wavelet shrinkage for image noise reductiondsp.vscht.cz/hostalke/upload/tcp08_presentation.pdf ·...
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![Page 1: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/1.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 1 / 24
WAVELET-BASEDHIDDEN MARKOV TREES
FOR IMAGE NOISE REDUCTION
Eva Hošt’álková & Aleš Procházka
Institute of Chemical Technology, PragueDept of Computing and Control Engineering
Technical Computing Prague 2008
![Page 2: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/2.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 2 / 24
Table of Contents
1 Introduction
2 Wavelet Shrinkage (WS)Wavelet Shrinkage SchemeNormalShrink Method
3 Noise Reduction via Hidden Markov Trees (HMTs)Persistence and Clustering PropertiesWavelet-Based HMT ModelsNoise Reduction
4 Results
5 Conclusions
6 Selected Bibliography
![Page 3: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/3.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 3 / 24
Table of Contents
1 Introduction
2 Wavelet Shrinkage (WS)Wavelet Shrinkage SchemeNormalShrink Method
3 Noise Reduction via Hidden Markov Trees (HMTs)Persistence and Clustering PropertiesWavelet-Based HMT ModelsNoise Reduction
4 Results
5 Conclusions
6 Selected Bibliography
![Page 4: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/4.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 4 / 24
Introduction
Applications of Noise ReductionImage enhancementPreprocessing step to other techniques(e.g. segmentation, edge detection)
Noise Reduction via Wavelet Shrinkage
+ Recovering signals from additive Gaussian noise- Thresholding w. coefficients without considering their
dependencies⇒ artifacts, blurred edges
Noise Reduction via Wavelet-Based HMTsHidden Markov Tree (HMT) modelsAiming to capture these dependencies
![Page 5: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/5.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 4 / 24
Introduction
Applications of Noise ReductionImage enhancementPreprocessing step to other techniques(e.g. segmentation, edge detection)
Noise Reduction via Wavelet Shrinkage
+ Recovering signals from additive Gaussian noise- Thresholding w. coefficients without considering their
dependencies⇒ artifacts, blurred edges
Noise Reduction via Wavelet-Based HMTsHidden Markov Tree (HMT) modelsAiming to capture these dependencies
![Page 6: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/6.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 4 / 24
Introduction
Applications of Noise ReductionImage enhancementPreprocessing step to other techniques(e.g. segmentation, edge detection)
Noise Reduction via Wavelet Shrinkage
+ Recovering signals from additive Gaussian noise- Thresholding w. coefficients without considering their
dependencies⇒ artifacts, blurred edges
Noise Reduction via Wavelet-Based HMTsHidden Markov Tree (HMT) modelsAiming to capture these dependencies
![Page 7: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/7.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 5 / 24
Introduction
Other Applications of Wavelet-Based HMTsImage segmentation (texture features)Edge detection, signal prediction etc.
Image Data
(a) MANDRILL IMAGE (b) CUT OUT (c) NOISY IMAGE
The mandrill image, a 240×240 cut out, corruption byiid Gaussian noise
![Page 8: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/8.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 5 / 24
Introduction
Other Applications of Wavelet-Based HMTsImage segmentation (texture features)Edge detection, signal prediction etc.
