A Comprehensive Study of WaveleA Comprehensive Study of Wavelet Transforms for SPIHTt Transforms for SPIHT

台北科技大學資工所台北科技大學資工所指導教授：楊士萱指導教授：楊士萱學生：廖武傑學生：廖武傑

2003/03/27

OutlineOutline

IntroductionIntroduction Compression performanceCompression performance ScalingScaling Finite length signal analysisFinite length signal analysis ConclusionConclusion

IntroductionIntroduction

TransformsTransforms

integer-to-integer (reversible)integer-to-integer (reversible)

real-to-real(irreversibel)real-to-real(irreversibel) SPIHTSPIHT

wavelet domain codingwavelet domain coding

zero-tree codingzero-tree coding

TransformsTransforms

Integer-to-integer transformInteger-to-integer transform::

Real-to-real transformReal-to-real transform:: Dot products between the two filter masks and the Dot products between the two filter masks and the

signal.signal.

2

)22()2()12()12(

nxnxnxny

4

2)12()12()2()2(

nynynxny

Wavelet filters for evaluation of coding Wavelet filters for evaluation of coding

Integer-to-integer:Integer-to-integer:

5/3, 9/7-M, 5/11-A, 5/11-C,13/7-T, 13/7-5/3, 9/7-M, 5/11-A, 5/11-C,13/7-T, 13/7-C, C, 9/7-F (biorthogonal)9/7-F (biorthogonal)

Real-to-real:Real-to-real:

9/7, 10/18 (biothogonal)9/7, 10/18 (biothogonal)

Haar, Daubechies 4 taps, 6 taps(orthogonHaar, Daubechies 4 taps, 6 taps(orthogonal)al)

ComplexityComplexity

Integer-to-integer:Integer-to-integer:

5/3:5/3:

9/7-F:9/7-F:

2

1

4

])1[][(][][

2

])[]1[(][][

0

00

ndndnsns

nsnsndnd

2

1

4096

])1[][(1817][][

2

1

128

])[]1[(113][][

2

1

4096

])1[][(217][][

2

1

128

])[]1[(203][][

1

111

1101

0001

ndndnsns

nsnsndnd

ndndnsns

nsnsndnd

ComplexityComplexity

Real-to-real:Real-to-real:

Haar:Haar:

9/7:9/7:

ii Low-Pass FilterLow-Pass Filter High-Pass FilterHigh-Pass Filter

00 11 -1-1

11 11 11

ii Low-Pass FilterLow-Pass Filter High-Pass FilterHigh-Pass Filter

00 0.60294901823635790.6029490182363579 1.1150870524569941.115087052456994

±1±1 0.26686411844287230.2668641184428723 -0.5912717631142470-0.5912717631142470±2±2 -0.078223266652898785-0.078223266652898785 -0.05754352622849957-0.05754352622849957

±3±3 -0.01686411844287495-0.01686411844287495 0.091271763114249480.09127176311424948

±4±4 0.026748757410809760.02674875741080976

SPIHT(set partitioning in hierarchical tSPIHT(set partitioning in hierarchical trees)rees)

Zero-tree coding:Zero-tree coding:

->inter-scaling correlation->inter-scaling correlation

->energy distribution->energy distribution

OutlineOutline

IntroductionIntroduction Compression performanceCompression performance ScalingScaling Finite length signal analysisFinite length signal analysis conclusionconclusion

Compression performanceCompression performance

Test images:Test images:

lena

baboon

pepper

F16

Compression performanceCompression performance5/3 9/7-F 9/7-M 5/11A 5/11-

C13/7C 13/7-

T10/18 9/7 Haar D4 D6

1/128 26.90 27.66 27.08 26.92 26.82 26.96 27.21 27.92 27.59 25.42 26.42 26.83

1/64 29.71 30.25 29.78 29.84 29.79 29.94 29.90 30.68 30.53 27.53 28.97 29.38

1/32 32.60 33.24 32.87 32.81 32.88 33.04 33.07 33.75 33.58 30.21 31.85 32.35

1/16 35.75 36.17 35.93 35.92 35.89 36.14 36.13 36.86 36.74 33.50 35.24 35.75

1/8 38.87 38.84 38.80 38.89 38.80 39.03 39.00 39.96 39.92 37.47 38.92 39.26

1/4 42.46 41.78 42.54 42.51 42.38 42.72 42.69 44.29 44.31 42.55 43.60 43.78

1/2 48.02 45.41 48.93 48.24 48.12 48.31 48.34 54.92 55.11 53.25 54.30 54.28

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

13/7C 13/7-T

10/18 9/7 Haar D4 D6

1/128 20.24 20.55 20.12 20.21 20.15 20.32 20.26 20.64 20.56 20.22 20.40 20.47

1/64 20.96 21.42 20.85 20.92 20.87 21.05 20.98 21.60 21.49 20.97 21.28 21.37

1/32 22.25 22.80 22.18 22.23 22.17 22.40 22.35 22.97 22.88 22.14 22.54 22.64

1/16 24.22 25.07 24.28 24.25 24.23 24.49 24.47 25.13 25.11 24.60 24.60 24.79

1/8 27.71 28.37 27.80 27.79 27.76 28.02 27.98 28.61 28.62 27.97 27.97 28.21

1/4 33.13 33.63 33.15 33.14 33.09 33.35 33.32 34.02 34.10 32.74 33.43 33.66

1/2 42.58 41.90 42.73 42.61 42.51 42.90 42.86 44.72 44.88 43.51 44.26 44.44

Compression performanceCompression performance5/3 9/7-F 9/7-M 5/11A 5/11-

C13/7C 13/7-

T10/18 9/7 Haar D4 D6

1/128 25.65 26.50 25.56 25.67 25.61 25.70 25.69 26.47 26.39 24.54 25.57 25.75

1/64 28.31 28.95 28.49 28.41 28.39 28.56 28.57 29.26 29.28 26.78 27.97 28.18

1/32 31.45 32.14 31.82 31.66 31.72 31.85 31.88 32.56 32.51 29.57 30.95 31.40

1/16 35.27 35.85 35.75 35.55 35.61 35.84 35.87 36.45 36.42 33.45 34.75 35.27

1/8 39.84 39.66 39.82 39.86 39.74 39.99 39.99 40.92 40.92 38.42 39.61 39.59

1/4 44.54 43.01 44.72 44.59 44.47 44.83 44.84 46.52 46.58 44.69 45.64 45.82

1/2 48.51 45.46 48.58 48.62 48.46 48.47 48.60 57.11 57.29 54.61 56.21 55.86

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

13/7C 13/7-T

10/18 9/7 Haar D4 D6

1/128 25.42 25.86 25.27 25.43 25.39 25.45 25.41 26.14 26.07 23.26 24.47 24.74

1/64 28.31 28.77 28.30 28.36 28.30 28.44 28.43 29.01 29.02 25.78 27.22 27.56

1/32 30.68 31.42 31.07 30.83 30.86 31.19 31.15 31.67 31.74 28.76 30.20 30.46

1/16 33.09 33.51 33.16 33.07 32.98 33.33 33.30 33.78 33.80 31.81 32.95 33.06

1/8 35.38 35.84 35.48 35.34 35.18 35.69 35.66 36.21 36.18 35.01 35.71 35.77

1/4 38.93 39.44 39.41 38.88 38.69 39.64 39.58 40.69 40.75 39.63 40.22 40.29

1/2 44.26 44.83 47.12 44.29 44.14 47.14 47.17 51.91 51.97 50.35 51.32 51.30

Energy of LL subband(%)Energy of LL subband(%)5/3 9/7-F 9/7-M 5/11-

A5/11-

C13/7-

C13/7-T 10/18 9/7 Haar D4 D6

Lena 78.27 96.12 82.14 78.09 77.74 81.88 82.10 97.42 97.98 97.79 97.37 97.28

Baboon

88.69 98.40 91.21 88.58 88.39 91.69 91.64 98.68 99.18 98.85 98.96 98.90

F16 88.65 98.06 91.10 88.60 88.45 90.91 90.10 98.33 98.98 98.77 98.76 98.74

pepper

68.53 92.14 73.46 68.71 68.62 72.54 73.10 94.27 95.71 94.73 94.72 94.08

OutlineOutline

IntroductionIntroduction Compression performanceCompression performance ScalingScaling Finite length signal analysisFinite length signal analysis ConclusionConclusion

