R. DOSIL, X. M. PARDO, A. MOSQUERA, D. CABELLO Grupo de Visión Artificial Departamento de Electrónica e Computación Universidade de Santiago de Compostela Curvature dependent diffusion for feature detection in 3D medical images

 Objectives  Calculus of gradient and curvature  Detection of boundaries and corners  Applications  Energy minimization techniques: definition of image potentials  Matching techniques: detection of characteristic features Feature detection in medical images

 Problems: Noise, textures,...  Erroneous calculus of gradient and curvature  Failure in boundary and corner detection  Typical solution: gaussian smoothing  Alteration of gradient and curvature values  Dislocation of boundaries and rounding of corners  Proposal: use of adaptive filtering based on diffusion processes Feature detection in medical images

I. Introduction II. Feature enhancement with diffusion  Tangential diffusion  Construction of the diffusion tensor  Threshold parameter III. Corner preserving diffusion  Previous works  Curvature dependent diffusivity IV. Results Outline

 Diffusion equation with Introduction

 Linear  C is a scalar constant  It blurs boundaries as gaussian filtering does  Nonlinear (Perona & Malik, 1990)  C depends on local image properties  If C is a decreasing function of ||  u|| Boundaries are not blurred Boundaries are not blurred  Noise is preserved at surfaces  Nonlinear anisotropic (Weickert, 1994)  C is a tensor  Flux vector is not parallel to gradient  Different diffusivity values i for different directions e i Introduction

 Tangential diffusion:  Diffusivity is reduced in the normal dir. at each point Boundaries are not blurred Boundaries are not blurred  Diffusion is maintained in the tangent plane Reduces noise by flattening surfaces Reduces noise by flattening surfaces  It rounds corners Feature enhancement with diffusion

 Construction of C  e i are the eigenvectors of the hessian matrix  i are their correspondent desired eigenvalues Eigenvectors e i Eigenvalues i Normal g (||  u||,  ) Max. curvature tangent 1 Min. curvature tangent 1 Feature enhancement with diffusion

 Threshold parameter   Represents the gradient threshold at which flux stops growing  Automatic estimation of  using robust statistics (Black, 1998) Feature enhancement with diffusion

 Previous work by Krissian, 1996  Diffusion in the max. curvature dir. is removed Eigenvectors e i Eigenvalues i Normal g (||  u||,  ) Max. curvature tangent 0 Min. curvature tangent 1 It avoids corner rounding It avoids corner rounding  Noise reduction is lower Corner preserving diffusion

 Curvature dependent diffusivity  Diffusion in the max. curvature direction depends on a corner measure Eigenvectors e i Eigenvalues i Normal g (||  u||,  ) Max. curvature tangent g (corner,  ) Min. curvature tangent 1  Diffusion in the max. curvature dir. is reduced on corners  Remainder surface regions are flattened in the tangent plane Corner preserving diffusion

II. II. +  Filtering with four different diffusion schemes EigenvectorsABCD Normal1 g (||  u||,  ) Max. curvature tang. 110 g (corner,  ) Min. curvature tang  Construction of a synthetic image with gaussian noise of variance  = 50 Results: Comparison of different schemes

ABCD smoothed gradient max. curvature surface

 Test with synthetic image with gaussian noise of variance  = 50 Original Max. curvature Gradient Smoothed gaussian anisotropic Surfaces Results: Anisotropic filter Vs Gaussian filter

 Surface points location Error in location of corners Error in sphere radius estimation Results: Anisotropic filter Vs Gaussian filter

 Curvature estimation Error in curvature estimation using gaussian filter Error in curvature estimation using anisotropic filter Results: Anisotropic filter Vs Gaussian filter

 MRI image of aorta Results: Medical image example Original Smoothed with gaussian filter Smoothed with anisotropic diffusion

 MRI image of aorta Gradient modulus Max. Curvature Gaussian filter Anisotropic filter Results: Medical image example

 Contributions  Use of diffusion techniques to improve gradient and curvature measures in 3D medical imaging: –definition of image potentials –feature detection  Design of corner preserving diffusion filter  Automatic estimation of filter parameters  Future work  Introduction of adaptive estimation of threshold parameters Conclusions

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