Image Data
(a) MANDRILL IMAGE (b) CUT OUT (c) NOISY IMAGE
The mandrill image, a 240×240 cut out, corruption byiid Gaussian noise
![Page 9: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/9.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 6 / 24
Table of Contents
1 Introduction
2 Wavelet Shrinkage (WS)Wavelet Shrinkage SchemeNormalShrink Method
3 Noise Reduction via Hidden Markov Trees (HMTs)Persistence and Clustering PropertiesWavelet-Based HMT ModelsNoise Reduction
4 Results
5 Conclusions
6 Selected Bibliography
![Page 10: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/10.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 7 / 24
Wavelet Shrinkage Scheme
Wavelet Shrinkage Algorithm
NoisyImage
NoisyImage
WaveletAnalysisWaveletAnalysisWaveletAnalysis
Image
HiHi1HiLo1
LoHi1LoLo1
HiHi2HiLo2
LoHi2LoLo2
Thresholdingwavelet
coefficients
Thresholdingwavelet
coefficients
Thresholdingwavelet
coefficients
The NormalShrink Method
WaveletSynthesisWavelet
SynthesisWavelet
Synthesis
Image
LoLo2
LoLo1LoHi1HiLo1HiHi1
LoHi2HiLo2HiHi2
The Haar wavelet transformThe Haar wavelet transform
DenoisedImage
DenoisedImage
![Page 11: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/11.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 7 / 24
Wavelet Shrinkage Scheme
Wavelet Shrinkage Algorithm
NoisyImage
NoisyImage
WaveletAnalysis
WaveletAnalysisWaveletAnalysis
Image
HiHi1HiLo1
LoHi1LoLo1
HiHi2HiLo2
LoHi2LoLo2
Thresholdingwavelet
coefficients
Thresholdingwavelet
coefficients
Thresholdingwavelet
coefficients
The NormalShrink Method
WaveletSynthesisWavelet
SynthesisWavelet
Synthesis
Image
LoLo2
LoLo1LoHi1HiLo1HiHi1
LoHi2HiLo2HiHi2
The Haar wavelet transformThe Haar wavelet transform
DenoisedImage
DenoisedImage
![Page 12: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/12.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 7 / 24
Wavelet Shrinkage Scheme
Wavelet Shrinkage Algorithm
NoisyImage
NoisyImage
WaveletAnalysis
WaveletAnalysisWaveletAnalysis
Image
HiHi1HiLo1
LoHi1LoLo1
HiHi2HiLo2
LoHi2LoLo2
Thresholdingwavelet
coefficients
Thresholdingwavelet
coefficients
Thresholdingwavelet
coefficients
The NormalShrink Method
WaveletSynthesisWavelet
SynthesisWavelet
Synthesis
Image
LoLo2
LoLo1LoHi1HiLo1HiHi1
LoHi2HiLo2HiHi2
The Haar wavelet transformThe Haar wavelet transform
DenoisedImage
DenoisedImage
![Page 13: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/13.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 7 / 24
Wavelet Shrinkage Scheme
Wavelet Shrinkage Algorithm
NoisyImage
NoisyImage
WaveletAnalysis
WaveletAnalysisWaveletAnalysis
Image
HiHi1HiLo1
LoHi1LoLo1
HiHi2HiLo2
LoHi2LoLo2
Thresholdingwavelet
coefficients
Thresholdingwavelet
coefficients
Thresholdingwavelet
coefficients
The NormalShrink Method
WaveletSynthesis
WaveletSynthesisWavelet
Synthesis
Image
LoLo2
LoLo1LoHi1HiLo1HiHi1
LoHi2HiLo2HiHi2
The Haar wavelet transformThe Haar wavelet transform
DenoisedImage
DenoisedImage
![Page 14: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/14.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 7 / 24
Wavelet Shrinkage Scheme
Wavelet Shrinkage Algorithm
NoisyImageNoisyImage
WaveletAnalysis
WaveletAnalysisWaveletAnalysis
Image
HiHi1HiLo1
LoHi1LoLo1
HiHi2HiLo2
LoHi2LoLo2
Thresholdingwavelet
coefficients
Thresholdingwavelet
coefficients
Thresholdingwavelet
coefficients
The NormalShrink Method
WaveletSynthesis
WaveletSynthesisWavelet
Synthesis
Image
LoLo2
LoLo1LoHi1HiLo1HiHi1
LoHi2HiLo2HiHi2
The Haar wavelet transformThe Haar wavelet transform
DenoisedImage
DenoisedImage
![Page 15: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/15.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 7 / 24
Wavelet Shrinkage Scheme
Wavelet Shrinkage Algorithm
NoisyImage
NoisyImage
WaveletAnalysis
WaveletAnalysis
WaveletAnalysis
Image
HiHi1HiLo1
LoHi1LoLo1
HiHi2HiLo2
LoHi2LoLo2
Thresholdingwavelet
coefficients
Thresholdingwavelet
coefficients
Thresholdingwavelet
coefficients
The NormalShrink Method
WaveletSynthesis
WaveletSynthesis
WaveletSynthesis
Image
LoLo2
LoLo1LoHi1HiLo1HiHi1
LoHi2HiLo2HiHi2
The Haar wavelet transform
The Haar wavelet transform
DenoisedImage
DenoisedImage
![Page 16: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/16.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 7 / 24
Wavelet Shrinkage Scheme
Wavelet Shrinkage Algorithm
NoisyImage
NoisyImage
WaveletAnalysisWaveletAnalysis
WaveletAnalysis
Image
HiHi1HiLo1
LoHi1LoLo1
HiHi2HiLo2
LoHi2LoLo2
Thresholdingwavelet
coefficients
Thresholdingwavelet
coefficients
Thresholdingwavelet
coefficients
The NormalShrink Method
WaveletSynthesisWavelet
Synthesis
WaveletSynthesis
Image
LoLo2
LoLo1LoHi1HiLo1HiHi1
LoHi2HiLo2HiHi2
The Haar wavelet transform
The Haar wavelet transform
DenoisedImage
DenoisedImage
![Page 17: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/17.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 8 / 24
NormalShrink Method
NormalShrinkSubband-adaptive threshold computation
MAD estimate of iid Gaussian noise std. deviation
σ̂nmad = median{|whh1
1 |, |whh12 |, . . . , |whh1
N/4|}/0.6745
whh1 . . . HiHi wavelet coefficient of level 1 (noisedominated), N . . . image sizeRobust against large deviations of noise variance
Soft shrinkage function
SOFT THRESHOLDING
δ(s)
−δ(s)
−3 δ(s) −2 δ(s) −δ(s) 0 δ(s) 2 δ(s) 3 δ(s)−3 δ(s)
−2 δ(s)
−δ(s)
0
δ(s)
2 δ(s)
3 δ(s)
before thr.after thr.
