantizationF u v F u v Q u v

Quantization

Dequantization

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12 12 14 19 26 58 60 55

14 13 16 24 40 57 69 56

14 17 22 29 51 87 80 62

18 22 37 56 68 109 103 77

24 35 55 64 81 104 113 92

49 64 78 87 103 121 120 101

72 92 95 98 112 100 103 99

YQ

17 18 24 47 99 99 99 99

18 21 26 66 99 99 99 99

24 26 56 99 99 99 99 99

47 66 99 99 99 99 99 99

99 99 99 99 99 99 99 99

99 99 99 99 99 99 99 99

99 99 99 99 99 99 99 99

99 99 99 99 99 99 99 99

CQ


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JPEG Compression

Exploiting Psychovisual Redundancy

Exploit variable sensitivity of humans to colors:

We’re more sensitive to differences between dark intensities than bright ones.

Encode log(intensity) instead of intensity.

We’re more sensitive to high spatial frequencies of green than red or blue.

Sample green at highest spatial frequency, blue at lowest.

We’re more sensitive to differences of intensity in green than red or blue.

Use variable quantization: devote most bits to green, fewest to blue.


49

JPEG Compression


NTSC Video

Y bandlimited to 4.2 MHzI to 1.6 MHzQ to .6 MHz


50

JPEG Compression


In JPEG and MPEG

Cb and Cr are sub-sampled

image compression

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