diffusion filters s. derin babacan department of electrical and computer engineering northwestern...
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Diffusion FiltersDiffusion Filters
S Derin BabacanS Derin BabacanDepartment of Electrical and Computer Department of Electrical and Computer
EngineeringEngineeringNorthwestern UniversityNorthwestern University
MarchMarch 77 200 20066
ECE 463 Term Project
IntroductionIntroduction
Filters for enhancement restoration Filters for enhancement restoration smoothing and feature extractionsmoothing and feature extraction
Varying for each data entry based on Varying for each data entry based on local featureslocal features u u = [u(0)u(1)u(n-1)u(n)]= [u(0)u(1)u(n-1)u(n)] ww = [ = [ww0 0 w w1 1 w wn-1 n-1 w wn n ]]
wwi i = [w= [wii(0) w(0) wii(1) w(1) wii(m-1) w(m-1) wii(m) ] (m) ]
Mean-value preservingMean-value preserving
Physics of DiffusionPhysics of Diffusion
Fickrsquos LawFickrsquos Law Continuity Continuity
Diffusion EquationDiffusion Equation
Gradient (uGradient (uxx u uyy) )
Divergence div Divergence div ΨΨ = = ΨΨ xx + + ΨΨ yy
Concentration replaced by data valuesConcentration replaced by data values
FluxDiffusion Tensor
Gradient
Concentration
Diffusion FiltersDiffusion Filters
D a scalar (flux and gradient are parallel)D a scalar (flux and gradient are parallel) Isotropic diffusion Isotropic diffusion D spatially invariant (fixed) Homogeneous D spatially invariant (fixed) Homogeneous
linear linear D = g(D = g() Inhomogeneous (linearnonlinear)) Inhomogeneous (linearnonlinear)
D a matrix (rotation andor scaling) flux D a matrix (rotation andor scaling) flux and gradient are generally not paralleland gradient are generally not parallel Anisotropic diffusionAnisotropic diffusion Eigenvectors of D determine the diffusion Eigenvectors of D determine the diffusion
directiondirection
Linear Diffusion FiltersLinear Diffusion Filters
Homogeneous case D = 1Homogeneous case D = 1
Initial condition u(x0) = f(x)Initial condition u(x0) = f(x) Unique solutionUnique solution
Linear Diffusion FiltersLinear Diffusion Filters
rarr
HL
IHL
Linear Diffusion FiltersLinear Diffusion Filters
Inhomogeneous case Inhomogeneous case Decrease blurring of the important Decrease blurring of the important
featuresfeatures Spatial adaptation rarr Reduce diffusion Spatial adaptation rarr Reduce diffusion
at the edgesat the edges D = g(D = g() monotonically decrease with ) monotonically decrease with
increasing parameterincreasing parameter Gradient of f as the fuzzy edge detectorGradient of f as the fuzzy edge detector
Result
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
Incorporate adaptation based on the current Incorporate adaptation based on the current filtered image instead of the initial imagefiltered image instead of the initial image Spatial and temporal adaptationSpatial and temporal adaptation
D = g(D = g()) g a function of ug a function of u
High contrast region low diffusionHigh contrast region low diffusion Low contrast region high diffusionLow contrast region high diffusion
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
CharbonnierCharbonnier
Perona-MalikPerona-Malik
λλ contrast parameter to discriminate contrast parameter to discriminate noise from edgesnoise from edges
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
High noise in gradient High noise in gradient Instability suppresion of important Instability suppresion of important
image featuresimage features Unnecessary suppression of diffusion Unnecessary suppression of diffusion
(noise remains)(noise remains) Regularization Smooth the gradient Regularization Smooth the gradient
before adaptationbefore adaptation
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
t = 1500t = 40 t = 400t = 0EEOriginal PM R-PM
R-PM
rarr
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
Canny PM
Original PM
Nonlinear Anisotropic Nonlinear Anisotropic FiltersFilters
So far adaptation is only done spatially So far adaptation is only done spatially andor temporally but diffusion direction is andor temporally but diffusion direction is fixedfixed
Adaptation of the diffusion orientationAdaptation of the diffusion orientation D is chosen as a matrix (scaling andor rotation)D is chosen as a matrix (scaling andor rotation)
Varying the diffusion direction over the Varying the diffusion direction over the image can preserveenhance localsemilocal image can preserveenhance localsemilocal featuresfeatures
Intraregion smoothing instead of Intraregion smoothing instead of interregion smoothinginterregion smoothing
Edge Enhancing FilterEdge Enhancing Filter
Change the orientation and the Change the orientation and the strength of the diffusion at an edgestrength of the diffusion at an edge
Smooth along the edge boundarySmooth along the edge boundary Diffusion along the edgesDiffusion along the edges
ReduceStop diffusion across the ReduceStop diffusion across the edge (eg Orthogonal to the edge)edge (eg Orthogonal to the edge)
Intraregion smoothingIntraregion smoothing
Edge Enhancing FilterEdge Enhancing Filter
Create a positive semidefinite diffusion Create a positive semidefinite diffusion tensor Dtensor D
Orthonormal eigenvectors of D Orthonormal eigenvectors of D υυ1 1 υυ2 2
Eigenvalues control the diffusion strengthEigenvalues control the diffusion strength 1)1)
2)2)
Edge Enhancing FilterEdge Enhancing Filter
t = 1500t = 40 t = 400t = 0
t = 3000t = 250 t = 875t = 0
R-PM
EE
larr
Coherence Enhancing Coherence Enhancing FilterFilter
Interrupted lines flows in addition Interrupted lines flows in addition to noiseto noise
Need to process line or coherent Need to process line or coherent flow-like structures complete gapsflow-like structures complete gaps
Coherence Enhancing Coherence Enhancing FilterFilter
Orientations instead of directions Orientations instead of directions (invariance to sign changes in (invariance to sign changes in gradient)gradient)
Orientation feature Orientation feature Structure tensorStructure tensor
Create diffusion tensor as a function Create diffusion tensor as a function of structure tensor instead of the of structure tensor instead of the gradientgradient
Coherence Enhancing Coherence Enhancing FilterFilter
JJρρ positive semidefinite rarr orthonormal positive semidefinite rarr orthonormal eigenvectors eigenvectors υυ1 1 υυ2 2 eigenvalues eigenvalues μμ11ge ge μμ22 ge ge 00
υυ1 1 orientation of the highest grey value orientation of the highest grey value fluctuationfluctuation
υυ2 2 preferred local orientation (lowest preferred local orientation (lowest fluctuation)fluctuation)
((μμ1 1 - - μμ22))22 a measure of local coherence a measure of local coherence Suppress diffusion along Suppress diffusion along υυ1 1
Smooth along the direction of Smooth along the direction of υυ2 2
Coherence Enhancing Coherence Enhancing FilterFilter
Choose the same set of eigenvectors Choose the same set of eigenvectors for Dfor D
Set eigenvalues Set eigenvalues λλ11 λλ22 as as
((μμ1 1 - - μμ22 ) )22 large large λλ22 asymp 1 asymp 1
((μμ1 1 - - μμ22 ) )22 small small λλ22 asymp asymp άά
Coherence Enhancing Coherence Enhancing FilterFilter
Coherence Enhancing Coherence Enhancing FilterFilter
PMOriginal CE
ConclusionsConclusions
Fast and stable filtersFast and stable filters Highly flexible Highly flexible
Varying the parameters of the diffusion Varying the parameters of the diffusion tensor (orientation rotation strength)tensor (orientation rotation strength)
Incorporation of new features (texture Incorporation of new features (texture color etc)color etc)
Separate filters for each data entrySeparate filters for each data entry High performanceHigh performance
Thank youThank you
Questions Questions
IntroductionIntroduction
Filters for enhancement restoration Filters for enhancement restoration smoothing and feature extractionsmoothing and feature extraction
Varying for each data entry based on Varying for each data entry based on local featureslocal features u u = [u(0)u(1)u(n-1)u(n)]= [u(0)u(1)u(n-1)u(n)] ww = [ = [ww0 0 w w1 1 w wn-1 n-1 w wn n ]]
wwi i = [w= [wii(0) w(0) wii(1) w(1) wii(m-1) w(m-1) wii(m) ] (m) ]
Mean-value preservingMean-value preserving
Physics of DiffusionPhysics of Diffusion
Fickrsquos LawFickrsquos Law Continuity Continuity
Diffusion EquationDiffusion Equation
Gradient (uGradient (uxx u uyy) )
Divergence div Divergence div ΨΨ = = ΨΨ xx + + ΨΨ yy
Concentration replaced by data valuesConcentration replaced by data values
FluxDiffusion Tensor
Gradient
Concentration
Diffusion FiltersDiffusion Filters
D a scalar (flux and gradient are parallel)D a scalar (flux and gradient are parallel) Isotropic diffusion Isotropic diffusion D spatially invariant (fixed) Homogeneous D spatially invariant (fixed) Homogeneous
linear linear D = g(D = g() Inhomogeneous (linearnonlinear)) Inhomogeneous (linearnonlinear)
D a matrix (rotation andor scaling) flux D a matrix (rotation andor scaling) flux and gradient are generally not paralleland gradient are generally not parallel Anisotropic diffusionAnisotropic diffusion Eigenvectors of D determine the diffusion Eigenvectors of D determine the diffusion
directiondirection
Linear Diffusion FiltersLinear Diffusion Filters
Homogeneous case D = 1Homogeneous case D = 1
Initial condition u(x0) = f(x)Initial condition u(x0) = f(x) Unique solutionUnique solution
Linear Diffusion FiltersLinear Diffusion Filters
rarr
HL
IHL
Linear Diffusion FiltersLinear Diffusion Filters
Inhomogeneous case Inhomogeneous case Decrease blurring of the important Decrease blurring of the important
featuresfeatures Spatial adaptation rarr Reduce diffusion Spatial adaptation rarr Reduce diffusion
at the edgesat the edges D = g(D = g() monotonically decrease with ) monotonically decrease with
increasing parameterincreasing parameter Gradient of f as the fuzzy edge detectorGradient of f as the fuzzy edge detector
Result
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
Incorporate adaptation based on the current Incorporate adaptation based on the current filtered image instead of the initial imagefiltered image instead of the initial image Spatial and temporal adaptationSpatial and temporal adaptation
D = g(D = g()) g a function of ug a function of u
High contrast region low diffusionHigh contrast region low diffusion Low contrast region high diffusionLow contrast region high diffusion
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
CharbonnierCharbonnier
Perona-MalikPerona-Malik
λλ contrast parameter to discriminate contrast parameter to discriminate noise from edgesnoise from edges
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
High noise in gradient High noise in gradient Instability suppresion of important Instability suppresion of important
image featuresimage features Unnecessary suppression of diffusion Unnecessary suppression of diffusion
(noise remains)(noise remains) Regularization Smooth the gradient Regularization Smooth the gradient
before adaptationbefore adaptation
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