![Page 18: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/18.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 8 / 24
NormalShrink Method
NormalShrinkSubband-adaptive threshold computation
MAD estimate of iid Gaussian noise std. deviation
σ̂nmad = median{|whh1
1 |, |whh12 |, . . . , |whh1
N/4|}/0.6745
whh1 . . . HiHi wavelet coefficient of level 1 (noisedominated), N . . . image sizeRobust against large deviations of noise variance
Soft shrinkage function
SOFT THRESHOLDING
δ(s)
−δ(s)
−3 δ(s) −2 δ(s) −δ(s) 0 δ(s) 2 δ(s) 3 δ(s)−3 δ(s)
−2 δ(s)
−δ(s)
0
δ(s)
2 δ(s)
3 δ(s)
before thr.after thr.
![Page 19: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/19.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 8 / 24
NormalShrink Method
NormalShrinkSubband-adaptive threshold computation
MAD estimate of iid Gaussian noise std. deviation
σ̂nmad = median{|whh1
1 |, |whh12 |, . . . , |whh1
N/4|}/0.6745
whh1 . . . HiHi wavelet coefficient of level 1 (noisedominated), N . . . image sizeRobust against large deviations of noise variance
Soft shrinkage functionSOFT THRESHOLDING
δ(s)
−δ(s)
−3 δ(s) −2 δ(s) −δ(s) 0 δ(s) 2 δ(s) 3 δ(s)−3 δ(s)
−2 δ(s)
−δ(s)
0
δ(s)
2 δ(s)
3 δ(s)
before thr.after thr.
![Page 20: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/20.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 9 / 24
Table of Contents
1 Introduction
2 Wavelet Shrinkage (WS)Wavelet Shrinkage SchemeNormalShrink Method
3 Noise Reduction via Hidden Markov Trees (HMTs)Persistence and Clustering PropertiesWavelet-Based HMT ModelsNoise Reduction
4 Results
5 Conclusions
6 Selected Bibliography
![Page 21: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/21.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 10 / 24
Persistence and Clustering Properties
Interdependencies of the DWT CoefficientsShrinkage methods assume the DWT to de-correlatesignals thoroughly (incorrect)DWT coefficients reveal clustering and persistence
Persistence & Clustering Properties
Clustering within scaleWe expect large (small) coefficients in the vicinity ofa large (small) coef.
Persistence across scaleParent-child relationsWe expect a large (small) parent coef. to have large(small) children
Both captured by the HMT models
![Page 22: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/22.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 10 / 24
Persistence and Clustering Properties
Interdependencies of the DWT CoefficientsShrinkage methods assume the DWT to de-correlatesignals thoroughly (incorrect)DWT coefficients reveal clustering and persistence
Persistence & Clustering Properties
Clustering within scaleWe expect large (small) coefficients in the vicinity ofa large (small) coef.
