Wavelets and Denoising Wavelets and Denoising Jun Ge Jun Ge and and Gagan Mirchandani Gagan Mirchandani Electrical and Computer Engineering Electrical and Computer Engineering Department Department The University of Vermont The University of Vermont October 10, 2003 October 10, 2003 Research day, Computer Science Research day, Computer Science Department, UVM Department, UVM
31
Embed
Wavelets and Denoising Jun Ge and Gagan Mirchandani Electrical and Computer Engineering Department The University of Vermont October 10, 2003 Research.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Wavelets and DenoisingWavelets and Denoising
Jun GeJun Ge andand Gagan MirchandaniGagan MirchandaniElectrical and Computer Engineering DepartmentElectrical and Computer Engineering Department
The University of VermontThe University of VermontOctober 10, 2003 October 10, 2003
– Remove Remove noisenoise– Preserve Preserve useful informationuseful information
• Applications:Applications:– Medical signal/image analysis (ECG, CT, MRI etc.)Medical signal/image analysis (ECG, CT, MRI etc.)– Data mining Data mining – Radio astronomy image analysisRadio astronomy image analysis
noise signal
noisy signal
Wiener filtering
noise signal
noisy signal
Wiener filtering Wavelet Shrinkage
1-D
noise signal
noisy signal
Wiener filtering Wavelet Shrinkage
1-D2-D (m-D)
Geometrical Analysis
Incorporating geometrical Incorporating geometrical structurestructureTwo possible solutions:Two possible solutions:
• Constructing Constructing non-separable non-separable parsimonious representationsparsimonious representations for two for two dimensional signals (e.g., ridgelets dimensional signals (e.g., ridgelets (Donoho et al.), edgelets (Vetterli et (Donoho et al.), edgelets (Vetterli et al.), bandlets (Mallat et al.), al.), bandlets (Mallat et al.), triangulation), triangulation), no fast algorithms yetno fast algorithms yet..
• Incorporating Incorporating geometrical informationgeometrical information (inter- and intra-scale correlation) in (inter- and intra-scale correlation) in the analysis because the analysis because wavelet wavelet decorrelation is not completedecorrelation is not complete. .
Local Covariance Analysis: Local Covariance Analysis: MotivationMotivation
• Idea: Capture Idea: Capture intra-scale correlationintra-scale correlation• Feature extraction (e.g., edge detection) is one of Feature extraction (e.g., edge detection) is one of
the most important areas of image analysis and the most important areas of image analysis and computer vision.computer vision.
• Edge Detection: intensity image Edge Detection: intensity image edge map ( a edge map ( a map of edge related pixel sites).map of edge related pixel sites).o Significance Measure (e.g., the magnitude of the Significance Measure (e.g., the magnitude of the
• False detections are unavoidable False detections are unavoidable • Looking for better significance measureLooking for better significance measure
Local Covariance AnalysisLocal Covariance Analysis
• Plessy corner detector (Noble 1988): a spatial average Plessy corner detector (Noble 1988): a spatial average of an outer product of the gradient vectorof an outer product of the gradient vector
• Image field categorization (Ando 2000): gradient Image field categorization (Ando 2000): gradient covariance form differential Gaussian Filterscovariance form differential Gaussian Filters
Cross correlation of the gradients along x- and y-Cross correlation of the gradients along x- and y-coordinates: coordinates:
Local Covariance AnalysisLocal Covariance Analysis• The covariance matrix is Hermitian and positive The covariance matrix is Hermitian and positive
semidefinite semidefinite the two eigenvalues are real and the two eigenvalues are real and positivepositive
• The two eigenvalues are the principle components of The two eigenvalues are the principle components of the (fx, fy) distribution.the (fx, fy) distribution.
• A dimensionless and normalized homogeneity A dimensionless and normalized homogeneity measure is defined as the ratio of the multiplicative measure is defined as the ratio of the multiplicative average to the additive average (Ando 2000)average to the additive average (Ando 2000)
• A significance measure is defined as A significance measure is defined as
A New Data-Driven Shrinkage A New Data-Driven Shrinkage MaskMask
• Experimental results indicate that the new mask Experimental results indicate that the new mask offers better performance only for relatively high offers better performance only for relatively high level (standard deviation) noise.level (standard deviation) noise.
• r is an empirical parameter which provides the r is an empirical parameter which provides the mixture of masks.mixture of masks.
olddkj
kjyykjxx
kjxykjyykjxx
olddkjs
olddkj
dkj
newdkj
maskSS
SSS
maskMmaskwmask
,,2
,,,,
2,,
2,,,,
,,
,,,
,,
)(
4)(
10),/exp(
)1( ,,
,,
,,
een
olddkj
newdkj
mixdkj
r
maskrmaskrmask
Comparison with several Comparison with several algorithmsalgorithms
• wiener2 in MATLABwiener2 in MATLAB
• Xu et al. (IEEE Trans. Image Processing, Xu et al. (IEEE Trans. Image Processing, 1994)1994)
• Strela (in 3Strela (in 3rdrd European Congress of European Congress of Mathematics, Barcelona, July 2000)Mathematics, Barcelona, July 2000)
• Portilla et al. (Technical Report, Computer Portilla et al. (Technical Report, Computer Science Dept., New York University, Sept. Science Dept., New York University, Sept. 2002)2002)
Experimental ResultsExperimental Results
Experimental ResultsExperimental Results
Experimental ResultsExperimental Results
AppendixAppendix
• What is a wavelet?What is a wavelet?
• What is good about wavelet analysis?What is good about wavelet analysis?
• What is denoising?What is denoising?
• Why choose wavelets to denoise?Why choose wavelets to denoise?
What is a wavelet?What is a wavelet?
A wavelet is an A wavelet is an elementary function elementary function
• which satisfies which satisfies certain admissible certain admissible conditionsconditions
• whose dilates and whose dilates and shifts give a Riesz shifts give a Riesz (stable) basis of (stable) basis of L^2(R)L^2(R)
What is good about wavelet analysis?What is good about wavelet analysis?
• Simultaneous time Simultaneous time and frequency and frequency localizationslocalizations
• Unconditional basis Unconditional basis for a variety of for a variety of classes of functions classes of functions spacesspaces
• Approximation powerApproximation power• A complement to A complement to
Fourier analysisFourier analysis
Why choose wavelets to Why choose wavelets to denoise?denoise?Wavelet Shrinkage (Donoho-Johnstone 1994)Wavelet Shrinkage (Donoho-Johnstone 1994)
• Unconditional basis:Unconditional basis:– Magnitude is an important significance measureMagnitude is an important significance measure– A binary classifier: A binary classifier:
Joint Models:Joint Models:• Hidden Markov Tree modelsHidden Markov Tree models
Denoising Algorithm using GSM Denoising Algorithm using GSM Model and a Bayes least squares Model and a Bayes least squares estimatorestimator (Portilla (Portilla et al.et al. 2002) 2002)