ScalingScaling

Optimal scaling factorOptimal scaling factor

->fixed scaling->fixed scaling

->variable scaling->variable scaling Modify SPIHT coding algorithmModify SPIHT coding algorithm

->variable sorting threshold->variable sorting threshold

Fixed scalingFixed scaling

Optimal scaling factor for all wavelet Optimal scaling factor for all wavelet decomposition is 1.41421 ,except 9/7-decomposition is 1.41421 ,except 9/7-F(1.1496) F(1.1496)

With proper scaling, the compression With proper scaling, the compression performance is much better for all wavelet performance is much better for all wavelet filter.filter.

Coding with or without scaling Coding with or without scaling (“Lena”)(“Lena”)

5/3 9/7-F

Coding with or without scaling Coding with or without scaling (“Lena”)(“Lena”)

13/7-T 13/7-C

Coding with or without scaling Coding with or without scaling (“Lena”)(“Lena”)

5/11-A 5/11-C

Finite length signal analysisFinite length signal analysis

Optimal signal extensionOptimal signal extension

->minimal the distortion of the ->minimal the distortion of the reconstructive signal reconstructive signal

Restriction of signal extensionRestriction of signal extension

->extension must match the filter-bank.->extension must match the filter-bank.

Extensions for various filtersExtensions for various filters

– Odd symmetric extension for odd taps filter.Odd symmetric extension for odd taps filter.– Even symmetric extension and anti-symmetric for Even symmetric extension and anti-symmetric for

even taps filter.even taps filter.– periodic extension for asymmetric filter. (circular periodic extension for asymmetric filter. (circular

convolution)convolution)

Only guarantee the forward-backward Only guarantee the forward-backward transform works.transform works.

Extension affects performance Extension affects performance

Symmetric extension

periodic extension

Performance (with proper and Performance (with proper and improper extension )improper extension )

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

13/7C 13/7-T

10/18 9/7 Haar D4 D6

1/128 26.90 27.66 27.08 26.92 26.82 26.96 27.21 27.92 27.59 25.42 26.42 26.83

1/64 29.71 30.25 29.78 29.84 29.79 29.94 29.90 30.68 30.53 27.53 28.97 29.38

1/32 32.60 33.24 32.87 32.81 32.88 33.04 33.07 33.75 33.58 30.21 31.85 32.35

1/16 35.75 36.17 35.93 35.92 35.89 36.14 36.13 36.86 36.74 33.50 35.24 35.75

1/8 38.87 38.84 38.80 38.89 38.80 39.03 39.00 39.96 39.92 37.47 38.92 39.26

1/4 42.46 41.78 42.54 42.51 42.38 42.72 42.69 44.29 44.31 42.55 43.60 43.78

1/2 48.02 45.41 48.93 48.24 48.12 48.31 48.34 54.92 55.11 53.25 54.30 54.28

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

13/7C 13/7-T

10/18 9/7 Haar D4 D6

1/128 26.48 27.12 26.40 26.57 26.51 26.51 26.51 27.59

1/64 29.20 29.86 29.40 29.33 29.32 29.53 29.53 30.53

1/32 32.12 32.88 32.53 32.36 32.41 32.70 32.69 33.58

1/16 35.50 35.96 35.73 35.72 35.72 35.90 35.89 36.74

1/8 38.76 38.74 38.71 38.78 38.70 38.92 38.88 39.92

1/4 42.38 41.70 42.43 42.42 42.28 42.59 42.58 44.31

1/2 47.98 45.44 48.37 48.22 48.09 48.27 48.33 55.11

OutlineOutline

IntroductionIntroduction Compression performanceCompression performance ScalingScaling Finite length signal analysisFinite length signal analysis ConclusionConclusion

OutlineOutline

IntroductionIntroduction Compression performanceCompression performance ScalingScaling Finite length signal analysisFinite length signal analysis ConclusionConclusion

ConclusionConclusion

Coding performance associated with filter:Coding performance associated with filter:– Properties of filterProperties of filter– Energy distribution of wavelet coefficientsEnergy distribution of wavelet coefficients– Some issues of implementationSome issues of implementation

The differences between fixed and floating The differences between fixed and floating point filtering computation.point filtering computation.