Image Compression - Nottinghampszqiu/Teaching/G52IVG/Topic04-Lectures.pdf · Image Compression Palletized color image (gif) r0 g0 b0 r1 g1 b1 r255 g255 b255 (r, g, b) Colour Table

1G52IVG, School of Computer Science, University of Nottingham

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 −== Lknnrp k

k


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


Image Compression


some pixel values more common than others


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


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


Image Compression

Fundamentals: Coding redundancyVariable length Coding: assign fewer bits to the more probable gray levels than to less probable ones can achieve data compression


Image Compression

Fundamentals: Interpixel (spatial) redundancy

neighboring pixels have similar values

Binary imagesGray scale image (later)


Image Compression

Fundamentals: Interpixel (spatial) redundancy: Binary images

Run-length codingMapping 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


Image Compression

Example: Run-length coding

decode

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16123456

111213141516

789

123456789

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

111213141516

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


Image Compression

Fundamentals: Psychovisualredundancy

some color differences are imperceptible


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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Image Compression

Fidelity criteria

erms and SNR are convenient objective measuresMost decompressed images are view by human beingsSubjective evaluation of compressed image quality by human observers are often more appropriate


Image Compression

Fidelity criteria


Image Compression

Image compression models


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.


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


Image Compression


Two common methods

Huffman coding and,LZW coding


Image Compression


Huffman Coding

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


Image Compression


Huffman Coding


Image Compression


Huffman Coding


Image Compression


Huffman Coding

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


Image Compression


Huffman Coding


Image Compression

Exploiting Spatial/InterpixelRedundancy

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


Image Compression

Exploiting Spatial/Interpixel Redundancy

Predictive CodingDifferential 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, ……


Image Compression


Predictive CodingKey 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)


Image Compression


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

1D Causal:

1D Non-causal:

2D Causal:

2D Non-Causal:


Image Compression


2D Causal:


Image Compression

Quantization

Quantization: Widely Used in Lossy CompressionRepresent certain image components with fewer bits (compression)With unavoidable distortions (lossy)

Quantizer DesignFind the best tradeoff between

maximal compression minimal distortion


Image Compression

Quantization

Scalar quantization

24 408 ... 248

∆ ∆∆ ∆

∆2∆1 ∆3 ∆4

Uniform scalar quantization:

Non-uniform scalar quantization:


Image Compression

Quantization

Scalar quantization


Image Compression

Quantization

Vector quantization and palletized images (gif format)


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



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 psychovisualredundancy


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


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?


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


Image Compression


R

G

B

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


Image Compression


R

G

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


Image Compression

R

G

B r0 g0 b0

r1 g1 b1

r255 g255 b255

Colour Table

Representative colours placed in the colour table


Image Compression

Discrete Cosine Transform (DCT)2D-DCT

Inverse 2D-DCT

∑∑−

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where


Image Compression

−−

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2D-DCTimage block

DCT block

low frequency

high frequency

low frequency

high frequency

DC component


JPEG Compression

169130173129

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DCT

scalar quantization

zig-zag scan

−−−

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0000

0000

0000

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0000

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1011

0111

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0123

2113

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JPEG Compression

DecodedImage

RG

BChrominanceUpsampling

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

8 X 8IDCT

YVU colorcoordinate

Dequantizer

QuantizationTable

De-zigzag

De-DC coding

HuffmanDecoding

HuffmanDecoding

Bit-stream


JPEG Compression

=

),(),(),('vuQvuXRoundvuX

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

• The Baseline System – Quantization


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”


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

( , )( , )( , )

QuantizationF u vF u v roundQ u v

=

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

Quantization

Dequantization


JPEG Compression

( , )( , )( , )

QuantizationF u vF u v roundQ u v

=

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

Quantization

Dequantization

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

YQ

=

17 18 24 47 99 99 99 9918 21 26 66 99 99 99 9924 26 56 99 99 99 99 9947 66 99 99 99 99 99 9999 99 99 99 99 99 99 9999 99 99 99 99 99 99 9999 99 99 99 99 99 99 9999 99 99 99 99 99 99 99

CQ

=


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.


JPEG Compression


NTSC Video

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


JPEG Compression


In JPEG and MPEG

Cb and Cr are sub-sampled

Image Compression - Nottinghampszqiu/Teaching/G52IVG/Topic04-Lectures.pdf · Image Compression Palletized color image (gif) r0 g0 b0 r1 g1 b1 r255 g255 b255 (r, g, b) Colour Table

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