t = 1500t = 40 t = 400t = 0EEOriginal PM R-PM
R-PM
rarr
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
Canny PM
Original PM
Nonlinear Anisotropic Nonlinear Anisotropic FiltersFilters
So far adaptation is only done spatially So far adaptation is only done spatially andor temporally but diffusion direction is andor temporally but diffusion direction is fixedfixed
Adaptation of the diffusion orientationAdaptation of the diffusion orientation D is chosen as a matrix (scaling andor rotation)D is chosen as a matrix (scaling andor rotation)
Varying the diffusion direction over the Varying the diffusion direction over the image can preserveenhance localsemilocal image can preserveenhance localsemilocal featuresfeatures
Intraregion smoothing instead of Intraregion smoothing instead of interregion smoothinginterregion smoothing
Edge Enhancing FilterEdge Enhancing Filter
Change the orientation and the Change the orientation and the strength of the diffusion at an edgestrength of the diffusion at an edge
Smooth along the edge boundarySmooth along the edge boundary Diffusion along the edgesDiffusion along the edges
ReduceStop diffusion across the ReduceStop diffusion across the edge (eg Orthogonal to the edge)edge (eg Orthogonal to the edge)
Intraregion smoothingIntraregion smoothing
Edge Enhancing FilterEdge Enhancing Filter
Create a positive semidefinite diffusion Create a positive semidefinite diffusion tensor Dtensor D
Orthonormal eigenvectors of D Orthonormal eigenvectors of D υυ1 1 υυ2 2
Eigenvalues control the diffusion strengthEigenvalues control the diffusion strength 1)1)
2)2)
Edge Enhancing FilterEdge Enhancing Filter
t = 1500t = 40 t = 400t = 0
t = 3000t = 250 t = 875t = 0
R-PM
EE
larr
Coherence Enhancing Coherence Enhancing FilterFilter
Interrupted lines flows in addition Interrupted lines flows in addition to noiseto noise
Need to process line or coherent Need to process line or coherent flow-like structures complete gapsflow-like structures complete gaps
Coherence Enhancing Coherence Enhancing FilterFilter
Orientations instead of directions Orientations instead of directions (invariance to sign changes in (invariance to sign changes in gradient)gradient)
Orientation feature Orientation feature Structure tensorStructure tensor
Create diffusion tensor as a function Create diffusion tensor as a function of structure tensor instead of the of structure tensor instead of the gradientgradient
Coherence Enhancing Coherence Enhancing FilterFilter
JJρρ positive semidefinite rarr orthonormal positive semidefinite rarr orthonormal eigenvectors eigenvectors υυ1 1 υυ2 2 eigenvalues eigenvalues μμ11ge ge μμ22 ge ge 00
υυ1 1 orientation of the highest grey value orientation of the highest grey value fluctuationfluctuation
υυ2 2 preferred local orientation (lowest preferred local orientation (lowest fluctuation)fluctuation)
((μμ1 1 - - μμ22))22 a measure of local coherence a measure of local coherence Suppress diffusion along Suppress diffusion along υυ1 1
Smooth along the direction of Smooth along the direction of υυ2 2
Coherence Enhancing Coherence Enhancing FilterFilter
Choose the same set of eigenvectors Choose the same set of eigenvectors for Dfor D
Set eigenvalues Set eigenvalues λλ11 λλ22 as as
((μμ1 1 - - μμ22 ) )22 large large λλ22 asymp 1 asymp 1
((μμ1 1 - - μμ22 ) )22 small small λλ22 asymp asymp άά
Coherence Enhancing Coherence Enhancing FilterFilter
Coherence Enhancing Coherence Enhancing FilterFilter
PMOriginal CE
ConclusionsConclusions
Fast and stable filtersFast and stable filters Highly flexible Highly flexible
Varying the parameters of the diffusion Varying the parameters of the diffusion tensor (orientation rotation strength)tensor (orientation rotation strength)
Incorporation of new features (texture Incorporation of new features (texture color etc)color etc)
Separate filters for each data entrySeparate filters for each data entry High performanceHigh performance
Thank youThank you
Questions Questions
Physics of DiffusionPhysics of Diffusion
Fickrsquos LawFickrsquos Law Continuity Continuity
Diffusion EquationDiffusion Equation
Gradient (uGradient (uxx u uyy) )
Divergence div Divergence div ΨΨ = = ΨΨ xx + + ΨΨ yy
Concentration replaced by data valuesConcentration replaced by data values
FluxDiffusion Tensor
Gradient
Concentration
Diffusion FiltersDiffusion Filters
D a scalar (flux and gradient are parallel)D a scalar (flux and gradient are parallel) Isotropic diffusion Isotropic diffusion D spatially invariant (fixed) Homogeneous D spatially invariant (fixed) Homogeneous
linear linear D = g(D = g() Inhomogeneous (linearnonlinear)) Inhomogeneous (linearnonlinear)
D a matrix (rotation andor scaling) flux D a matrix (rotation andor scaling) flux and gradient are generally not paralleland gradient are generally not parallel Anisotropic diffusionAnisotropic diffusion Eigenvectors of D determine the diffusion Eigenvectors of D determine the diffusion
directiondirection
Linear Diffusion FiltersLinear Diffusion Filters
Homogeneous case D = 1Homogeneous case D = 1
Initial condition u(x0) = f(x)Initial condition u(x0) = f(x) Unique solutionUnique solution
Linear Diffusion FiltersLinear Diffusion Filters
rarr
HL
IHL
Linear Diffusion FiltersLinear Diffusion Filters
Inhomogeneous case Inhomogeneous case Decrease blurring of the important Decrease blurring of the important
featuresfeatures Spatial adaptation rarr Reduce diffusion Spatial adaptation rarr Reduce diffusion
at the edgesat the edges D = g(D = g() monotonically decrease with ) monotonically decrease with
increasing parameterincreasing parameter Gradient of f as the fuzzy edge detectorGradient of f as the fuzzy edge detector
Result
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
Incorporate adaptation based on the current Incorporate adaptation based on the current filtered image instead of the initial imagefiltered image instead of the initial image Spatial and temporal adaptationSpatial and temporal adaptation
D = g(D = g()) g a function of ug a function of u
High contrast region low diffusionHigh contrast region low diffusion Low contrast region high diffusionLow contrast region high diffusion
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
CharbonnierCharbonnier
Perona-MalikPerona-Malik
λλ contrast parameter to discriminate contrast parameter to discriminate noise from edgesnoise from edges
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
High noise in gradient High noise in gradient Instability suppresion of important Instability suppresion of important
image featuresimage features Unnecessary suppression of diffusion Unnecessary suppression of diffusion
(noise remains)(noise remains) Regularization Smooth the gradient Regularization Smooth the gradient
before adaptationbefore adaptation
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
t = 1500t = 40 t = 400t = 0EEOriginal PM R-PM
R-PM
rarr
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
Canny PM
Original PM
Nonlinear Anisotropic Nonlinear Anisotropic FiltersFilters
So far adaptation is only done spatially So far adaptation is only done spatially andor temporally but diffusion direction is andor temporally but diffusion direction is fixedfixed
Adaptation of the diffusion orientationAdaptation of the diffusion orientation D is chosen as a matrix (scaling andor rotation)D is chosen as a matrix (scaling andor rotation)
Varying the diffusion direction over the Varying the diffusion direction over the image can preserveenhance localsemilocal image can preserveenhance localsemilocal featuresfeatures
Intraregion smoothing instead of Intraregion smoothing instead of interregion smoothinginterregion smoothing
Edge Enhancing FilterEdge Enhancing Filter
Change the orientation and the Change the orientation and the strength of the diffusion at an edgestrength of the diffusion at an edge
Smooth along the edge boundarySmooth along the edge boundary Diffusion along the edgesDiffusion along the edges
ReduceStop diffusion across the ReduceStop diffusion across the edge (eg Orthogonal to the edge)edge (eg Orthogonal to the edge)
Intraregion smoothingIntraregion smoothing
Edge Enhancing FilterEdge Enhancing Filter
Create a positive semidefinite diffusion Create a positive semidefinite diffusion tensor Dtensor D
Orthonormal eigenvectors of D Orthonormal eigenvectors of D υυ1 1 υυ2 2
Eigenvalues control the diffusion strengthEigenvalues control the diffusion strength 1)1)
2)2)
Edge Enhancing FilterEdge Enhancing Filter
t = 1500t = 40 t = 400t = 0
t = 3000t = 250 t = 875t = 0
R-PM
EE
larr
Coherence Enhancing Coherence Enhancing FilterFilter
Interrupted lines flows in addition Interrupted lines flows in addition to noiseto noise
Need to process line or coherent Need to process line or coherent flow-like structures complete gapsflow-like structures complete gaps
Coherence Enhancing Coherence Enhancing FilterFilter
Orientations instead of directions Orientations instead of directions (invariance to sign changes in (invariance to sign changes in gradient)gradient)
Orientation feature Orientation feature Structure tensorStructure tensor
Create diffusion tensor as a function Create diffusion tensor as a function of structure tensor instead of the of structure tensor instead of the gradientgradient
Coherence Enhancing Coherence Enhancing FilterFilter
JJρρ positive semidefinite rarr orthonormal positive semidefinite rarr orthonormal eigenvectors eigenvectors υυ1 1 υυ2 2 eigenvalues eigenvalues μμ11ge ge μμ22 ge ge 00
υυ1 1 orientation of the highest grey value orientation of the highest grey value fluctuationfluctuation
υυ2 2 preferred local orientation (lowest preferred local orientation (lowest fluctuation)fluctuation)
((μμ1 1 - - μμ22))22 a measure of local coherence a measure of local coherence Suppress diffusion along Suppress diffusion along υυ1 1
Smooth along the direction of Smooth along the direction of υυ2 2
Coherence Enhancing Coherence Enhancing FilterFilter
Choose the same set of eigenvectors Choose the same set of eigenvectors for Dfor D
Set eigenvalues Set eigenvalues λλ11 λλ22 as as
((μμ1 1 - - μμ22 ) )22 large large λλ22 asymp 1 asymp 1
((μμ1 1 - - μμ22 ) )22 small small λλ22 asymp asymp άά
Coherence Enhancing Coherence Enhancing FilterFilter
Coherence Enhancing Coherence Enhancing FilterFilter
PMOriginal CE
ConclusionsConclusions
Fast and stable filtersFast and stable filters Highly flexible Highly flexible
Varying the parameters of the diffusion Varying the parameters of the diffusion tensor (orientation rotation strength)tensor (orientation rotation strength)
Incorporation of new features (texture Incorporation of new features (texture color etc)color etc)
Separate filters for each data entrySeparate filters for each data entry High performanceHigh performance
Thank youThank you
Questions Questions
Diffusion FiltersDiffusion Filters
D a scalar (flux and gradient are parallel)D a scalar (flux and gradient are parallel) Isotropic diffusion Isotropic diffusion D spatially invariant (fixed) Homogeneous D spatially invariant (fixed) Homogeneous