Persistence across scaleParent-child relationsWe expect a large (small) parent coef. to have large(small) children
Both captured by the HMT models
![Page 23: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/23.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 11 / 24
Persistence and Clustering Properties
Persistence and Clustering
LL3
LH2
HH2
HL1
HH1LH
1
HL2
2D: each parent coefficient p(i) has four children iHMT connects the hidden states Si , Sp(i) - not theactual coefficients values wi , wp(i)
![Page 24: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/24.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 12 / 24
Wavelet-Based HMT Models
Histograms of Wavelet Coefficients
−0.5 0 0.50
0.5
1
1.5
2
(a) LH1 COEFFS HISTOGRAM
Histogram State S=1 State S=2 Marginal PDF
−0.5 0 0.50
0.5
1
1.5
2
2.5
(b) HL1 COEFFS HISTOGRAM
−0.5 0 0.50
0.5
1
1.5
2
2.5
(c) HH1 COEFFS HISTOGRAM
Probability Distribution of Wavelet CoefficientsNon-Gaussian distribution (peaky and heavy tailed)M-component mixture of conditional G. distributionsG(µi,m, σ
2i,m) associated with hidden states Si = m
For M = 2 (2-state models) m = 1,2
![Page 25: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/25.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 12 / 24
Wavelet-Based HMT Models
Histograms of Wavelet Coefficients
−0.5 0 0.50
0.5
1
1.5
2
(a) LH1 COEFFS HISTOGRAM
Histogram State S=1 State S=2 Marginal PDF
−0.5 0 0.50
0.5
1
1.5
2
2.5
(b) HL1 COEFFS HISTOGRAM
−0.5 0 0.50
0.5
1
1.5
2
2.5
(c) HH1 COEFFS HISTOGRAM
Probability Distribution of Wavelet CoefficientsNon-Gaussian distribution (peaky and heavy tailed)M-component mixture of conditional G. distributionsG(µi,m, σ
2i,m) associated with hidden states Si = m
For M = 2 (2-state models) m = 1,2
![Page 26: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/26.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 13 / 24
Wavelet-Based HMT Models
The Overall PDF
f (wi)=p(Si=m) f (wi |Si=m)
p(Si=m) . . . PMF of the hidden states∑M
m=1 p(Si=m) = 1
f (wi |Si = m) . . . conditional probability of the coefficientsvalue wi given the state Si corresponds to G(µi,m, σ
2i,m)
Transition ProbabilitiesChildren hidden states Si given the parent state Sp(i)
f (Si =m | Sp(i) =n)=
[f (Si =1 | Sp(i) =1) f (Si =1 | Sp(i) =2)f (Si =2 | Sp(i) =1) f (Si =2 | Sp(i) =2)
]For M =2 (2-state model)Persistence⇒ f1,1>> f2,1, f2,2>> f1,2
![Page 27: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/27.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 13 / 24
Wavelet-Based HMT Models
The Overall PDF
f (wi)=p(Si=m) f (wi |Si=m)
p(Si=m) . . . PMF of the hidden states∑M
m=1 p(Si=m) = 1
f (wi |Si = m) . . . conditional probability of the coefficientsvalue wi given the state Si corresponds to G(µi,m, σ
2i,m)
Transition ProbabilitiesChildren hidden states Si given the parent state Sp(i)
f (Si =m | Sp(i) =n)=
[f (Si =1 | Sp(i) =1) f (Si =1 | Sp(i) =2)f (Si =2 | Sp(i) =1) f (Si =2 | Sp(i) =2)
]For M =2 (2-state model)Persistence⇒ f1,1>> f2,1, f2,2>> f1,2
![Page 28: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/28.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 14 / 24
Wavelet-Based HMT Models
HMT Training
Tying within subbands to prevent model overfitting⇒3 independent HMT treesExpectation Maximization (EM) training algorithm
![Page 29: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/29.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 15 / 24
Noise Reduction
Noise Reduction via HMTs
NoisyImage
WaveletAnalysis
Alteringwavelet
coefficients
Via HMT models
WaveletSynthesis
The Haar wavelet transform
DenoisedImage
![Page 30: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/30.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 16 / 24
Noise Reduction
Additional Noise ModelWavelet domain, a noisy w. coefficient observation wi
wi = yi + ni
y . . . desired noise-free signal, n . . . iid Gaussian noise
Noise ReductionConditional mean estimate of yi , given the observedcoefficients w and the HMT model parameters θ
E [yi |w, θ] =M∑
m=1
p(Si = m) ·σ2
i,m
σ2n + σ2
i,m· wi (1)
p(Si =m), σi,m . . . obtained from the HMT modelσn . . . noise std. deviation (MAD estimate)
![Page 31: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/31.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 16 / 24
Noise Reduction
Additional Noise ModelWavelet domain, a noisy w. coefficient observation wi
wi = yi + ni
y . . . desired noise-free signal, n . . . iid Gaussian noise
