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Erschienen in:
2015 | OriginalPaper | Buchkapitel
Data-Driven Sub-Riemannian Geodesics in SE(2)
verfasst von : E. J. Bekkers, R. Duits, A. Mashtakov, G. R. Sanguinetti
Erschienen in: Scale Space and Variational Methods in Computer Vision
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We present a new flexible wavefront propagation algorithm for the boundary value problem for sub-Riemannian (SR) geodesics in the roto-translation group
$$SE(2) = \mathbb {R}^2 \rtimes S^1$$
with a metric tensor depending on a smooth external cost
$$\mathcal {C}:SE(2) \rightarrow [\delta ,1]$$
,
$$\delta >0$$
, computed from image data. The method consists of a first step where geodesically equidistant surfaces are 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. We show that our method produces geodesically equidistant surfaces. For
$$\mathcal {C}=1$$
we show that our method produces the global minimizers, and comparison with exact solutions shows a remarkable accuracy of the SR-spheres/geodesics. Finally, trackings in synthetic and retinal images show the potential of including the SR-geometry.