( Adaptive ) enhancement of HARDI images

(Adaptive) enhancement of HARDI images

Eric Creusen

Supervisor:

Remco Duits

Diffusion of Water

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Diffusion is dependent on orientation

*Graphic borrowed from Thomas Schultz

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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Embed in group

• Embed position and orientation into translations and rotation group

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Parametrization of Sphere


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!

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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


nA3

A1

A2

n

z

yx

Rn (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


In the same direction as normal

Perpendicular (make random choice here)

Angular derivatives

Should bezero

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)


Some simple examples

• Isotropic Spatial Gaussian

smoothing: • Angular diffusion:


Not interesting, only spatial OR only angular diffusion

More examples


Convection term

Angular diffusion Spatial diffusion

“Contour enhancement”

“Contour completion”

Simulation of PDEs

• Convolutions with kernels• Finite difference methods


Creating convolution kernels

• Get approximations with complicated math……. OR• Start with oriented delta peak• Simulate PDE with finite difference methods


Finite difference methods: enhancement

• Directly use finite difference methods to simulate PDE


enhancement

Finite difference: contour completion


completion

Adaptive filtering

• Make smoothing dependant on data• Finite difference methods neccesary


Perona-Malik type diffusion

• Goal: Edge preserving smoothing• Contour enhancement, but stop diffusion across

edges

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“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


Questions?


Documents

( Adaptive ) enhancement of HARDI images