R. DOSIL, X. M. PARDO, A. MOSQUERA, D. CABELLOR. DOSIL, X. M. PARDO, A. MOSQUERA, D. CABELLO

Grupo de Visión ArtificialGrupo de Visión ArtificialDepartamento de Electrónica e Departamento de Electrónica e

ComputaciónComputaciónUniversidade de Santiago de CompostelaUniversidade de Santiago de Compostela

Curvature dependent diffusion forCurvature dependent diffusion forfeature detection in 3D medical feature detection in 3D medical

imagesimages

ObjectivesObjectives Calculus of gradient and curvatureCalculus of gradient and curvature Detection of boundaries and cornersDetection of boundaries and corners

ApplicationsApplications EEnergy minimization techniquesnergy minimization techniques: d: definition of image efinition of image

potentialspotentials Matching techniques: detection of characteristic Matching techniques: detection of characteristic

featuresfeatures

Feature detection in medical Feature detection in medical imagesimages

Problems: Noise, textures,Problems: Noise, textures, ...... Erroneous calculus of gradient and curvatureErroneous calculus of gradient and curvature FailureFailure in boundary and corner detection in boundary and corner detection

Typical solution: gaussian smoothingTypical solution: gaussian smoothing Alteration of gradient and curvature valuesAlteration of gradient and curvature values Dislocation of boundaries and Dislocation of boundaries and rounding of rounding of cornerscorners

Proposal: use of adaptive filtering based on Proposal: use of adaptive filtering based on diffusion processesdiffusion processes

Feature detection in medical Feature detection in medical imagesimages

I.I. IntroductionIntroduction

II.II. Feature enhancement with diffusionFeature enhancement with diffusion Tangential diffusionTangential diffusion Construction of the diffusion tensorConstruction of the diffusion tensor Threshold parameterThreshold parameter

III.III. Corner preserving diffusionCorner preserving diffusion Previous woPrevious worrksks Curvature dependent diffusivityCurvature dependent diffusivity

IV.IV. ResultsResults

OutlineOutline

Diffusion equationDiffusion equation

productinner

normalouter

frontier its and domain image

timeat of versionfiltered

image original

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IntroductionIntroduction

LinearLinear CC is a scalar constant is a scalar constant It blurs boundaries as gaussian filteringIt blurs boundaries as gaussian filtering does does

Nonlinear (Perona & Malik, 1990)Nonlinear (Perona & Malik, 1990) C C depends on local image propertiesdepends on local image properties If If CC is a decreasing function of is a decreasing function of ||||u||u||

Boundaries are not blurredBoundaries are not blurred NNoise is preservedoise is preserved at surfaces at surfaces

Nonlinear anisotropic (Weickert, 1994)Nonlinear anisotropic (Weickert, 1994) CC is a tensor is a tensor Flux vector is not parallel to Flux vector is not parallel to

gradientgradient Different diffusivity values Different diffusivity values ii for different directions for different directions

ee ii

IntroductionIntroduction

Tangential diffusion:Tangential diffusion:

Diffusivity is reduced in Diffusivity is reduced in the normal dir. at each the normal dir. at each pointpoint

Boundaries are not blurredBoundaries are not blurred

Diffusion is maintained in Diffusion is maintained in the tangent planethe tangent plane

Reduces noise by Reduces noise by flatflattteningening surfaces surfaces

It rounds cornersIt rounds corners

Feature enhancement with Feature enhancement with diffusiondiffusion

Construction of Construction of CC

ee ii are the eigenvectors of the hessian are the eigenvectors of the hessian matrix matrix

ii are their correspondent desired are their correspondent desired eigenvalueseigenvaluesEigenvectorsEigenvectors ee ii EEigenvaluesigenvalues ii

NormalNormal g g (||(||uu||||, , ))

Max. curvature tangentMax. curvature tangent 11

Min. curvature tangentMin. curvature tangent 11

uuug tanh

Feature enhancement with Feature enhancement with diffusiondiffusion

TeeediageeeC 321321321 ||,,||

Threshold parameter Threshold parameter Represents the gradient threshold at which flux stops Represents the gradient threshold at which flux stops

growinggrowing Automatic estimation of Automatic estimation of using robust statistics using robust statistics

