Locally Adaptive Differential frames and Sub-Riemannian geodesics via the Left Cartan Connection on the Roto-Translation Group SE(d).

Jeudi 9 juin 2016 14:15-15:15 - Remco Duits - CASA & BMIa, TU/e Eindhoven

Résumé : Locally adaptive differential frames are used in differential invariants and PDE-flows on d-D images.
However, at complex structures, these frames are not well-defined. Therefore, we propose locally adaptive frames on (invertible) data representations
U defined on the homogeneous space of positions and orientations, which is a Lie group quotient in the roto-translation group SE(d), d=2,3.
This allows for multiple well-defined frames per position, one for each orientation.
In our paper [1] we compute these frames via local exponential curve fits,
minimizing a first/second order variational problem solved by spectral decomposition of the structure tensor/Hessian on SE(d),
expressed by covariant derivatives of the left Cartan connection.
The same connection appears in the computation of (sub)-Riemannian geodesics in SE(d), cf. [2,3,4] for d=2, and [5] for d=3.
Due to non-vanishing torsion of the (partial) left Cartan connection, the ‘straight curves’, i.e. auto-parallels, are (horizontal) exponential curves,
do not coincide with the `shortest curves’, i.e. the (sub)-Riemannian geodesics which have parallel momentum.
We compute the shortest curves via a geometric back-tracing applied on a distance map solving the (sub-)Riemannian eikonal equation [6],
which can also be solved [7] via an anisotropic fast-marching approach [8].
Applications include crossing-preserving diffusion, vesselness filtering, curvature-based biomarkers, vessel tracking
and vessel segmentation in noisy medical images. The methods perform well in comparisons to other state-of-the-art methods.
Literature :
[1] Duits & Janssen, Hannink, Sanguinetti. “Locally Adaptive Frames in the Roto-translation Group and their Applications in Medical Imaging”, JMIV, 2016.
[2] Moiseev & Sachkov, “Maxwell strata in sub-Riemannian problem on the group of motions of a plane”, ESAIM-COCV, 2010.
[3] Boscain, Duits, Rossi, Sachkov. “Curve Cuspless Reconstruction via sub-Riemannian Geometry”, ESAIM-COCV, 2015.
[4] Duits, Boscain, Rossi, Sachkov. “Association Fields via Cuspless Sub-Riemannian Geodesics in SE(2).”, JMIV, 2014.
[5] Duits, Ghosh, Dela Haije, Mashtakov. “On Sub-Riemannian geodesics in SE(3) whose Spatial Projections do not have Cusps”,
Accepted for publication in JDCS, to appear in 2016. http://bmia.bmt.tue.nl/people/RDuit...
[6] Bekkers & Duits & Mashtakov & Sanguinetti, “A PDE Approach to Data-Driven Sub-Riemannian Geodesics”, SIIMS, 2015.
[7] Sanguinetti, Bekkers, Duits, Janssen, Mashtakov, Mirebeau “Sub-Riemannian Fast Marching in SE(2)”, CIARP, 2015.
[8] Mirebeau, “Anisotropic Fast-Marching on Cartesian grids using Lattice Basis Reduction.”, 2014.

Lieu : Salle 113-115, Bâtiment 425

Locally Adaptive Differential frames and Sub-Riemannian geodesics via the Left Cartan Connection on the Roto-Translation Group SE(d).  Version PDF