Noise ReductionConditional mean estimate of yi , given the observedcoefficients w and the HMT model parameters θ
E [yi |w, θ] =M∑
m=1
p(Si = m) ·σ2
i,m
σ2n + σ2
i,m· wi (1)
p(Si =m), σi,m . . . obtained from the HMT modelσn . . . noise std. deviation (MAD estimate)
![Page 32: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/32.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 17 / 24
Noise Reduction
Altering Wavelet Coefficients
Noise reduction via HMT and NormalShrink
![Page 33: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/33.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 18 / 24
Table of Contents
1 Introduction
2 Wavelet Shrinkage (WS)Wavelet Shrinkage SchemeNormalShrink Method
3 Noise Reduction via Hidden Markov Trees (HMTs)Persistence and Clustering PropertiesWavelet-Based HMT ModelsNoise Reduction
4 Results
5 Conclusions
6 Selected Bibliography
![Page 34: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/34.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 19 / 24
Results
Residual Images Parameters in Our Experiments
Noise NormalShrink HMT
µn [10−2] σ2n [10−2] µ [10−2] σ2 [10−2] µ [10−2] σ2 [10−2]
5.00 3.00 0.04 2.18 1.12 0.600.00 1.00 0.00 1.12 0.16 0.325.00 1.00 0.46 1.04 1.00 0.32
Residual image - difference between the noisereduction result and the original image
![Page 35: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/35.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 20 / 24
Results
Noise Reduction Results (a) ORIGINAL
(f) ABS. DIFFERENCE NORMALSHRINK (e) ABS. DIFFERENCE HMT (d) NOISY IMAGE
(c) DENOISED VIA NORMALSHRINK (b) DENOISED VIA HMT
![Page 36: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/36.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 21 / 24
Table of Contents
1 Introduction
2 Wavelet Shrinkage (WS)Wavelet Shrinkage SchemeNormalShrink Method
3 Noise Reduction via Hidden Markov Trees (HMTs)Persistence and Clustering PropertiesWavelet-Based HMT ModelsNoise Reduction
4 Results
5 Conclusions
6 Selected Bibliography
![Page 37: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/37.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 22 / 24
Conclusions
Noise Reduction ExperimentsHMT models outperform the NormalShrink method(at the expense of greater computation cost)NormalShrink causes artifacts and blurs edges
Future WorkOur experiment have been very limited so farNext step: Carry out more experiments onbiomedical image data
![Page 38: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/38.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 22 / 24
Conclusions
Noise Reduction ExperimentsHMT models outperform the NormalShrink method(at the expense of greater computation cost)NormalShrink causes artifacts and blurs edges
Future WorkOur experiment have been very limited so farNext step: Carry out more experiments onbiomedical image data
![Page 39: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/39.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 23 / 24
Table of Contents
1 Introduction
2 Wavelet Shrinkage (WS)Wavelet Shrinkage SchemeNormalShrink Method
3 Noise Reduction via Hidden Markov Trees (HMTs)Persistence and Clustering PropertiesWavelet-Based HMT ModelsNoise Reduction
4 Results
5 Conclusions
6 Selected Bibliography
![Page 40: Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme](https://reader031.vdocuments.net/reader031/viewer/2022022508/5ace9af67f8b9a56098c023d/html5/thumbnails/40.jpg)
WAVELET HMTFOR NOISEREDUCTION
http://dsp.vscht.cz/
ICT Prague
Introduction
Wavelet ShrinkageWS Scheme
NormalShrink
Noise Reductionvia HMTsPersistence & Clustering
Wavelet-Based HMTs
Noise Reduction
Results
Conclusions
Bibliography
ICTC Prague 2008 24 / 24
Further Reading
M. S. Crouse, R. D. Nowak, and R. G. Baraniuk.Wavelet-Based Statistical Signal Processing Using Hidden MarkovModels.IEEE Trans. on Signal Processing, 46(4):886–902, April, 1998.
H. Choi and R. G. Baraniuk.Multiscale Image Segmentation Using Wavelet Domain HiddenMarkov Models.Int. Conf. on Image Processing, 1309–1321, IEEE, 2001.
C. W. Shaffrey, N. G. Kingsbury, and I. H. Jermyn.Unsupervised Image Segmentation via Markov Trees and ComplexWavelets.Int. Conf. on Image Processing, USA, 801–804, IEEE, 2002.
D. B. Percival and A. T. Walden.Wavelet Methods for Time Series Analysis.Cambridge University Press, USA, 2006.
L. Kaur, S. Gupta and R. C. ChauhanImage Denoising Using Wavelet Thresholding.3rd Conf. on Computer Vision, India, 1 – 4, 2002.