Download - ( Adaptive ) enhancement of HARDI images
![Page 1: ( Adaptive ) enhancement of HARDI images](https://reader035.vdocument.in/reader035/viewer/2022062321/5681341c550346895d9b09e5/html5/thumbnails/1.jpg)
(Adaptive) enhancement of HARDI images
Eric Creusen
Supervisor:
Remco Duits
![Page 2: ( Adaptive ) enhancement of HARDI images](https://reader035.vdocument.in/reader035/viewer/2022062321/5681341c550346895d9b09e5/html5/thumbnails/2.jpg)
Diffusion of Water
/ BIOMIM PAGE 2
Diffusion is dependent on orientation
*Graphic borrowed from Thomas Schultz
![Page 3: ( Adaptive ) enhancement of HARDI images](https://reader035.vdocument.in/reader035/viewer/2022062321/5681341c550346895d9b09e5/html5/thumbnails/3.jpg)
Diffusion of water: Modelling
BIOMIM PAGE 3
![Page 4: ( Adaptive ) enhancement of HARDI images](https://reader035.vdocument.in/reader035/viewer/2022062321/5681341c550346895d9b09e5/html5/thumbnails/4.jpg)
Domain
• Diffusion weighted images are functions on position and orientation
• Both HARDI and DTI!
/ Communicatie Expertise Centrum PAGE 4mei 2008
![Page 5: ( Adaptive ) enhancement of HARDI images](https://reader035.vdocument.in/reader035/viewer/2022062321/5681341c550346895d9b09e5/html5/thumbnails/5.jpg)
Embed in group
• Embed position and orientation into translations and rotation group
/ BIOMIM PAGE 5
![Page 6: ( Adaptive ) enhancement of HARDI images](https://reader035.vdocument.in/reader035/viewer/2022062321/5681341c550346895d9b09e5/html5/thumbnails/6.jpg)
Parametrization of Sphere
/ Communicatie Expertise Centrum PAGE 6mei 2008
Euler Angles
![Page 7: ( Adaptive ) enhancement of HARDI images](https://reader035.vdocument.in/reader035/viewer/2022062321/5681341c550346895d9b09e5/html5/thumbnails/7.jpg)
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
![Page 8: ( Adaptive ) enhancement of HARDI images](https://reader035.vdocument.in/reader035/viewer/2022062321/5681341c550346895d9b09e5/html5/thumbnails/8.jpg)
Convection-diffusion processes
• See function as distribution of oriented particles• Particles can translate and rotate
/ BIOMIM PAGE 8
![Page 9: ( Adaptive ) enhancement of HARDI images](https://reader035.vdocument.in/reader035/viewer/2022062321/5681341c550346895d9b09e5/html5/thumbnails/9.jpg)
Rotated coordinate system
/ Communicatie Expertise Centrum PAGE 9mei 2008
nA3
A1
A2
n
z
yx
Rn (α)
![Page 10: ( Adaptive ) enhancement of HARDI images](https://reader035.vdocument.in/reader035/viewer/2022062321/5681341c550346895d9b09e5/html5/thumbnails/10.jpg)
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 bezero
![Page 11: ( Adaptive ) enhancement of HARDI images](https://reader035.vdocument.in/reader035/viewer/2022062321/5681341c550346895d9b09e5/html5/thumbnails/11.jpg)
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
![Page 12: ( Adaptive ) enhancement of HARDI images](https://reader035.vdocument.in/reader035/viewer/2022062321/5681341c550346895d9b09e5/html5/thumbnails/12.jpg)
Some simple examples
• Isotropic Spatial Gaussian
smoothing: • Angular diffusion:
/ Communicatie Expertise Centrum PAGE 12mei 2008
Not interesting, only spatial OR only angular diffusion
![Page 13: ( Adaptive ) enhancement of HARDI images](https://reader035.vdocument.in/reader035/viewer/2022062321/5681341c550346895d9b09e5/html5/thumbnails/13.jpg)
More examples
/ Communicatie Expertise Centrum PAGE 13mei 2008
Convection term
Angular diffusion Spatial diffusion
“Contour enhancement”
“Contour completion”
![Page 14: ( Adaptive ) enhancement of HARDI images](https://reader035.vdocument.in/reader035/viewer/2022062321/5681341c550346895d9b09e5/html5/thumbnails/14.jpg)
Simulation of PDEs
• Convolutions with kernels• Finite difference methods
/ Communicatie Expertise Centrum PAGE 14mei 2008
![Page 15: ( Adaptive ) enhancement of HARDI images](https://reader035.vdocument.in/reader035/viewer/2022062321/5681341c550346895d9b09e5/html5/thumbnails/15.jpg)
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
![Page 16: ( Adaptive ) enhancement of HARDI images](https://reader035.vdocument.in/reader035/viewer/2022062321/5681341c550346895d9b09e5/html5/thumbnails/16.jpg)
Finite difference methods: enhancement
• Directly use finite difference methods to simulate PDE
/ Communicatie Expertise Centrum PAGE 16mei 2008
enhancement
![Page 17: ( Adaptive ) enhancement of HARDI images](https://reader035.vdocument.in/reader035/viewer/2022062321/5681341c550346895d9b09e5/html5/thumbnails/17.jpg)
Finite difference: contour completion
/ Communicatie Expertise Centrum PAGE 17mei 2008
completion
![Page 18: ( Adaptive ) enhancement of HARDI images](https://reader035.vdocument.in/reader035/viewer/2022062321/5681341c550346895d9b09e5/html5/thumbnails/18.jpg)
Adaptive filtering
• Make smoothing dependant on data• Finite difference methods neccesary
/ Communicatie Expertise Centrum PAGE 18mei 2008
![Page 19: ( Adaptive ) enhancement of HARDI images](https://reader035.vdocument.in/reader035/viewer/2022062321/5681341c550346895d9b09e5/html5/thumbnails/19.jpg)
Perona-Malik type diffusion
• Goal: Edge preserving smoothing• Contour enhancement, but stop diffusion across
edges
/ BIOMIM PAGE 19
“Contour enhancement”
![Page 20: ( Adaptive ) enhancement of HARDI images](https://reader035.vdocument.in/reader035/viewer/2022062321/5681341c550346895d9b09e5/html5/thumbnails/20.jpg)
/ BIOMIM PAGE 20
![Page 21: ( Adaptive ) enhancement of HARDI images](https://reader035.vdocument.in/reader035/viewer/2022062321/5681341c550346895d9b09e5/html5/thumbnails/21.jpg)
/ BIOMIM PAGE 21
![Page 22: ( Adaptive ) enhancement of HARDI images](https://reader035.vdocument.in/reader035/viewer/2022062321/5681341c550346895d9b09e5/html5/thumbnails/22.jpg)
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
![Page 23: ( Adaptive ) enhancement of HARDI images](https://reader035.vdocument.in/reader035/viewer/2022062321/5681341c550346895d9b09e5/html5/thumbnails/23.jpg)
Questions?
/ Communicatie Expertise Centrum PAGE 23mei 2008