wavelets and denoising jun ge and gagan mirchandani electrical and computer engineering department...

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

Research day, Computer Science Department, UVMResearch day, Computer Science Department, UVM

signal

noise signal

noisy signal

What is denoising?What is denoising?• Goal:Goal:

– 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


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. .

noise signal

noisy signal


1-D2-D (m-D)

Statistical Approach(Bayesian, parametric)


noise signal

noisy signal


1-D2-D (m-D)


Deterministic/Statistical Approach(non-parametric)


noise signal

noisy signal


1-D2-D (m-D)



Nonseparable basis


noise signal

noisy signal


1-D2-D (m-D)



Nonseparable basis


Geometrical Decorrelation

noise signal

noisy signal


1-D2-D (m-D)



Nonseparable basis

Inter-scale (MPM)



Multiscale Product MethodMultiscale Product Method

• Idea: capture Idea: capture inter-scale correlationinter-scale correlation

• Nonlinear edge detection (Rosenfeld Nonlinear edge detection (Rosenfeld 1970)1970)

• Noise reduction for medical images Noise reduction for medical images (Xu et al. 1994)(Xu et al. 1994)

• Analyzed by Sadler and Swami Analyzed by Sadler and Swami (1999)(1999)


The algorithm:The algorithm:save a copy of the W (m, n) to WW (m, n)save a copy of the W (m, n) to WW (m, n)

loop for each wavelet scale m {loop for each wavelet scale m {

loop for the iteration process {loop for the iteration process {

calculate the power of Corr2(m, n) and W (m, n)calculate the power of Corr2(m, n) and W (m, n)

rescale he power of Corr2(m, n) to that of W (m, n)rescale he power of Corr2(m, n) to that of W (m, n)

for each pixel n {for each pixel n {

if |Corr2(m,n)| > |W (m, n)|if |Corr2(m,n)| > |W (m, n)|

mask (m, n) = 1, Corr2(m, n) = 0, W (m, n) mask (m, n) = 1, Corr2(m, n) = 0, W (m, n) = 0 }= 0 }

} iterate until the power of W (m, n) < the noise threshold T (m)} iterate until the power of W (m, n) < the noise threshold T (m)

apply the “spatial filter mask” to the saved WW (m, n)}apply the “spatial filter mask” to the saved WW (m, n)}

noise signal

noisy signal


1-D2-D (m-D)



Nonseparable basis

Inter-scale (MPM)



Intra-scale (LCA)

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

directional gradient)directional gradient)o Thresholding (e.g., Canny’s hysteresis thresholding)Thresholding (e.g., Canny’s hysteresis thresholding)

• Canny Edge Detectors | Mallat’s quadratic spline Canny Edge Detectors | Mallat’s quadratic spline wavelet wavelet

• 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)

• Donoho (IEEE Trans. Inform. Theory, 1995)Donoho (IEEE Trans. Inform. Theory, 1995)

• 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

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:

Wavelet coefficients Wavelet coefficients {signal, noise} {signal, noise}– generalization: Bayesian approachgeneralization: Bayesian approach

• Approximation power:Approximation power:– n-term nonlinear approximationn-term nonlinear approximation– generalization: restricted nonlinear generalization: restricted nonlinear

approximationapproximation

Statistical ModelingStatistical Modeling

Gaussian Markov Random FieldsGaussian Markov Random Fields Statistical modeling of wavelet Statistical modeling of wavelet

coefficients:coefficients: Marginal Models:Marginal Models:

• Generalized Gaussian distributionsGeneralized Gaussian distributions• Gaussian Scale MixturesGaussian Scale Mixtures

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)

wavelets and denoising jun ge and gagan mirchandani electrical and computer engineering department...

Documents

dstatistical approachbayesian

pixel n

wavelet scale

power of w

power of corr2m

intensity image edge

computer vision

wavelet decorrelation