(Adaptive) enhancement of
HARDI images
Eric Creusen
Supervisor:
Remco Duits
Diffusion of Water
Diffusion is dependent on orientation
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*Graphic borrowed from Thomas Schultz
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Diffusion of water: Modelling
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Domain
• Diffusion weighted images are functions on position
and orientation
• Both HARDI and DTI!
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mei 2008
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Embed in group
• Embed position and orientation into translations and
rotation group
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Parametrization of Sphere
Euler Angles
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mei 2008
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Embedding problem
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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!
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Convection-diffusion processes
• See function as distribution of oriented particles
• Particles can translate and rotate
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Rotated coordinate system
A1
z
A3
n
n
Rn (α)
y
x
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A2
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Now that we have a basis: derivatives!
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Derivatives can be calculated
Directions dependant on orientation n
3 spatial directions
2 angular directions (+1 dummy direction)
For n=ez
Perpendicular
(make random
choice here)
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In the same
direction as normal
Angular
derivatives
Should be
zero
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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)
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Some simple examples
• Isotropic Spatial Gaussian
smoothing:
• Angular diffusion:
Not interesting, only
spatial OR only angular
diffusion
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mei 2008
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More examples
“Contour enhancement”
Spatial diffusion
Angular diffusion
“Contour completion”
Convection
term
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Simulation of PDEs
• Convolutions with kernels
• Finite difference methods
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mei 2008
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Creating convolution kernels
• Get approximations with complicated math……. OR
• Start with oriented delta peak
• Simulate PDE with finite difference methods
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mei 2008
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Finite difference methods: enhancement
• Directly use finite difference methods to simulate
PDE
enhancement
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Finite difference: contour completion
completion
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Adaptive filtering
• Make smoothing dependant on data
• Finite difference methods neccesary
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Perona-Malik type diffusion
• Goal: Edge preserving smoothing
• Contour enhancement, but stop diffusion across
edges
“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
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mei 2008
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Questions?
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mei 2008
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