linear linear D = g(D = g() Inhomogeneous (linearnonlinear)) Inhomogeneous (linearnonlinear)
D a matrix (rotation andor scaling) flux D a matrix (rotation andor scaling) flux and gradient are generally not paralleland gradient are generally not parallel Anisotropic diffusionAnisotropic diffusion Eigenvectors of D determine the diffusion Eigenvectors of D determine the diffusion
directiondirection
Linear Diffusion FiltersLinear Diffusion Filters
Homogeneous case D = 1Homogeneous case D = 1
Initial condition u(x0) = f(x)Initial condition u(x0) = f(x) Unique solutionUnique solution
Linear Diffusion FiltersLinear Diffusion Filters
rarr
HL
IHL
Linear Diffusion FiltersLinear Diffusion Filters
Inhomogeneous case Inhomogeneous case Decrease blurring of the important Decrease blurring of the important
featuresfeatures Spatial adaptation rarr Reduce diffusion Spatial adaptation rarr Reduce diffusion
at the edgesat the edges D = g(D = g() monotonically decrease with ) monotonically decrease with
increasing parameterincreasing parameter Gradient of f as the fuzzy edge detectorGradient of f as the fuzzy edge detector
Result
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
Incorporate adaptation based on the current Incorporate adaptation based on the current filtered image instead of the initial imagefiltered image instead of the initial image Spatial and temporal adaptationSpatial and temporal adaptation
D = g(D = g()) g a function of ug a function of u
High contrast region low diffusionHigh contrast region low diffusion Low contrast region high diffusionLow contrast region high diffusion
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
CharbonnierCharbonnier
Perona-MalikPerona-Malik
λλ contrast parameter to discriminate contrast parameter to discriminate noise from edgesnoise from edges
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
High noise in gradient High noise in gradient Instability suppresion of important Instability suppresion of important
image featuresimage features Unnecessary suppression of diffusion Unnecessary suppression of diffusion
(noise remains)(noise remains) Regularization Smooth the gradient Regularization Smooth the gradient
before adaptationbefore adaptation
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
t = 1500t = 40 t = 400t = 0EEOriginal PM R-PM
R-PM
rarr
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
Canny PM
Original PM
Nonlinear Anisotropic Nonlinear Anisotropic FiltersFilters
So far adaptation is only done spatially So far adaptation is only done spatially andor temporally but diffusion direction is andor temporally but diffusion direction is fixedfixed
Adaptation of the diffusion orientationAdaptation of the diffusion orientation D is chosen as a matrix (scaling andor rotation)D is chosen as a matrix (scaling andor rotation)
Varying the diffusion direction over the Varying the diffusion direction over the image can preserveenhance localsemilocal image can preserveenhance localsemilocal featuresfeatures
Intraregion smoothing instead of Intraregion smoothing instead of interregion smoothinginterregion smoothing
Edge Enhancing FilterEdge Enhancing Filter
Change the orientation and the Change the orientation and the strength of the diffusion at an edgestrength of the diffusion at an edge
Smooth along the edge boundarySmooth along the edge boundary Diffusion along the edgesDiffusion along the edges
ReduceStop diffusion across the ReduceStop diffusion across the edge (eg Orthogonal to the edge)edge (eg Orthogonal to the edge)
Intraregion smoothingIntraregion smoothing
Edge Enhancing FilterEdge Enhancing Filter
Create a positive semidefinite diffusion Create a positive semidefinite diffusion tensor Dtensor D
Orthonormal eigenvectors of D Orthonormal eigenvectors of D υυ1 1 υυ2 2
Eigenvalues control the diffusion strengthEigenvalues control the diffusion strength 1)1)
2)2)
Edge Enhancing FilterEdge Enhancing Filter
t = 1500t = 40 t = 400t = 0
t = 3000t = 250 t = 875t = 0
R-PM
EE
larr
Coherence Enhancing Coherence Enhancing FilterFilter
Interrupted lines flows in addition Interrupted lines flows in addition to noiseto noise
Need to process line or coherent Need to process line or coherent flow-like structures complete gapsflow-like structures complete gaps
Coherence Enhancing Coherence Enhancing FilterFilter
Orientations instead of directions Orientations instead of directions (invariance to sign changes in (invariance to sign changes in gradient)gradient)
Orientation feature Orientation feature Structure tensorStructure tensor
Create diffusion tensor as a function Create diffusion tensor as a function of structure tensor instead of the of structure tensor instead of the gradientgradient
Coherence Enhancing Coherence Enhancing FilterFilter
JJρρ positive semidefinite rarr orthonormal positive semidefinite rarr orthonormal eigenvectors eigenvectors υυ1 1 υυ2 2 eigenvalues eigenvalues μμ11ge ge μμ22 ge ge 00
υυ1 1 orientation of the highest grey value orientation of the highest grey value fluctuationfluctuation
υυ2 2 preferred local orientation (lowest preferred local orientation (lowest fluctuation)fluctuation)
((μμ1 1 - - μμ22))22 a measure of local coherence a measure of local coherence Suppress diffusion along Suppress diffusion along υυ1 1
Smooth along the direction of Smooth along the direction of υυ2 2
Coherence Enhancing Coherence Enhancing FilterFilter
Choose the same set of eigenvectors Choose the same set of eigenvectors for Dfor D
Set eigenvalues Set eigenvalues λλ11 λλ22 as as
((μμ1 1 - - μμ22 ) )22 large large λλ22 asymp 1 asymp 1
((μμ1 1 - - μμ22 ) )22 small small λλ22 asymp asymp άά
Coherence Enhancing Coherence Enhancing FilterFilter
Coherence Enhancing Coherence Enhancing FilterFilter
PMOriginal CE
ConclusionsConclusions
Fast and stable filtersFast and stable filters Highly flexible Highly flexible
Varying the parameters of the diffusion Varying the parameters of the diffusion tensor (orientation rotation strength)tensor (orientation rotation strength)
Incorporation of new features (texture Incorporation of new features (texture color etc)color etc)
Separate filters for each data entrySeparate filters for each data entry High performanceHigh performance
Thank youThank you
Questions Questions
Linear Diffusion FiltersLinear Diffusion Filters
Homogeneous case D = 1Homogeneous case D = 1
Initial condition u(x0) = f(x)Initial condition u(x0) = f(x) Unique solutionUnique solution
Linear Diffusion FiltersLinear Diffusion Filters
rarr
HL
IHL
Linear Diffusion FiltersLinear Diffusion Filters
Inhomogeneous case Inhomogeneous case Decrease blurring of the important Decrease blurring of the important
featuresfeatures Spatial adaptation rarr Reduce diffusion Spatial adaptation rarr Reduce diffusion
at the edgesat the edges D = g(D = g() monotonically decrease with ) monotonically decrease with
increasing parameterincreasing parameter Gradient of f as the fuzzy edge detectorGradient of f as the fuzzy edge detector
Result
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
Incorporate adaptation based on the current Incorporate adaptation based on the current filtered image instead of the initial imagefiltered image instead of the initial image Spatial and temporal adaptationSpatial and temporal adaptation
D = g(D = g()) g a function of ug a function of u
High contrast region low diffusionHigh contrast region low diffusion Low contrast region high diffusionLow contrast region high diffusion
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
CharbonnierCharbonnier
Perona-MalikPerona-Malik
λλ contrast parameter to discriminate contrast parameter to discriminate noise from edgesnoise from edges
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
High noise in gradient High noise in gradient Instability suppresion of important Instability suppresion of important
image featuresimage features Unnecessary suppression of diffusion Unnecessary suppression of diffusion
(noise remains)(noise remains) Regularization Smooth the gradient Regularization Smooth the gradient
before adaptationbefore adaptation
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
t = 1500t = 40 t = 400t = 0EEOriginal PM R-PM
R-PM
rarr
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
Canny PM
Original PM
Nonlinear Anisotropic Nonlinear Anisotropic FiltersFilters
So far adaptation is only done spatially So far adaptation is only done spatially andor temporally but diffusion direction is andor temporally but diffusion direction is fixedfixed
Adaptation of the diffusion orientationAdaptation of the diffusion orientation D is chosen as a matrix (scaling andor rotation)D is chosen as a matrix (scaling andor rotation)
Varying the diffusion direction over the Varying the diffusion direction over the image can preserveenhance localsemilocal image can preserveenhance localsemilocal featuresfeatures
Intraregion smoothing instead of Intraregion smoothing instead of interregion smoothinginterregion smoothing
Edge Enhancing FilterEdge Enhancing Filter
Change the orientation and the Change the orientation and the strength of the diffusion at an edgestrength of the diffusion at an edge
Smooth along the edge boundarySmooth along the edge boundary Diffusion along the edgesDiffusion along the edges
ReduceStop diffusion across the ReduceStop diffusion across the edge (eg Orthogonal to the edge)edge (eg Orthogonal to the edge)
Intraregion smoothingIntraregion smoothing
Edge Enhancing FilterEdge Enhancing Filter
Create a positive semidefinite diffusion Create a positive semidefinite diffusion tensor Dtensor D
Orthonormal eigenvectors of D Orthonormal eigenvectors of D υυ1 1 υυ2 2
Eigenvalues control the diffusion strengthEigenvalues control the diffusion strength 1)1)
2)2)
Edge Enhancing FilterEdge Enhancing Filter
t = 1500t = 40 t = 400t = 0
t = 3000t = 250 t = 875t = 0
R-PM
EE
larr
Coherence Enhancing Coherence Enhancing FilterFilter
Interrupted lines flows in addition Interrupted lines flows in addition to noiseto noise
Need to process line or coherent Need to process line or coherent flow-like structures complete gapsflow-like structures complete gaps
Coherence Enhancing Coherence Enhancing FilterFilter
Orientations instead of directions Orientations instead of directions (invariance to sign changes in (invariance to sign changes in gradient)gradient)
Orientation feature Orientation feature Structure tensorStructure tensor
Create diffusion tensor as a function Create diffusion tensor as a function of structure tensor instead of the of structure tensor instead of the gradientgradient
Coherence Enhancing Coherence Enhancing FilterFilter
JJρρ positive semidefinite rarr orthonormal positive semidefinite rarr orthonormal eigenvectors eigenvectors υυ1 1 υυ2 2 eigenvalues eigenvalues μμ11ge ge μμ22 ge ge 00
υυ1 1 orientation of the highest grey value orientation of the highest grey value fluctuationfluctuation
υυ2 2 preferred local orientation (lowest preferred local orientation (lowest fluctuation)fluctuation)
