571 research outputs found
A PDE Approach to Data-driven Sub-Riemannian Geodesics in SE(2)
We present a new flexible wavefront propagation algorithm for the boundary
value problem for sub-Riemannian (SR) geodesics in the roto-translation group
with a metric tensor depending on a smooth
external cost , , computed from
image data. The method consists of a first step where a SR-distance map is
computed as a viscosity solution of a Hamilton-Jacobi-Bellman (HJB) system
derived via Pontryagin's Maximum Principle (PMP). Subsequent backward
integration, again relying on PMP, gives the SR-geodesics. For
we show that our method produces the global minimizers. Comparison with exact
solutions shows a remarkable accuracy of the SR-spheres and the SR-geodesics.
We present numerical computations of Maxwell points and cusp points, which we
again verify for the uniform cost case . Regarding image
analysis applications, tracking of elongated structures in retinal and
synthetic images show that our line tracking generically deals with crossings.
We show the benefits of including the sub-Riemannian geometry.Comment: Extended version of SSVM 2015 conference article "Data-driven
Sub-Riemannian Geodesics in SE(2)
Sub-Riemannian Fast Marching in SE(2)
We propose a Fast Marching based implementation for computing sub-Riemanninan
(SR) geodesics in the roto-translation group SE(2), with a metric depending on
a cost induced by the image data. The key ingredient is a Riemannian
approximation of the SR-metric. Then, a state of the art Fast Marching solver
that is able to deal with extreme anisotropies is used to compute a SR-distance
map as the solution of a corresponding eikonal equation. Subsequent
backtracking on the distance map gives the geodesics. To validate the method,
we consider the uniform cost case in which exact formulas for SR-geodesics are
known and we show remarkable accuracy of the numerically computed SR-spheres.
We also show a dramatic decrease in computational time with respect to a
previous PDE-based iterative approach. Regarding image analysis applications,
we show the potential of considering these data adaptive geodesics for a fully
automated retinal vessel tree segmentation.Comment: CIARP 201
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