(Adaptive) enhancement of HARDI images Eric Creusen Supervisor: Remco Duits

Diffusion of Water / BIOMIM PAGE 2 Diffusion is dependent on orientation *Graphic borrowed from Thomas Schultz

Diffusion of water: Modelling BIOMIM PAGE 3

Domain Diffusion weighted images are functions on position and orientation Both HARDI and DTI! / Communicatie Expertise Centrum PAGE 4mei 2008

Embed in group Embed position and orientation into translations and rotation group / BIOMIM PAGE 5

Parametrization of Sphere / Communicatie Expertise Centrum PAGE 6mei 2008 Euler Angles

Embedding problem SE(3) has 3 angles(α,β,γ) Positions and orientation only 2 (β,γ) Orientations are independent of α Introduce equivalence classes on SE(3) Processes should preserve α-invariance! / BIOMIM PAGE 7

Convection-diffusion processes See function as distribution of oriented particles Particles can translate and rotate / BIOMIM PAGE 8

Rotated coordinate system / Communicatie Expertise Centrum PAGE 9mei 2008 n A3A3 A1A1 A2A2 n z y x R n (α)

Now that we have a basis: derivatives! Derivatives can be calculated Directions dependant on orientation n 3 spatial directions 2 angular directions (+1 dummy direction) For n=ez / Communicatie Expertise Centrum PAGE 10mei 2008 In the same direction as normal Perpendicular (make random choice here) Angular derivatives Should be zero

Smoothing processes With derivatives, we can simulate diffusion processes by simulating PDE’s General diffusion processes are given by the PDE: (with parameter D and simulation time t) / Communicatie Expertise Centrum PAGE 11mei 2008

Some simple examples Isotropic Spatial Gaussian smoothing: Angular diffusion: / Communicatie Expertise Centrum PAGE 12mei 2008 Not interesting, only spatial OR only angular diffusion

More examples / Communicatie Expertise Centrum PAGE 13mei 2008 Convection term Angular diffusionSpatial diffusion “Contour enhancement” “Contour completion”

Simulation of PDEs Convolutions with kernels Finite difference methods / Communicatie Expertise Centrum PAGE 14mei 2008

Creating convolution kernels Get approximations with complicated math……. OR Start with oriented delta peak Simulate PDE with finite difference methods / Communicatie Expertise Centrum PAGE 15mei 2008

Finite difference methods: enhancement Directly use finite difference methods to simulate PDE / Communicatie Expertise Centrum PAGE 16mei 2008 enhancement

Finite difference: contour completion / Communicatie Expertise Centrum PAGE 17mei 2008 completion

Adaptive filtering Make smoothing dependant on data Finite difference methods neccesary / Communicatie Expertise Centrum PAGE 18mei 2008

Perona-Malik type diffusion Goal: Edge preserving smoothing Contour enhancement, but stop diffusion across edges / BIOMIM PAGE 19 “Contour enhancement”

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Future work Adaptive diffusion processes Use curvature and torsion for adaptive processes Combine diffusion and erosion processes Start looking into practical applications / Communicatie Expertise Centrum PAGE 22mei 2008

Questions? / Communicatie Expertise Centrum PAGE 23mei 2008