((μμ1 1 - - μμ22))22 a measure of local coherence a measure of local coherence Suppress diffusion along Suppress diffusion along υυ1 1
Smooth along the direction of Smooth along the direction of υυ2 2
Coherence Enhancing Coherence Enhancing FilterFilter
Choose the same set of eigenvectors Choose the same set of eigenvectors for Dfor D
Set eigenvalues Set eigenvalues λλ11 λλ22 as as
((μμ1 1 - - μμ22 ) )22 large large λλ22 asymp 1 asymp 1
((μμ1 1 - - μμ22 ) )22 small small λλ22 asymp asymp άά
Coherence Enhancing Coherence Enhancing FilterFilter
Coherence Enhancing Coherence Enhancing FilterFilter
PMOriginal CE
ConclusionsConclusions
Fast and stable filtersFast and stable filters Highly flexible Highly flexible
Varying the parameters of the diffusion Varying the parameters of the diffusion tensor (orientation rotation strength)tensor (orientation rotation strength)
Incorporation of new features (texture Incorporation of new features (texture color etc)color etc)
Separate filters for each data entrySeparate filters for each data entry High performanceHigh performance
Thank youThank you
Questions Questions
Linear Diffusion FiltersLinear Diffusion Filters
rarr
HL
IHL
Linear Diffusion FiltersLinear Diffusion Filters
Inhomogeneous case Inhomogeneous case Decrease blurring of the important Decrease blurring of the important
featuresfeatures Spatial adaptation rarr Reduce diffusion Spatial adaptation rarr Reduce diffusion
at the edgesat the edges D = g(D = g() monotonically decrease with ) monotonically decrease with
increasing parameterincreasing parameter Gradient of f as the fuzzy edge detectorGradient of f as the fuzzy edge detector
Result
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
Incorporate adaptation based on the current Incorporate adaptation based on the current filtered image instead of the initial imagefiltered image instead of the initial image Spatial and temporal adaptationSpatial and temporal adaptation
D = g(D = g()) g a function of ug a function of u
High contrast region low diffusionHigh contrast region low diffusion Low contrast region high diffusionLow contrast region high diffusion
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
CharbonnierCharbonnier
Perona-MalikPerona-Malik
λλ contrast parameter to discriminate contrast parameter to discriminate noise from edgesnoise from edges
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
High noise in gradient High noise in gradient Instability suppresion of important Instability suppresion of important
image featuresimage features Unnecessary suppression of diffusion Unnecessary suppression of diffusion
(noise remains)(noise remains) Regularization Smooth the gradient Regularization Smooth the gradient
before adaptationbefore adaptation
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
t = 1500t = 40 t = 400t = 0EEOriginal PM R-PM
R-PM
rarr
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
Canny PM
Original PM
Nonlinear Anisotropic Nonlinear Anisotropic FiltersFilters
So far adaptation is only done spatially So far adaptation is only done spatially andor temporally but diffusion direction is andor temporally but diffusion direction is fixedfixed
Adaptation of the diffusion orientationAdaptation of the diffusion orientation D is chosen as a matrix (scaling andor rotation)D is chosen as a matrix (scaling andor rotation)
Varying the diffusion direction over the Varying the diffusion direction over the image can preserveenhance localsemilocal image can preserveenhance localsemilocal featuresfeatures
Intraregion smoothing instead of Intraregion smoothing instead of interregion smoothinginterregion smoothing
Edge Enhancing FilterEdge Enhancing Filter
Change the orientation and the Change the orientation and the strength of the diffusion at an edgestrength of the diffusion at an edge
Smooth along the edge boundarySmooth along the edge boundary Diffusion along the edgesDiffusion along the edges
ReduceStop diffusion across the ReduceStop diffusion across the edge (eg Orthogonal to the edge)edge (eg Orthogonal to the edge)
Intraregion smoothingIntraregion smoothing
Edge Enhancing FilterEdge Enhancing Filter
Create a positive semidefinite diffusion Create a positive semidefinite diffusion tensor Dtensor D
Orthonormal eigenvectors of D Orthonormal eigenvectors of D υυ1 1 υυ2 2
Eigenvalues control the diffusion strengthEigenvalues control the diffusion strength 1)1)
2)2)
Edge Enhancing FilterEdge Enhancing Filter
t = 1500t = 40 t = 400t = 0
t = 3000t = 250 t = 875t = 0
R-PM
EE
larr
Coherence Enhancing Coherence Enhancing FilterFilter
Interrupted lines flows in addition Interrupted lines flows in addition to noiseto noise
Need to process line or coherent Need to process line or coherent flow-like structures complete gapsflow-like structures complete gaps
Coherence Enhancing Coherence Enhancing FilterFilter
Orientations instead of directions Orientations instead of directions (invariance to sign changes in (invariance to sign changes in gradient)gradient)
Orientation feature Orientation feature Structure tensorStructure tensor
Create diffusion tensor as a function Create diffusion tensor as a function of structure tensor instead of the of structure tensor instead of the gradientgradient
Coherence Enhancing Coherence Enhancing FilterFilter
JJρρ positive semidefinite rarr orthonormal positive semidefinite rarr orthonormal eigenvectors eigenvectors υυ1 1 υυ2 2 eigenvalues eigenvalues μμ11ge ge μμ22 ge ge 00
υυ1 1 orientation of the highest grey value orientation of the highest grey value fluctuationfluctuation
υυ2 2 preferred local orientation (lowest preferred local orientation (lowest fluctuation)fluctuation)
((μμ1 1 - - μμ22))22 a measure of local coherence a measure of local coherence Suppress diffusion along Suppress diffusion along υυ1 1
Smooth along the direction of Smooth along the direction of υυ2 2
Coherence Enhancing Coherence Enhancing FilterFilter
Choose the same set of eigenvectors Choose the same set of eigenvectors for Dfor D
Set eigenvalues Set eigenvalues λλ11 λλ22 as as
((μμ1 1 - - μμ22 ) )22 large large λλ22 asymp 1 asymp 1
((μμ1 1 - - μμ22 ) )22 small small λλ22 asymp asymp άά
Coherence Enhancing Coherence Enhancing FilterFilter
Coherence Enhancing Coherence Enhancing FilterFilter
PMOriginal CE
ConclusionsConclusions
Fast and stable filtersFast and stable filters Highly flexible Highly flexible
Varying the parameters of the diffusion Varying the parameters of the diffusion tensor (orientation rotation strength)tensor (orientation rotation strength)
Incorporation of new features (texture Incorporation of new features (texture color etc)color etc)
Separate filters for each data entrySeparate filters for each data entry High performanceHigh performance
Thank youThank you
Questions Questions
Linear Diffusion FiltersLinear Diffusion Filters
Inhomogeneous case Inhomogeneous case Decrease blurring of the important Decrease blurring of the important
featuresfeatures Spatial adaptation rarr Reduce diffusion Spatial adaptation rarr Reduce diffusion
at the edgesat the edges D = g(D = g() monotonically decrease with ) monotonically decrease with
increasing parameterincreasing parameter Gradient of f as the fuzzy edge detectorGradient of f as the fuzzy edge detector
Result
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
Incorporate adaptation based on the current Incorporate adaptation based on the current filtered image instead of the initial imagefiltered image instead of the initial image Spatial and temporal adaptationSpatial and temporal adaptation
D = g(D = g()) g a function of ug a function of u
High contrast region low diffusionHigh contrast region low diffusion Low contrast region high diffusionLow contrast region high diffusion
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
CharbonnierCharbonnier
Perona-MalikPerona-Malik
λλ contrast parameter to discriminate contrast parameter to discriminate noise from edgesnoise from edges
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
High noise in gradient High noise in gradient Instability suppresion of important Instability suppresion of important
image featuresimage features Unnecessary suppression of diffusion Unnecessary suppression of diffusion
(noise remains)(noise remains) Regularization Smooth the gradient Regularization Smooth the gradient
before adaptationbefore adaptation
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
t = 1500t = 40 t = 400t = 0EEOriginal PM R-PM
R-PM
rarr
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
Canny PM
Original PM
Nonlinear Anisotropic Nonlinear Anisotropic FiltersFilters
So far adaptation is only done spatially So far adaptation is only done spatially andor temporally but diffusion direction is andor temporally but diffusion direction is fixedfixed
Adaptation of the diffusion orientationAdaptation of the diffusion orientation D is chosen as a matrix (scaling andor rotation)D is chosen as a matrix (scaling andor rotation)
Varying the diffusion direction over the Varying the diffusion direction over the image can preserveenhance localsemilocal image can preserveenhance localsemilocal featuresfeatures
Intraregion smoothing instead of Intraregion smoothing instead of interregion smoothinginterregion smoothing
Edge Enhancing FilterEdge Enhancing Filter
Change the orientation and the Change the orientation and the strength of the diffusion at an edgestrength of the diffusion at an edge
Smooth along the edge boundarySmooth along the edge boundary Diffusion along the edgesDiffusion along the edges
ReduceStop diffusion across the ReduceStop diffusion across the edge (eg Orthogonal to the edge)edge (eg Orthogonal to the edge)
Intraregion smoothingIntraregion smoothing
Edge Enhancing FilterEdge Enhancing Filter
Create a positive semidefinite diffusion Create a positive semidefinite diffusion tensor Dtensor D
Orthonormal eigenvectors of D Orthonormal eigenvectors of D υυ1 1 υυ2 2
Eigenvalues control the diffusion strengthEigenvalues control the diffusion strength 1)1)
2)2)
Edge Enhancing FilterEdge Enhancing Filter
t = 1500t = 40 t = 400t = 0
t = 3000t = 250 t = 875t = 0
R-PM
EE
larr
Coherence Enhancing Coherence Enhancing FilterFilter
Interrupted lines flows in addition Interrupted lines flows in addition to noiseto noise
Need to process line or coherent Need to process line or coherent flow-like structures complete gapsflow-like structures complete gaps
Coherence Enhancing Coherence Enhancing FilterFilter
Orientations instead of directions Orientations instead of directions (invariance to sign changes in (invariance to sign changes in gradient)gradient)
Orientation feature Orientation feature Structure tensorStructure tensor
Create diffusion tensor as a function Create diffusion tensor as a function of structure tensor instead of the of structure tensor instead of the gradientgradient
Coherence Enhancing Coherence Enhancing FilterFilter