(Black, 1998)(Black, 1998) 6745.06745.0 III II medianmedianMAD

Feature enhancement with Feature enhancement with diffusiondiffusion

Previous work by Krissian, 1996Previous work by Krissian, 1996 Diffusion in the max. curvature dir. is removedDiffusion in the max. curvature dir. is removed

EigenvectorsEigenvectors ee ii EEigenvaluesigenvalues ii

NormalNormal g g (||(||uu||||, , ))

Max. curvature tangentMax. curvature tangent 00

Min. curvature tangentMin. curvature tangent 11

It avoids corner roundingIt avoids corner rounding Noise reduction is lowerNoise reduction is lower

Corner preserving diffusionCorner preserving diffusion

Curvature dependent diffusivityCurvature dependent diffusivity Diffusion in the max. curvature direction depends on Diffusion in the max. curvature direction depends on

a corner measurea corner measure

EigenvectorsEigenvectors ee ii EEigenvaluesigenvalues ii

NormalNormal g g (||(||uu||||, , ))

Max. curvature tangentMax. curvature tangent g g ((cornercorner, , ))

Min. curvature tangentMin. curvature tangent 11

maxkucorner

Diffusion in the max. curvature dir. is reduced on Diffusion in the max. curvature dir. is reduced on cornerscorners

Remainder surface regions are Remainder surface regions are flattenedflattened in the tangent in the tangent planeplane

Corner preserving diffusionCorner preserving diffusion

II. +

Filtering with four different diffusion schemesFiltering with four different diffusion schemes

EigenvectorsEigenvectors AA BB CC DD

NormalNormal 11 g g (||(||uu||||, , )) g g (||(||uu||||, , )) g g (||(||uu||||, , ))

Max. curvatureMax. curvature tang.tang.

11 11 00 g g ((cornercorner, , ))

Min. curvatureMin. curvature tang.tang.

11 11 11 11

Construction of a synthetic image with Construction of a synthetic image with gaussian noise of variance gaussian noise of variance = 50 = 50

Results:Results:Comparison of different Comparison of different

schemesschemes

AA BB CC DD

smoothedsmoothed

gradientgradient

max.max.curvaturecurvature

surfacesurface

Test with sTest with synthetic imageynthetic image with with gaussian noise gaussian noise ofof variance variance = 50 = 50

Original Max. curvatureMax. curvatureGradientGradientSmoothedSmoothed

gaussiangaussian

anisotropicanisotropic

Surfaces

Results:Results:Anisotropic filter Vs Gaussian Anisotropic filter Vs Gaussian

filterfilter

Surface points locationSurface points location

Error in location of cornersError in location of corners Error in sphere radius estimationError in sphere radius estimation

Results:Results:Anisotropic filter Vs Gaussian Anisotropic filter Vs Gaussian

filterfilter

Curvature estimationCurvature estimation

Error in curvature estimation Error in curvature estimation using gaussian filterusing gaussian filter

Error in curvature estimation Error in curvature estimation using anisotropic filterusing anisotropic filter

Results:Results:Anisotropic filter Vs Gaussian Anisotropic filter Vs Gaussian

filterfilter

MRI image of aortaMRI image of aorta

Results:Results:Medical image exampleMedical image example

Original Smoothed withSmoothed withgaussian filter gaussian filter

Smoothed with anisotropic diffusion

MRI image of aorta

Gradient modulusGradient modulus Max. CurvatureMax. Curvature

Gaussian filterGaussian filter

Anisotropic filterAnisotropic filter

Results: Results: Medical image exampleMedical image example

ContributionsContributions Use of diffusion techniques to improve gradient and Use of diffusion techniques to improve gradient and

curvature measures in 3D medical imaging:curvature measures in 3D medical imaging:– definition of image potentials definition of image potentials – feature detectionfeature detection

Design of corner preserving diffusion filterDesign of corner preserving diffusion filter Automatic estimation of filter parametersAutomatic estimation of filter parameters

Future workFuture work Introduction of adaptive estimation of threshold Introduction of adaptive estimation of threshold

parametersparameters

ConclusionsConclusions

EndEnd