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Published byEddie Hardick Modified about 1 year ago

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

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

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Particular Focus on Diffusion weighted MRI

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Invertible Orientation Scores imagekernelorientation score invertible

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Brownian motion of water molecules along fibers fibertracking DTIHARDI Extend to New Medical Image Modalities

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Adaptive Left Invariant HJB-Equations on HARDI/DTI InputViscosity solution

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Diffusion & Erosion on DTI

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Challenges The viscosity solutions of HJK are solved by morphological convolution. Analytic/exact solutions ? Use HJB-eqs for fiber tracking via Charpit’s equations Canonical equations on contact manifold ? The left-invariant PDE’s take place along autoparallels w.r.t. Cartan connection. So Non-linear PDE’s by best exp-curve fit to data ? Exact solutions of geodesics

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

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

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

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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 :

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Cartan’s Geometry

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Commutator Table Lie brackets for the Lie algebra given by

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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

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

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

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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,

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

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Solution for Curvature The Pfaffian system produces the following

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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).

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Transformed Curve Choose s.t.

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Solution of Transformed Curve where

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We get back the original curve by the following relation

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Results

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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!

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

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

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Thank you!

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