JJρρ positive semidefinite rarr orthonormal positive semidefinite rarr orthonormal eigenvectors eigenvectors υυ1 1 υυ2 2 eigenvalues eigenvalues μμ11ge ge μμ22 ge ge 00
υυ1 1 orientation of the highest grey value orientation of the highest grey value fluctuationfluctuation
υυ2 2 preferred local orientation (lowest preferred local orientation (lowest fluctuation)fluctuation)
((μμ1 1 - - μμ22))22 a measure of local coherence a measure of local coherence Suppress diffusion along Suppress diffusion along υυ1 1
Smooth along the direction of Smooth along the direction of υυ2 2
Coherence Enhancing Coherence Enhancing FilterFilter
Choose the same set of eigenvectors Choose the same set of eigenvectors for Dfor D
Set eigenvalues Set eigenvalues λλ11 λλ22 as as
((μμ1 1 - - μμ22 ) )22 large large λλ22 asymp 1 asymp 1
((μμ1 1 - - μμ22 ) )22 small small λλ22 asymp asymp άά
Coherence Enhancing Coherence Enhancing FilterFilter
Coherence Enhancing Coherence Enhancing FilterFilter
PMOriginal CE
ConclusionsConclusions
Fast and stable filtersFast and stable filters Highly flexible Highly flexible
Varying the parameters of the diffusion Varying the parameters of the diffusion tensor (orientation rotation strength)tensor (orientation rotation strength)
Incorporation of new features (texture Incorporation of new features (texture color etc)color etc)
Separate filters for each data entrySeparate filters for each data entry High performanceHigh performance
Thank youThank you
Questions Questions
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
Incorporate adaptation based on the current Incorporate adaptation based on the current filtered image instead of the initial imagefiltered image instead of the initial image Spatial and temporal adaptationSpatial and temporal adaptation
D = g(D = g()) g a function of ug a function of u
High contrast region low diffusionHigh contrast region low diffusion Low contrast region high diffusionLow contrast region high diffusion
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
CharbonnierCharbonnier
Perona-MalikPerona-Malik
λλ contrast parameter to discriminate contrast parameter to discriminate noise from edgesnoise from edges
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
High noise in gradient High noise in gradient Instability suppresion of important Instability suppresion of important
image featuresimage features Unnecessary suppression of diffusion Unnecessary suppression of diffusion
(noise remains)(noise remains) Regularization Smooth the gradient Regularization Smooth the gradient
before adaptationbefore adaptation
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
t = 1500t = 40 t = 400t = 0EEOriginal PM R-PM
R-PM
rarr
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
Canny PM
Original PM
Nonlinear Anisotropic Nonlinear Anisotropic FiltersFilters
So far adaptation is only done spatially So far adaptation is only done spatially andor temporally but diffusion direction is andor temporally but diffusion direction is fixedfixed
Adaptation of the diffusion orientationAdaptation of the diffusion orientation D is chosen as a matrix (scaling andor rotation)D is chosen as a matrix (scaling andor rotation)
Varying the diffusion direction over the Varying the diffusion direction over the image can preserveenhance localsemilocal image can preserveenhance localsemilocal featuresfeatures
Intraregion smoothing instead of Intraregion smoothing instead of interregion smoothinginterregion smoothing
Edge Enhancing FilterEdge Enhancing Filter
Change the orientation and the Change the orientation and the strength of the diffusion at an edgestrength of the diffusion at an edge
Smooth along the edge boundarySmooth along the edge boundary Diffusion along the edgesDiffusion along the edges
ReduceStop diffusion across the ReduceStop diffusion across the edge (eg Orthogonal to the edge)edge (eg Orthogonal to the edge)
Intraregion smoothingIntraregion smoothing
Edge Enhancing FilterEdge Enhancing Filter
Create a positive semidefinite diffusion Create a positive semidefinite diffusion tensor Dtensor D
Orthonormal eigenvectors of D Orthonormal eigenvectors of D υυ1 1 υυ2 2
Eigenvalues control the diffusion strengthEigenvalues control the diffusion strength 1)1)
2)2)
Edge Enhancing FilterEdge Enhancing Filter
t = 1500t = 40 t = 400t = 0
t = 3000t = 250 t = 875t = 0
R-PM
EE
larr
Coherence Enhancing Coherence Enhancing FilterFilter
Interrupted lines flows in addition Interrupted lines flows in addition to noiseto noise
Need to process line or coherent Need to process line or coherent flow-like structures complete gapsflow-like structures complete gaps
Coherence Enhancing Coherence Enhancing FilterFilter
Orientations instead of directions Orientations instead of directions (invariance to sign changes in (invariance to sign changes in gradient)gradient)
Orientation feature Orientation feature Structure tensorStructure tensor
Create diffusion tensor as a function Create diffusion tensor as a function of structure tensor instead of the of structure tensor instead of the gradientgradient
Coherence Enhancing Coherence Enhancing FilterFilter
JJρρ positive semidefinite rarr orthonormal positive semidefinite rarr orthonormal eigenvectors eigenvectors υυ1 1 υυ2 2 eigenvalues eigenvalues μμ11ge ge μμ22 ge ge 00
υυ1 1 orientation of the highest grey value orientation of the highest grey value fluctuationfluctuation
υυ2 2 preferred local orientation (lowest preferred local orientation (lowest fluctuation)fluctuation)
((μμ1 1 - - μμ22))22 a measure of local coherence a measure of local coherence Suppress diffusion along Suppress diffusion along υυ1 1
Smooth along the direction of Smooth along the direction of υυ2 2
Coherence Enhancing Coherence Enhancing FilterFilter
Choose the same set of eigenvectors Choose the same set of eigenvectors for Dfor D
Set eigenvalues Set eigenvalues λλ11 λλ22 as as
((μμ1 1 - - μμ22 ) )22 large large λλ22 asymp 1 asymp 1
((μμ1 1 - - μμ22 ) )22 small small λλ22 asymp asymp άά
Coherence Enhancing Coherence Enhancing FilterFilter
Coherence Enhancing Coherence Enhancing FilterFilter
PMOriginal CE
ConclusionsConclusions
Fast and stable filtersFast and stable filters Highly flexible Highly flexible
Varying the parameters of the diffusion Varying the parameters of the diffusion tensor (orientation rotation strength)tensor (orientation rotation strength)
Incorporation of new features (texture Incorporation of new features (texture color etc)color etc)
Separate filters for each data entrySeparate filters for each data entry High performanceHigh performance
Thank youThank you
Questions Questions
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
CharbonnierCharbonnier
Perona-MalikPerona-Malik
λλ contrast parameter to discriminate contrast parameter to discriminate noise from edgesnoise from edges
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
High noise in gradient High noise in gradient Instability suppresion of important Instability suppresion of important
image featuresimage features Unnecessary suppression of diffusion Unnecessary suppression of diffusion
(noise remains)(noise remains) Regularization Smooth the gradient Regularization Smooth the gradient
before adaptationbefore adaptation
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
t = 1500t = 40 t = 400t = 0EEOriginal PM R-PM
R-PM
rarr
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
Canny PM
Original PM
Nonlinear Anisotropic Nonlinear Anisotropic FiltersFilters
So far adaptation is only done spatially So far adaptation is only done spatially andor temporally but diffusion direction is andor temporally but diffusion direction is fixedfixed
Adaptation of the diffusion orientationAdaptation of the diffusion orientation D is chosen as a matrix (scaling andor rotation)D is chosen as a matrix (scaling andor rotation)
Varying the diffusion direction over the Varying the diffusion direction over the image can preserveenhance localsemilocal image can preserveenhance localsemilocal featuresfeatures
Intraregion smoothing instead of Intraregion smoothing instead of interregion smoothinginterregion smoothing
Edge Enhancing FilterEdge Enhancing Filter
Change the orientation and the Change the orientation and the strength of the diffusion at an edgestrength of the diffusion at an edge
Smooth along the edge boundarySmooth along the edge boundary Diffusion along the edgesDiffusion along the edges
ReduceStop diffusion across the ReduceStop diffusion across the edge (eg Orthogonal to the edge)edge (eg Orthogonal to the edge)
Intraregion smoothingIntraregion smoothing
Edge Enhancing FilterEdge Enhancing Filter
Create a positive semidefinite diffusion Create a positive semidefinite diffusion tensor Dtensor D
Orthonormal eigenvectors of D Orthonormal eigenvectors of D υυ1 1 υυ2 2
Eigenvalues control the diffusion strengthEigenvalues control the diffusion strength 1)1)
2)2)
Edge Enhancing FilterEdge Enhancing Filter
t = 1500t = 40 t = 400t = 0
t = 3000t = 250 t = 875t = 0
R-PM
EE
larr
Coherence Enhancing Coherence Enhancing FilterFilter
Interrupted lines flows in addition Interrupted lines flows in addition to noiseto noise
Need to process line or coherent Need to process line or coherent flow-like structures complete gapsflow-like structures complete gaps
Coherence Enhancing Coherence Enhancing FilterFilter
Orientations instead of directions Orientations instead of directions (invariance to sign changes in (invariance to sign changes in gradient)gradient)
Orientation feature Orientation feature Structure tensorStructure tensor
Create diffusion tensor as a function Create diffusion tensor as a function of structure tensor instead of the of structure tensor instead of the gradientgradient
Coherence Enhancing Coherence Enhancing FilterFilter
JJρρ positive semidefinite rarr orthonormal positive semidefinite rarr orthonormal eigenvectors eigenvectors υυ1 1 υυ2 2 eigenvalues eigenvalues μμ11ge ge μμ22 ge ge 00
υυ1 1 orientation of the highest grey value orientation of the highest grey value fluctuationfluctuation
υυ2 2 preferred local orientation (lowest preferred local orientation (lowest fluctuation)fluctuation)
((μμ1 1 - - μμ22))22 a measure of local coherence a measure of local coherence Suppress diffusion along Suppress diffusion along υυ1 1
Smooth along the direction of Smooth along the direction of υυ2 2
Coherence Enhancing Coherence Enhancing FilterFilter
Choose the same set of eigenvectors Choose the same set of eigenvectors for Dfor D
Set eigenvalues Set eigenvalues λλ11 λλ22 as as
((μμ1 1 - - μμ22 ) )22 large large λλ22 asymp 1 asymp 1
((μμ1 1 - - μμ22 ) )22 small small λλ22 asymp asymp άά
Coherence Enhancing Coherence Enhancing FilterFilter
Coherence Enhancing Coherence Enhancing FilterFilter
PMOriginal CE
ConclusionsConclusions
Fast and stable filtersFast and stable filters Highly flexible Highly flexible
Varying the parameters of the diffusion Varying the parameters of the diffusion tensor (orientation rotation strength)tensor (orientation rotation strength)
Incorporation of new features (texture Incorporation of new features (texture color etc)color etc)
