Evaluation of SPIHT Coding Parameters

Shih-Hsuan Yang and Wu-Jie Liao

Department of Computer Science and Information Engineering

National Taipei University of Technology

Taipei, Taiwan, ROC

December 15, 2003December 15, 2003

Outline

Wavelet Transform SPIHT Quantization Experimental Result Conclusion

Wavelet Transform

Time domain pixels Transform domain coefficients

Wavelet Transform (1D)

h0

h1

2

2

g0

g12

2

LPF LPF

HPF HPF

X(n) y(n)

If X(n) = y(n), this called perfect reconstruction

Effects of Wavelet Filters

Properties of wavelets:Desirable time-frequency localization.Compact support.Orthogonality.Smoothness, regularity, or vanishing moment

s.Symmetry (linear-phase constraint).

Wavelet Filters for Evaluation

Real to real transform : (irreversible)5/3, 9/7-F, 9/7-M, 5/11-A, 5/11-C, 13/7-T, 13/

7-C (biorthogonal) Integer to integer transform : (reversible)

Haar wavelet (D2, orthogonal)Daubechies 4 and 6 tap (D4, D6, orthogonal)9/7, 10/18 (biorthogonal)

Real to Real Transform(RWT)

Conventional transform (convolves the input signal with the wavelet filter kernel.)

Computational complexity is proportion to the length of filter kernel.

Real to Real Transform (RWT)index D2(h0) D4(h0) D6(h0)

0 0.7071 0.4830 0.332705

1 0.7071 0.8365 0.806915

2 0.2241 0.459877

3 -0.1294 -0.135011

4 -0.0854412

5 0.0352263

index 9/7(h0) 9/7(g0) 10/18(h0) 10/18(g0)

0 0.852699 0.788486 0.75890773 0.62335964

1 0.377402 0.418092 .07679049 0.163368

2 -0.110624 -0.040689 -0.157526 -0.0856619

3 -0.023849 -0.064539 0.0000824478 -0.013765

4 0.037828 0.0288525 0.03083373

5 -0.002528037

6 -0.0094524629

7 -0.00000272719

8 0.0009544

Integer to Integer Transform (IWT)

Fixed-point approximation to conventional transform (RWT).

Suitable for lossy and lossless coding. Computational complexity is proportion to l

ifting steps required.

Integer to Integer Transform (IWT)

Lifting step

5/3:

2

1])1[][(

4

1][][

])[]1[(2

1][][

0

000

ndndnsns

nsnsndnd

9/7-F:

2

1]))1[][(217(

4096

1][][

2

1]))[]1[(203(

128

1][][

1101

0001

ndndnsns

nsnsndnd

2

1]))1[][(1817(

4096

1][][

2

1]))[]1[(113(

128

1][][

111

011

ndndnsns

nsnsndnd

Computational complexity

RWT

D2 D4 D6 9/7 10/18

1.00 1.56 2.06 2.03 3.59

IWT

5/3 9/7-F 9/7-M 5/11A 5/11-C 13/7C 13/7-T

1.00 1.94 1.01 1.52 1.52 1.13 1.13

Relative computation time required for transformation

The simulation is conducted on Pentium-4 2.4GHz PC

Effects of Extension TypesPeriodic extension Odd-symmetric (for odd-tap filter)

even-symmetric (for even-tap filter) anti-symmetric (for even-tap filter)

SPIHT Quantization

Wavelet coefficients c[i] Bit plane of c[i]

Significant : | c[i] | >= k=0,1,2,…,n

n212 n

22 n

02

k2

SPIHT Quantization(cont.)Example of Parent-Offspring dependencies

(i,j) root

O(i,j) offspring of root

D(i,j) descendant of root

L(i,j) = D(i,j) - O(i,j)

Type A

Type B

SPIHT Algorithm

Sorting passRefinement

pass

LIS : list of insignificant sets

LIP : list of insignificant pixels

LSP : list of significant pixels

for LSPencoding symbol :

0 or 1

T/ 2

T : threshold

Initialization

Experiments

ImagesLena and baboon.

