A Comprehensive Study of W A Comprehensive Study of W avelet Transforms for SPIH avelet Transforms for SPIH T T 台台台台台台台台台 台台台台台台台台台 台台台台 台台台 : 台台台台 台台台 : 台台 台台台 : 台台 台台台 : 2003/03/27
Jan 04, 2016
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.