FIRST (FP7/ Marie-Curie ) F ronts and I nterfaces in S cience & T echnlogy

1. FIRST (FP7/Marie-Curie) Fronts and Interfaces in Science & Technlogy Adaptive and directional local processing in image processing ArpanGhosh Supervisor: R.Duits

2. Directional processing image processing neural fibers in brain bone-structure catheters muscle cells hart retinal bloodvessels Crack-Detection collagen fibres Challenge: Deal with crossings and fiber-context

3. Particular Focus onDiffusionweighted MRI

4. Invertible Orientation Scores image kernel orientation score invertible

5. Extend to New Medical Image Modalities fibertracking fibertracking DTI HARDI Brownian motion of water molecules along fibers

6. Adaptive Left Invariant HJB-Equations on HARDI/DTI Input Viscosity solution

7. Diffusion & Erosion on DTI

8. Challenges • The viscositysolutions of HJK are solvedby • morphologicalconvolution. Analytic/exact solutions ? • UseHJB-eqsforfiber tracking via Charpit’sequations • Canonicalequationson contact manifold ? • The left-invariantPDE’stake place alongautoparallels • w.r.t. Cartanconnection. SoNon-linearPDE’sby • best exp-curve fit to data ? • Exact solutions of geodesics

9. Problem considered currently • Let curve Curvature function • Corresponding energy functional • The challenge is to find for given end points and directions s.t.

10. The Setup • 5D manifold of positions & directions ℝ³×S² not a group! • Consider embedding of ℝ³×S² into the Lie group SE(3) ≔ ℝ³⋊SO(3) • By the quotient: ℝ³ ⋊ S² ≔ SE(3)/({0} ⋊SO(2)) • Consider Euler angle parameterization of S² for chart.

11. Differential Geometry Tools • SE(3) : Lie Group with group product • Unity element • Lie algebra

12. Left Invariant VF’s • A moving frame of reference using left invariant vector fields • The space of left invariant vector fields on SE(3) • Duals :

13. Cartan’s Geometry

14. CommutatorTable • Lie brackets for the Lie algebra given by

15. Horizontal Curves in SE(3) • Curve is horizontal iff • Using the Frenet-Serret formulas, curvature and torsion of the spatial part of the curve are given by

16. Problem in the Moving Frame • The energy functional to minimize over all curves in SE(3): subject to the constraints along the curve :

17. 15D Manifold • Consider the sub-Riemannian manifold • The constraints are 1 forms

18. Lagrangian 1-form on T*(Z): • Consider perturbations of stationary horizontal curves • Then, using Stoke’s theorem • For stationary curves, the above is 0 for all t. So,

19. Pfaffian System • By inserting 15 independent vectors from the sub-Riemannian manifold T(Z) and using Cartan’s structural formulas, we get

21. Solution for Curvature • The Pfaffian system produces the following

22. The Geodesics • Can get curve by solving Frenet frame equations • The last 6 equations from the system gives the equation for the geodesic where μ is a constant 6D row vector and m(g) is the 6×6 matrix representation of g∈ SE(3).

23. Transformed Curve • Choose s.t.

24. Solution of Transformed Curve where

25. We get back the original curve by the following relation

26. Results

29. Boundary value problem • The boundary problem is solved by solving for 5 unknowns namely the following system • 6 dependent polynomial equations in the 5 variables above, of degree upto 8!

30. Current Focus • Solution of the boundary problem • Surjectivity of the geodesics • Study of the second variation of energy functional to prove optimality

31. Next target • Geometric fiber tracking based on the same Hamiltonian and Lagrangian framework with a cost coming from the enhanced data sets.

32. Thank you!