Separate filters for each data entrySeparate filters for each data entry High performanceHigh performance
Thank youThank you
Questions Questions
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
High noise in gradient High noise in gradient Instability suppresion of important Instability suppresion of important
image featuresimage features Unnecessary suppression of diffusion Unnecessary suppression of diffusion
(noise remains)(noise remains) Regularization Smooth the gradient Regularization Smooth the gradient
before adaptationbefore adaptation
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
t = 1500t = 40 t = 400t = 0EEOriginal PM R-PM
R-PM
rarr
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
Canny PM
Original PM
Nonlinear Anisotropic Nonlinear Anisotropic FiltersFilters
So far adaptation is only done spatially So far adaptation is only done spatially andor temporally but diffusion direction is andor temporally but diffusion direction is fixedfixed
Adaptation of the diffusion orientationAdaptation of the diffusion orientation D is chosen as a matrix (scaling andor rotation)D is chosen as a matrix (scaling andor rotation)
Varying the diffusion direction over the Varying the diffusion direction over the image can preserveenhance localsemilocal image can preserveenhance localsemilocal featuresfeatures
Intraregion smoothing instead of Intraregion smoothing instead of interregion smoothinginterregion smoothing
Edge Enhancing FilterEdge Enhancing Filter
Change the orientation and the Change the orientation and the strength of the diffusion at an edgestrength of the diffusion at an edge
Smooth along the edge boundarySmooth along the edge boundary Diffusion along the edgesDiffusion along the edges
ReduceStop diffusion across the ReduceStop diffusion across the edge (eg Orthogonal to the edge)edge (eg Orthogonal to the edge)
Intraregion smoothingIntraregion smoothing
Edge Enhancing FilterEdge Enhancing Filter
Create a positive semidefinite diffusion Create a positive semidefinite diffusion tensor Dtensor D
Orthonormal eigenvectors of D Orthonormal eigenvectors of D υυ1 1 υυ2 2
Eigenvalues control the diffusion strengthEigenvalues control the diffusion strength 1)1)
2)2)
Edge Enhancing FilterEdge Enhancing Filter
t = 1500t = 40 t = 400t = 0
t = 3000t = 250 t = 875t = 0
R-PM
EE
larr
Coherence Enhancing Coherence Enhancing FilterFilter
Interrupted lines flows in addition Interrupted lines flows in addition to noiseto noise
Need to process line or coherent Need to process line or coherent flow-like structures complete gapsflow-like structures complete gaps
Coherence Enhancing Coherence Enhancing FilterFilter
Orientations instead of directions Orientations instead of directions (invariance to sign changes in (invariance to sign changes in gradient)gradient)
Orientation feature Orientation feature Structure tensorStructure tensor
Create diffusion tensor as a function Create diffusion tensor as a function of structure tensor instead of the of structure tensor instead of the gradientgradient
Coherence Enhancing Coherence Enhancing FilterFilter
JJρρ positive semidefinite rarr orthonormal positive semidefinite rarr orthonormal eigenvectors eigenvectors υυ1 1 υυ2 2 eigenvalues eigenvalues μμ11ge ge μμ22 ge ge 00
υυ1 1 orientation of the highest grey value orientation of the highest grey value fluctuationfluctuation
υυ2 2 preferred local orientation (lowest preferred local orientation (lowest fluctuation)fluctuation)
((μμ1 1 - - μμ22))22 a measure of local coherence a measure of local coherence Suppress diffusion along Suppress diffusion along υυ1 1
Smooth along the direction of Smooth along the direction of υυ2 2
Coherence Enhancing Coherence Enhancing FilterFilter
Choose the same set of eigenvectors Choose the same set of eigenvectors for Dfor D
Set eigenvalues Set eigenvalues λλ11 λλ22 as as
((μμ1 1 - - μμ22 ) )22 large large λλ22 asymp 1 asymp 1
((μμ1 1 - - μμ22 ) )22 small small λλ22 asymp asymp άά
Coherence Enhancing Coherence Enhancing FilterFilter
Coherence Enhancing Coherence Enhancing FilterFilter
PMOriginal CE
ConclusionsConclusions
Fast and stable filtersFast and stable filters Highly flexible Highly flexible
Varying the parameters of the diffusion Varying the parameters of the diffusion tensor (orientation rotation strength)tensor (orientation rotation strength)
Incorporation of new features (texture Incorporation of new features (texture color etc)color etc)
Separate filters for each data entrySeparate filters for each data entry High performanceHigh performance
Thank youThank you
Questions Questions
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
t = 1500t = 40 t = 400t = 0EEOriginal PM R-PM
R-PM
rarr
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
Canny PM
Original PM
Nonlinear Anisotropic Nonlinear Anisotropic FiltersFilters
So far adaptation is only done spatially So far adaptation is only done spatially andor temporally but diffusion direction is andor temporally but diffusion direction is fixedfixed
Adaptation of the diffusion orientationAdaptation of the diffusion orientation D is chosen as a matrix (scaling andor rotation)D is chosen as a matrix (scaling andor rotation)
Varying the diffusion direction over the Varying the diffusion direction over the image can preserveenhance localsemilocal image can preserveenhance localsemilocal featuresfeatures
Intraregion smoothing instead of Intraregion smoothing instead of interregion smoothinginterregion smoothing
Edge Enhancing FilterEdge Enhancing Filter
Change the orientation and the Change the orientation and the strength of the diffusion at an edgestrength of the diffusion at an edge
Smooth along the edge boundarySmooth along the edge boundary Diffusion along the edgesDiffusion along the edges
ReduceStop diffusion across the ReduceStop diffusion across the edge (eg Orthogonal to the edge)edge (eg Orthogonal to the edge)
Intraregion smoothingIntraregion smoothing
Edge Enhancing FilterEdge Enhancing Filter
Create a positive semidefinite diffusion Create a positive semidefinite diffusion tensor Dtensor D
Orthonormal eigenvectors of D Orthonormal eigenvectors of D υυ1 1 υυ2 2
Eigenvalues control the diffusion strengthEigenvalues control the diffusion strength 1)1)
2)2)
Edge Enhancing FilterEdge Enhancing Filter
t = 1500t = 40 t = 400t = 0
t = 3000t = 250 t = 875t = 0
R-PM
EE
larr
Coherence Enhancing Coherence Enhancing FilterFilter
Interrupted lines flows in addition Interrupted lines flows in addition to noiseto noise
Need to process line or coherent Need to process line or coherent flow-like structures complete gapsflow-like structures complete gaps
Coherence Enhancing Coherence Enhancing FilterFilter
Orientations instead of directions Orientations instead of directions (invariance to sign changes in (invariance to sign changes in gradient)gradient)
Orientation feature Orientation feature Structure tensorStructure tensor
Create diffusion tensor as a function Create diffusion tensor as a function of structure tensor instead of the of structure tensor instead of the gradientgradient
Coherence Enhancing Coherence Enhancing FilterFilter
JJρρ positive semidefinite rarr orthonormal positive semidefinite rarr orthonormal eigenvectors eigenvectors υυ1 1 υυ2 2 eigenvalues eigenvalues μμ11ge ge μμ22 ge ge 00
υυ1 1 orientation of the highest grey value orientation of the highest grey value fluctuationfluctuation
υυ2 2 preferred local orientation (lowest preferred local orientation (lowest fluctuation)fluctuation)
((μμ1 1 - - μμ22))22 a measure of local coherence a measure of local coherence Suppress diffusion along Suppress diffusion along υυ1 1
Smooth along the direction of Smooth along the direction of υυ2 2
Coherence Enhancing Coherence Enhancing FilterFilter
Choose the same set of eigenvectors Choose the same set of eigenvectors for Dfor D
Set eigenvalues Set eigenvalues λλ11 λλ22 as as
((μμ1 1 - - μμ22 ) )22 large large λλ22 asymp 1 asymp 1
((μμ1 1 - - μμ22 ) )22 small small λλ22 asymp asymp άά
Coherence Enhancing Coherence Enhancing FilterFilter
Coherence Enhancing Coherence Enhancing FilterFilter
PMOriginal CE
ConclusionsConclusions
Fast and stable filtersFast and stable filters Highly flexible Highly flexible
Varying the parameters of the diffusion Varying the parameters of the diffusion tensor (orientation rotation strength)tensor (orientation rotation strength)
Incorporation of new features (texture Incorporation of new features (texture color etc)color etc)
Separate filters for each data entrySeparate filters for each data entry High performanceHigh performance
Thank youThank you
Questions Questions
Nonlinear Isotropic Nonlinear Isotropic FiltersFilters
Canny PM
Original PM
Nonlinear Anisotropic Nonlinear Anisotropic FiltersFilters
So far adaptation is only done spatially So far adaptation is only done spatially andor temporally but diffusion direction is andor temporally but diffusion direction is fixedfixed
Adaptation of the diffusion orientationAdaptation of the diffusion orientation D is chosen as a matrix (scaling andor rotation)D is chosen as a matrix (scaling andor rotation)
Varying the diffusion direction over the Varying the diffusion direction over the image can preserveenhance localsemilocal image can preserveenhance localsemilocal featuresfeatures
Intraregion smoothing instead of Intraregion smoothing instead of interregion smoothinginterregion smoothing
Edge Enhancing FilterEdge Enhancing Filter
Change the orientation and the Change the orientation and the strength of the diffusion at an edgestrength of the diffusion at an edge
Smooth along the edge boundarySmooth along the edge boundary Diffusion along the edgesDiffusion along the edges
ReduceStop diffusion across the ReduceStop diffusion across the edge (eg Orthogonal to the edge)edge (eg Orthogonal to the edge)
Intraregion smoothingIntraregion smoothing
Edge Enhancing FilterEdge Enhancing Filter
Create a positive semidefinite diffusion Create a positive semidefinite diffusion tensor Dtensor D
Orthonormal eigenvectors of D Orthonormal eigenvectors of D υυ1 1 υυ2 2
Eigenvalues control the diffusion strengthEigenvalues control the diffusion strength 1)1)
2)2)
Edge Enhancing FilterEdge Enhancing Filter
t = 1500t = 40 t = 400t = 0
t = 3000t = 250 t = 875t = 0
R-PM
EE
larr
Coherence Enhancing Coherence Enhancing FilterFilter
Interrupted lines flows in addition Interrupted lines flows in addition to noiseto noise
Need to process line or coherent Need to process line or coherent flow-like structures complete gapsflow-like structures complete gaps
Coherence Enhancing Coherence Enhancing FilterFilter
Orientations instead of directions Orientations instead of directions (invariance to sign changes in (invariance to sign changes in gradient)gradient)
Orientation feature Orientation feature Structure tensorStructure tensor
Create diffusion tensor as a function Create diffusion tensor as a function of structure tensor instead of the of structure tensor instead of the gradientgradient
