Image Coding/ CompressionImage Coding/ Compression

David HemmertPradeep SuthramTammo Heeren

All Mathcad files [MCD/PDF] can be found on:http://webpages.acs.ttu.edu/theeren

OverviewOverview

ReviewDCT (Discrete Cosine Transform)JPEG compression/ decompressionWavelet compression/ decompression

ReviewReviewLinear QuantizationLinear Quantization

0 17 34 51 68 85 102 119 136 153 170 187 204 221 238 2550

17

34

51

68

85

102

119

136

153

170

187

204

221

238

255Quantization steps

Grayscale levels

quan

tized

gra

ysca

le le

vels

Quantization of gray levels in equidistance quantization steps

ReviewReviewadaptive Quantizationadaptive Quantization

0 17 34 51 68 85 102 119 136 153 170 187 204 221 238 2550

17

34

51

68

85

102

119

136

153

170

187

204

221

238

255Quantization steps

Grayscale levels

quan

tized

gra

ysca

le le

vels

0 17 34 51 68 85 102 119 136 153 170 187 204 221 238 2550

2000

4000

6000Image histogram

DCTDCT

Discrete Cosine transform Transformation of spatial image information into

its spatial frequency components

f

f0

DCT MathDCT Math

DCTfy fx G

fy fx0

X 1

x 0

Y 1

y

imagey x kx x fx( ) ky y fy( )

IDCTy x

0

X 1

fx 0

Y 1

fy

Gfy fx DCT

fy fx kx x fx( ) ky y fy( )

kx x fx( ) cos 2 x 1( )fx 2 X

ky y fy( ) cos 2 y 1( )fy 2 Y

DCTDCT

Essentially taking the 2D fourier transform and only keeping the real part of the coefficients

Works with any orthogonal kernel (e.g. in wavelet compression/ decompression)

DCT used in JPEG coding/ decoding

DCT ResultsDCT Results

1 10 100 1 1030

10

20

30

40

50

60SNR vs. Compression Ratio

Compression Ration

SN

R

0.005%/ 22000 / 8.8 dB 0.1%/ 864/ 10.3 dB 0.8%/ 128/ 13.5 dB 2%/ 49/ 15.2 dB

13%/ 7.7 / 20.4 dB

5%/ 20 / 17.5 dB

82%/ 1.2 / 34.4 dB

SNR and visual artifactSNR and visual artifact

Procedure/ TransformSNR of no

visual artifactsCompression

ratio

Linear quantization 35 dB 1.6

Adaptive quantization 31 dB 2

DCT 34 dB 2

JPEG

Wavelet 35 dB 9.3

JPEG compression of LennaJPEG compression of Lenna

• 512 X 512 pixels

•1 pixel = 8 bits

• 64 bytes = 8 x 8 submatrix = block

• 4096 submatrices

• 262144 total/elements total

8x8 pixel block

DCT QuantizerLevel-shift Encoder Data

• Discrete Cosine Transform of every element

• Gray scale image level-shifted by –128

• for n = 8, 2^(n-1) = 128

JPEG AlgorithmJPEG Algorithm

Quantization

using a typical normalization matrix

[ 16 11 10 16 24 40 51 61

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 ]

JPEG AlgorithmJPEG Algorithm

• Normalization using a standard table

JPEG AlgorithmJPEG Algorithm

Zig-zag pattern Removal of zeros Convert to binary Compare the number

of bits used

- Orthogonal Basis

- Area of basis equals zero

- Low pass / High pass filtering scheme to generate basis coefficients

- Compression by reducing the number of coefficient (zeroing least significant coefficients)

Discrete Wavelet Transform Discrete Wavelet Transform (DWT)(DWT)

Haar wavelet(averaging)

Mexican Hat wavelet(2nd derivative of Gaussian distribution)

Daub4 wavelet(most common used)

2

2

g Mex_Hat x( )

33 x

2 0 2

2

1

1

21.1

1.1

g Haar x( )

1.10 x

0 0.5 1

1

1

0.15

0.12

Wi

512200 i

200 300 400 500

0.2

0.1

0.1

Common Orthogonal Wavelet Common Orthogonal Wavelet BasesBases

A Row 1

C 0

C 3

0

0

0

0

C 2

C 1

C 1

C 2

0

0

0

0

C 3

C 0

C 2

C 1

C 0

C 3

0

0

0

0

C 3

C 0

C 1

C 2

0

0

0

0

0

0

C 2

C 1

C 0

C 3

0

0

0

0

C 3

C 0

C 1

C 2

0

0

0

0

0

0

C 2

C 1

C 0

C 3

0

0

0

0

C 3

C 0

C 1

C 2

a

b

c

d

e

f

g

h

Low 1

Hi 1

Low 2

Hi 2

Low 3

Hi 3

Low 4

Hi 4

Low 1

Low 2

Low 3

Low 4

Hi 1

Hi 2

Hi 3

Hi 4

Lowpass

Highpass

DWT(Daub4 Nth order matrix)

Row 1Pixels(l to r)

Row 1Coefficients

Row 1C 0

1 3

4 2C 1

3 3

4 2

C 23 3

4 2C 3

1 3

4 2

Low 1A

Low 1B

Low 2A

Low 2B

Hi 1A

Hi 1B

Hi 2A

Hi 2B

N-1times

Filtering SchemeFiltering Scheme

Coefficients for First Row Coefficients for First Row DWT TransformationDWT Transformation

- DWT each row- Regroup coefficients into Low/Hi subvectors

- DWT all columns of transformed matrix- Regroup coefficients into Low/Hi subvectors

WavA

Wav

Hi-Hi

Low-Low

Wav comp

90 % of thecoefficientszeroed

Generating CoefficientsGenerating Coefficients

original 50% coefficients 10% coefficients

2% coefficients 0.5% coefficients 0.1% coefficients

Application to “Lenna”Application to “Lenna”

2%

Signal to Noise RatioSignal to Noise Ratio

ReferencesReferences

1. Rafael C. Gonzalez, Richard E. Wood, “Digital Image Processing”, Addison Wesley, 1993

2. Geoffrey M. Davis, Aria Nosratinia, “Wavelet-based Image Coding: An Overview”, http://www.geoffdavis.net/

3. Subhasis, Saha, “Image Compression - from DCT to Wavelets : A Review”, http://www.acm.org/crossroads/xrds6-3/sahaimgcoding.html

4. Weidong Kou, “Digital Image Compression Algorithms and Standards,” Kluwer Academic Publishers, 1995.

5. “Selected Papers on Image Coding and Compression,” Majid Rabbani, Ed., Brian J. Thompson, Gen. Ed., SPIE Milestone Series, Vol MS-48, SPIE Optical Engineering Press, 1992.

6. “Fractal Image Compression Theory and Application,” Yuval Fisher, Ed., Springer-Verlag New York, 1995.

7. Bernd Jaehne, “Digital Image Processing”, Third Edition, Springer-Verlag, New York 1995