Wavelet filters IWT and RWT.

Extension types Periodic and symmetric.

Test images & Visual Quality Measurement (MSE, PSNR)

MSE

2

10

255log10PSNR

MSE

2

10

255log10PSNR

MSE

2

10

255log10PSNR

M

iii yx

M 1

2)(1

MSE

lena baboon

Compression Results (“lena”)bpp

RWT

D2 D4 D6 9/7 10/18

0.125 27.53 28.97 29.38 30.53 30.68

0.25 30.21 31.85 32.35 33.58 33.75

0.5 33.50 35.24 35.75 36.74 36.86

1.0 37.47 38.92 39.26 39.92 39.96

bppIWT

5/3 9/7F 9/7M 5/11A 5/11C 13/7C 13/7T

0.125 29.71 30.25 29.78 29.84 29.79 29.94 29.90

0.25 32.60 33.24 32.87 32.81 32.88 33.04 33.07

0.5 35.75 36.17 35.93 35.92 35.89 36.14 36.13

1.0 38.87 38.84 38.80 38.89 38.80 39.03 39.00

Compression Results (“baboon”)

bppRWT

D2 D4 D6 9/7 10/18

0.125 20.97 21.28 21.37 21.49 21.60

0.25 22.14 22.54 22.64 22.88 22.97

0.5 24.08 24.60 24.79 25.11 25.13

1.0 27.31 27.97 28.21 28.62 28.61

bppIWT

5/3 9/7F 9/7M 5/11A 5/11C 13/7C 13/7T

0.125 20.96 21.42 20.85 20.92 20.87 21.05 20.98

0.25 22.25 22.80 22.18 22.23 22.17 22.40 22.35

0.5 24.22 25.07 24.28 24.25 24.23 24.49 24.47

1.0 27.71 28.37 27.80 27.79 27.76 28.02 27.98

Energy Compaction (“lena”)

RWT IWT

D2 D4 D6 9/7 10/18 5/3 9/7-F 9/7-M 5/11A 5/11C 13/7C 13/7T

97 97 97 97 97 78 96 82 78 77 81 82

Energy percentage of DC subband (%,5 level decomposition)

9/7-F 5/3

Energy Compaction (“baboon”)

RWT IWT

D2 D4 D6 9/7 10/18 5/3 9/7-F 9/7-M 5/11A 5/11C 13/7C 13/7T

98 98 98 99 98 88 98 91 88 88 91 91

Energy percentage of DC subband (%,5 level decomposition)

9/7-F 5/3

Compression Results for Period/Symmetric Extension (lena)bpp RWT

D2 D4 D6 9/7 10/18

0.125 27.53 28.97 29.38 30.06/30.53 30.20/30.68

0.25 30.21 31.85 32.35 33.21/33.58 33.58/33.75

0.5 33.50 35.24 35.75 36.52/36.74 36.46/36.86

1.0 37.47 38.92 39.26 39.77/39.92 39.75/39.96

bpp IWT

5/3 9/7F 9/7M 5/11A 5/11C 13/7C 13/7T

0.125 29.20/29.71

29.86/30.25

29.40/29.78

29.33/29.84

29.32/29.79

29.53/29.94

29.53/29.90

0.25 32.12/32.60

32.88/33.24

32.53/32.87

32.36/32.81

32.41/32.88

32.70/33.04

32.69/33.07

0.5 35.50/35.75

35.96/36.17

35.73/35.93

35.72/35.92

35.72/35.89

35.90/36.14

35.89/36.13

1.0 38.76/38.87

38.74/38.84

38.71/38.80

38.78/38.89

38.70/38.80

38.92/39.03

38.88/39.00

Conclusions

9/7 and 10/18 biothogonal wavelets with symmetric extension provide the best compression performance, but highest complexity.

5/3 filter may be reasonably choice for low complexity codecs.

Conclusions

This work investigate several parameters of wavelet transform for SPIHT.

Provide guidelines for the best tradeoff of a SPIHT-based image compression system.