Coherence Enhancing Coherence Enhancing FilterFilter
JJρρ positive semidefinite rarr orthonormal positive semidefinite rarr orthonormal eigenvectors eigenvectors υυ1 1 υυ2 2 eigenvalues eigenvalues μμ11ge ge μμ22 ge ge 00
υυ1 1 orientation of the highest grey value orientation of the highest grey value fluctuationfluctuation
υυ2 2 preferred local orientation (lowest preferred local orientation (lowest fluctuation)fluctuation)
((μμ1 1 - - μμ22))22 a measure of local coherence a measure of local coherence Suppress diffusion along Suppress diffusion along υυ1 1
Smooth along the direction of Smooth along the direction of υυ2 2
Coherence Enhancing Coherence Enhancing FilterFilter
Choose the same set of eigenvectors Choose the same set of eigenvectors for Dfor D
Set eigenvalues Set eigenvalues λλ11 λλ22 as as
((μμ1 1 - - μμ22 ) )22 large large λλ22 asymp 1 asymp 1
((μμ1 1 - - μμ22 ) )22 small small λλ22 asymp asymp άά
Coherence Enhancing Coherence Enhancing FilterFilter
Coherence Enhancing Coherence Enhancing FilterFilter
PMOriginal CE
ConclusionsConclusions
Fast and stable filtersFast and stable filters Highly flexible Highly flexible
Varying the parameters of the diffusion Varying the parameters of the diffusion tensor (orientation rotation strength)tensor (orientation rotation strength)
Incorporation of new features (texture Incorporation of new features (texture color etc)color etc)
Separate filters for each data entrySeparate filters for each data entry High performanceHigh performance
Thank youThank you
Questions Questions
Nonlinear Anisotropic Nonlinear Anisotropic FiltersFilters
So far adaptation is only done spatially So far adaptation is only done spatially andor temporally but diffusion direction is andor temporally but diffusion direction is fixedfixed
Adaptation of the diffusion orientationAdaptation of the diffusion orientation D is chosen as a matrix (scaling andor rotation)D is chosen as a matrix (scaling andor rotation)
Varying the diffusion direction over the Varying the diffusion direction over the image can preserveenhance localsemilocal image can preserveenhance localsemilocal featuresfeatures
Intraregion smoothing instead of Intraregion smoothing instead of interregion smoothinginterregion smoothing
Edge Enhancing FilterEdge Enhancing Filter
Change the orientation and the Change the orientation and the strength of the diffusion at an edgestrength of the diffusion at an edge
Smooth along the edge boundarySmooth along the edge boundary Diffusion along the edgesDiffusion along the edges
ReduceStop diffusion across the ReduceStop diffusion across the edge (eg Orthogonal to the edge)edge (eg Orthogonal to the edge)
Intraregion smoothingIntraregion smoothing
Edge Enhancing FilterEdge Enhancing Filter
Create a positive semidefinite diffusion Create a positive semidefinite diffusion tensor Dtensor D
Orthonormal eigenvectors of D Orthonormal eigenvectors of D υυ1 1 υυ2 2
Eigenvalues control the diffusion strengthEigenvalues control the diffusion strength 1)1)
2)2)
Edge Enhancing FilterEdge Enhancing Filter
t = 1500t = 40 t = 400t = 0
t = 3000t = 250 t = 875t = 0
R-PM
EE
larr
Coherence Enhancing Coherence Enhancing FilterFilter
Interrupted lines flows in addition Interrupted lines flows in addition to noiseto noise
Need to process line or coherent Need to process line or coherent flow-like structures complete gapsflow-like structures complete gaps
Coherence Enhancing Coherence Enhancing FilterFilter
Orientations instead of directions Orientations instead of directions (invariance to sign changes in (invariance to sign changes in gradient)gradient)
Orientation feature Orientation feature Structure tensorStructure tensor
Create diffusion tensor as a function Create diffusion tensor as a function of structure tensor instead of the of structure tensor instead of the gradientgradient
Coherence Enhancing Coherence Enhancing FilterFilter
JJρρ positive semidefinite rarr orthonormal positive semidefinite rarr orthonormal eigenvectors eigenvectors υυ1 1 υυ2 2 eigenvalues eigenvalues μμ11ge ge μμ22 ge ge 00
υυ1 1 orientation of the highest grey value orientation of the highest grey value fluctuationfluctuation
υυ2 2 preferred local orientation (lowest preferred local orientation (lowest fluctuation)fluctuation)
((μμ1 1 - - μμ22))22 a measure of local coherence a measure of local coherence Suppress diffusion along Suppress diffusion along υυ1 1
Smooth along the direction of Smooth along the direction of υυ2 2
Coherence Enhancing Coherence Enhancing FilterFilter
Choose the same set of eigenvectors Choose the same set of eigenvectors for Dfor D
Set eigenvalues Set eigenvalues λλ11 λλ22 as as
((μμ1 1 - - μμ22 ) )22 large large λλ22 asymp 1 asymp 1
((μμ1 1 - - μμ22 ) )22 small small λλ22 asymp asymp άά
Coherence Enhancing Coherence Enhancing FilterFilter
Coherence Enhancing Coherence Enhancing FilterFilter
PMOriginal CE
ConclusionsConclusions
Fast and stable filtersFast and stable filters Highly flexible Highly flexible
Varying the parameters of the diffusion Varying the parameters of the diffusion tensor (orientation rotation strength)tensor (orientation rotation strength)
Incorporation of new features (texture Incorporation of new features (texture color etc)color etc)
Separate filters for each data entrySeparate filters for each data entry High performanceHigh performance
Thank youThank you
Questions Questions
Edge Enhancing FilterEdge Enhancing Filter
Change the orientation and the Change the orientation and the strength of the diffusion at an edgestrength of the diffusion at an edge
Smooth along the edge boundarySmooth along the edge boundary Diffusion along the edgesDiffusion along the edges
ReduceStop diffusion across the ReduceStop diffusion across the edge (eg Orthogonal to the edge)edge (eg Orthogonal to the edge)
Intraregion smoothingIntraregion smoothing
Edge Enhancing FilterEdge Enhancing Filter
Create a positive semidefinite diffusion Create a positive semidefinite diffusion tensor Dtensor D
Orthonormal eigenvectors of D Orthonormal eigenvectors of D υυ1 1 υυ2 2
Eigenvalues control the diffusion strengthEigenvalues control the diffusion strength 1)1)
2)2)
Edge Enhancing FilterEdge Enhancing Filter
t = 1500t = 40 t = 400t = 0
t = 3000t = 250 t = 875t = 0
R-PM
EE
larr
Coherence Enhancing Coherence Enhancing FilterFilter
Interrupted lines flows in addition Interrupted lines flows in addition to noiseto noise
Need to process line or coherent Need to process line or coherent flow-like structures complete gapsflow-like structures complete gaps
Coherence Enhancing Coherence Enhancing FilterFilter
Orientations instead of directions Orientations instead of directions (invariance to sign changes in (invariance to sign changes in gradient)gradient)
Orientation feature Orientation feature Structure tensorStructure tensor
Create diffusion tensor as a function Create diffusion tensor as a function of structure tensor instead of the of structure tensor instead of the gradientgradient
Coherence Enhancing Coherence Enhancing FilterFilter
JJρρ positive semidefinite rarr orthonormal positive semidefinite rarr orthonormal eigenvectors eigenvectors υυ1 1 υυ2 2 eigenvalues eigenvalues μμ11ge ge μμ22 ge ge 00
υυ1 1 orientation of the highest grey value orientation of the highest grey value fluctuationfluctuation
υυ2 2 preferred local orientation (lowest preferred local orientation (lowest fluctuation)fluctuation)
((μμ1 1 - - μμ22))22 a measure of local coherence a measure of local coherence Suppress diffusion along Suppress diffusion along υυ1 1
Smooth along the direction of Smooth along the direction of υυ2 2
Coherence Enhancing Coherence Enhancing FilterFilter
Choose the same set of eigenvectors Choose the same set of eigenvectors for Dfor D
Set eigenvalues Set eigenvalues λλ11 λλ22 as as
((μμ1 1 - - μμ22 ) )22 large large λλ22 asymp 1 asymp 1
((μμ1 1 - - μμ22 ) )22 small small λλ22 asymp asymp άά
Coherence Enhancing Coherence Enhancing FilterFilter
Coherence Enhancing Coherence Enhancing FilterFilter
PMOriginal CE
ConclusionsConclusions
Fast and stable filtersFast and stable filters Highly flexible Highly flexible
Varying the parameters of the diffusion Varying the parameters of the diffusion tensor (orientation rotation strength)tensor (orientation rotation strength)
Incorporation of new features (texture Incorporation of new features (texture color etc)color etc)
Separate filters for each data entrySeparate filters for each data entry High performanceHigh performance
Thank youThank you
Questions Questions
Edge Enhancing FilterEdge Enhancing Filter
Create a positive semidefinite diffusion Create a positive semidefinite diffusion tensor Dtensor D
Orthonormal eigenvectors of D Orthonormal eigenvectors of D υυ1 1 υυ2 2
Eigenvalues control the diffusion strengthEigenvalues control the diffusion strength 1)1)
2)2)
Edge Enhancing FilterEdge Enhancing Filter
t = 1500t = 40 t = 400t = 0
t = 3000t = 250 t = 875t = 0
R-PM
EE
larr
Coherence Enhancing Coherence Enhancing FilterFilter
Interrupted lines flows in addition Interrupted lines flows in addition to noiseto noise
Need to process line or coherent Need to process line or coherent flow-like structures complete gapsflow-like structures complete gaps
Coherence Enhancing Coherence Enhancing FilterFilter
Orientations instead of directions Orientations instead of directions (invariance to sign changes in (invariance to sign changes in gradient)gradient)
Orientation feature Orientation feature Structure tensorStructure tensor
Create diffusion tensor as a function Create diffusion tensor as a function of structure tensor instead of the of structure tensor instead of the gradientgradient
Coherence Enhancing Coherence Enhancing FilterFilter
JJρρ positive semidefinite rarr orthonormal positive semidefinite rarr orthonormal eigenvectors eigenvectors υυ1 1 υυ2 2 eigenvalues eigenvalues μμ11ge ge μμ22 ge ge 00
υυ1 1 orientation of the highest grey value orientation of the highest grey value fluctuationfluctuation
υυ2 2 preferred local orientation (lowest preferred local orientation (lowest fluctuation)fluctuation)
((μμ1 1 - - μμ22))22 a measure of local coherence a measure of local coherence Suppress diffusion along Suppress diffusion along υυ1 1
Smooth along the direction of Smooth along the direction of υυ2 2
Coherence Enhancing Coherence Enhancing FilterFilter
Choose the same set of eigenvectors Choose the same set of eigenvectors for Dfor D
Set eigenvalues Set eigenvalues λλ11 λλ22 as as
((μμ1 1 - - μμ22 ) )22 large large λλ22 asymp 1 asymp 1
((μμ1 1 - - μμ22 ) )22 small small λλ22 asymp asymp άά
Coherence Enhancing Coherence Enhancing FilterFilter
Coherence Enhancing Coherence Enhancing FilterFilter
PMOriginal CE
ConclusionsConclusions
Fast and stable filtersFast and stable filters Highly flexible Highly flexible
Varying the parameters of the diffusion Varying the parameters of the diffusion tensor (orientation rotation strength)tensor (orientation rotation strength)
Incorporation of new features (texture Incorporation of new features (texture color etc)color etc)
Separate filters for each data entrySeparate filters for each data entry High performanceHigh performance
Thank youThank you
Questions Questions
Edge Enhancing FilterEdge Enhancing Filter
t = 1500t = 40 t = 400t = 0
t = 3000t = 250 t = 875t = 0
R-PM
EE
larr
Coherence Enhancing Coherence Enhancing FilterFilter
Interrupted lines flows in addition Interrupted lines flows in addition to noiseto noise
Need to process line or coherent Need to process line or coherent flow-like structures complete gapsflow-like structures complete gaps
Coherence Enhancing Coherence Enhancing FilterFilter
Orientations instead of directions Orientations instead of directions (invariance to sign changes in (invariance to sign changes in gradient)gradient)
Orientation feature Orientation feature Structure tensorStructure tensor
Create diffusion tensor as a function Create diffusion tensor as a function of structure tensor instead of the of structure tensor instead of the gradientgradient
Coherence Enhancing Coherence Enhancing FilterFilter
JJρρ positive semidefinite rarr orthonormal positive semidefinite rarr orthonormal eigenvectors eigenvectors υυ1 1 υυ2 2 eigenvalues eigenvalues μμ11ge ge μμ22 ge ge 00
υυ1 1 orientation of the highest grey value orientation of the highest grey value fluctuationfluctuation
υυ2 2 preferred local orientation (lowest preferred local orientation (lowest fluctuation)fluctuation)
((μμ1 1 - - μμ22))22 a measure of local coherence a measure of local coherence Suppress diffusion along Suppress diffusion along υυ1 1
Smooth along the direction of Smooth along the direction of υυ2 2
Coherence Enhancing Coherence Enhancing FilterFilter
Choose the same set of eigenvectors Choose the same set of eigenvectors for Dfor D
Set eigenvalues Set eigenvalues λλ11 λλ22 as as
((μμ1 1 - - μμ22 ) )22 large large λλ22 asymp 1 asymp 1
((μμ1 1 - - μμ22 ) )22 small small λλ22 asymp asymp άά
Coherence Enhancing Coherence Enhancing FilterFilter
Coherence Enhancing Coherence Enhancing FilterFilter
PMOriginal CE
ConclusionsConclusions
Fast and stable filtersFast and stable filters Highly flexible Highly flexible
Varying the parameters of the diffusion Varying the parameters of the diffusion tensor (orientation rotation strength)tensor (orientation rotation strength)
Incorporation of new features (texture Incorporation of new features (texture color etc)color etc)
Separate filters for each data entrySeparate filters for each data entry High performanceHigh performance
Thank youThank you
Questions Questions
Coherence Enhancing Coherence Enhancing FilterFilter
Interrupted lines flows in addition Interrupted lines flows in addition to noiseto noise
Need to process line or coherent Need to process line or coherent flow-like structures complete gapsflow-like structures complete gaps
Coherence Enhancing Coherence Enhancing FilterFilter
Orientations instead of directions Orientations instead of directions (invariance to sign changes in (invariance to sign changes in gradient)gradient)
Orientation feature Orientation feature Structure tensorStructure tensor
Create diffusion tensor as a function Create diffusion tensor as a function of structure tensor instead of the of structure tensor instead of the gradientgradient
Coherence Enhancing Coherence Enhancing FilterFilter
JJρρ positive semidefinite rarr orthonormal positive semidefinite rarr orthonormal eigenvectors eigenvectors υυ1 1 υυ2 2 eigenvalues eigenvalues μμ11ge ge μμ22 ge ge 00
υυ1 1 orientation of the highest grey value orientation of the highest grey value fluctuationfluctuation
υυ2 2 preferred local orientation (lowest preferred local orientation (lowest fluctuation)fluctuation)
((μμ1 1 - - μμ22))22 a measure of local coherence a measure of local coherence Suppress diffusion along Suppress diffusion along υυ1 1
Smooth along the direction of Smooth along the direction of υυ2 2
Coherence Enhancing Coherence Enhancing FilterFilter
Choose the same set of eigenvectors Choose the same set of eigenvectors for Dfor D
Set eigenvalues Set eigenvalues λλ11 λλ22 as as
((μμ1 1 - - μμ22 ) )22 large large λλ22 asymp 1 asymp 1
((μμ1 1 - - μμ22 ) )22 small small λλ22 asymp asymp άά
Coherence Enhancing Coherence Enhancing FilterFilter
Coherence Enhancing Coherence Enhancing FilterFilter
PMOriginal CE
ConclusionsConclusions
Fast and stable filtersFast and stable filters Highly flexible Highly flexible
Varying the parameters of the diffusion Varying the parameters of the diffusion tensor (orientation rotation strength)tensor (orientation rotation strength)
Incorporation of new features (texture Incorporation of new features (texture color etc)color etc)
Separate filters for each data entrySeparate filters for each data entry High performanceHigh performance
Thank youThank you
Questions Questions
Coherence Enhancing Coherence Enhancing FilterFilter
Orientations instead of directions Orientations instead of directions (invariance to sign changes in (invariance to sign changes in gradient)gradient)
Orientation feature Orientation feature Structure tensorStructure tensor
Create diffusion tensor as a function Create diffusion tensor as a function of structure tensor instead of the of structure tensor instead of the gradientgradient
Coherence Enhancing Coherence Enhancing FilterFilter
JJρρ positive semidefinite rarr orthonormal positive semidefinite rarr orthonormal eigenvectors eigenvectors υυ1 1 υυ2 2 eigenvalues eigenvalues μμ11ge ge μμ22 ge ge 00
υυ1 1 orientation of the highest grey value orientation of the highest grey value fluctuationfluctuation
υυ2 2 preferred local orientation (lowest preferred local orientation (lowest fluctuation)fluctuation)
((μμ1 1 - - μμ22))22 a measure of local coherence a measure of local coherence Suppress diffusion along Suppress diffusion along υυ1 1
Smooth along the direction of Smooth along the direction of υυ2 2
Coherence Enhancing Coherence Enhancing FilterFilter
Choose the same set of eigenvectors Choose the same set of eigenvectors for Dfor D
Set eigenvalues Set eigenvalues λλ11 λλ22 as as
((μμ1 1 - - μμ22 ) )22 large large λλ22 asymp 1 asymp 1
((μμ1 1 - - μμ22 ) )22 small small λλ22 asymp asymp άά
Coherence Enhancing Coherence Enhancing FilterFilter
Coherence Enhancing Coherence Enhancing FilterFilter
PMOriginal CE
ConclusionsConclusions
Fast and stable filtersFast and stable filters Highly flexible Highly flexible
Varying the parameters of the diffusion Varying the parameters of the diffusion tensor (orientation rotation strength)tensor (orientation rotation strength)
Incorporation of new features (texture Incorporation of new features (texture color etc)color etc)
Separate filters for each data entrySeparate filters for each data entry High performanceHigh performance
Thank youThank you
Questions Questions
Coherence Enhancing Coherence Enhancing FilterFilter
JJρρ positive semidefinite rarr orthonormal positive semidefinite rarr orthonormal eigenvectors eigenvectors υυ1 1 υυ2 2 eigenvalues eigenvalues μμ11ge ge μμ22 ge ge 00
υυ1 1 orientation of the highest grey value orientation of the highest grey value fluctuationfluctuation
υυ2 2 preferred local orientation (lowest preferred local orientation (lowest fluctuation)fluctuation)
((μμ1 1 - - μμ22))22 a measure of local coherence a measure of local coherence Suppress diffusion along Suppress diffusion along υυ1 1
Smooth along the direction of Smooth along the direction of υυ2 2
Coherence Enhancing Coherence Enhancing FilterFilter
Choose the same set of eigenvectors Choose the same set of eigenvectors for Dfor D
Set eigenvalues Set eigenvalues λλ11 λλ22 as as
((μμ1 1 - - μμ22 ) )22 large large λλ22 asymp 1 asymp 1
((μμ1 1 - - μμ22 ) )22 small small λλ22 asymp asymp άά
Coherence Enhancing Coherence Enhancing FilterFilter
Coherence Enhancing Coherence Enhancing FilterFilter
PMOriginal CE
ConclusionsConclusions
Fast and stable filtersFast and stable filters Highly flexible Highly flexible
Varying the parameters of the diffusion Varying the parameters of the diffusion tensor (orientation rotation strength)tensor (orientation rotation strength)
Incorporation of new features (texture Incorporation of new features (texture color etc)color etc)
Separate filters for each data entrySeparate filters for each data entry High performanceHigh performance
Thank youThank you
Questions Questions
Coherence Enhancing Coherence Enhancing FilterFilter
Choose the same set of eigenvectors Choose the same set of eigenvectors for Dfor D
Set eigenvalues Set eigenvalues λλ11 λλ22 as as
((μμ1 1 - - μμ22 ) )22 large large λλ22 asymp 1 asymp 1
((μμ1 1 - - μμ22 ) )22 small small λλ22 asymp asymp άά
Coherence Enhancing Coherence Enhancing FilterFilter
Coherence Enhancing Coherence Enhancing FilterFilter
PMOriginal CE
ConclusionsConclusions
Fast and stable filtersFast and stable filters Highly flexible Highly flexible
Varying the parameters of the diffusion Varying the parameters of the diffusion tensor (orientation rotation strength)tensor (orientation rotation strength)
Incorporation of new features (texture Incorporation of new features (texture color etc)color etc)
Separate filters for each data entrySeparate filters for each data entry High performanceHigh performance
Thank youThank you
Questions Questions
Coherence Enhancing Coherence Enhancing FilterFilter
Coherence Enhancing Coherence Enhancing FilterFilter
PMOriginal CE
ConclusionsConclusions
Fast and stable filtersFast and stable filters Highly flexible Highly flexible
Varying the parameters of the diffusion Varying the parameters of the diffusion tensor (orientation rotation strength)tensor (orientation rotation strength)
Incorporation of new features (texture Incorporation of new features (texture color etc)color etc)
Separate filters for each data entrySeparate filters for each data entry High performanceHigh performance
Thank youThank you
Questions Questions
Coherence Enhancing Coherence Enhancing FilterFilter
PMOriginal CE
ConclusionsConclusions
Fast and stable filtersFast and stable filters Highly flexible Highly flexible
Varying the parameters of the diffusion Varying the parameters of the diffusion tensor (orientation rotation strength)tensor (orientation rotation strength)
Incorporation of new features (texture Incorporation of new features (texture color etc)color etc)
Separate filters for each data entrySeparate filters for each data entry High performanceHigh performance
Thank youThank you
Questions Questions
ConclusionsConclusions
Fast and stable filtersFast and stable filters Highly flexible Highly flexible
Varying the parameters of the diffusion Varying the parameters of the diffusion tensor (orientation rotation strength)tensor (orientation rotation strength)
Incorporation of new features (texture Incorporation of new features (texture color etc)color etc)
Separate filters for each data entrySeparate filters for each data entry High performanceHigh performance
Thank youThank you
Questions Questions
Thank youThank you
Questions Questions