Abstract. We present a new flexible wavefront propagation algorithm for the boundary value problem for subRiemannian (SR) geodesics in the roto-translation group SE(2) = R 2 S 1 with a metric tensor depending on a smooth external cost C : SE(2) → [δ, 1], δ > 0, computed from image data. The method consists of a first step where an SR-distance map is computed as a viscosity solution of a Hamilton-Jacobi-Bellman system derived via Pontryagin's maximum principle (PMP). Subsequent backward integration, again relying on PMP, gives the SR-geodesics. For C = 1 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 C = 1. Regarding image analysis applications, trackings of elongated structures in retinal and synthetic images show that our line tracking generically deals with crossings. We show the benefits of including the SR-geometry.
Abstract. We propose a robust and fully automatic method for the analysis of vessel tortuosity. Our method does not rely on pre-segmentation of vessels, but instead acts directly on retinal image data. The method is based on theory of best-fit exponential curves in the roto-translation group SE(2). We lift 2D images to 3D functions called orientation scores by including an orientation dimension in the domain. In the extended domain of positions and orientations (identified with SE(2)) we study exponential curves, whose spatial projections have constant curvature. By locally fitting such curves to data in orientation scores, via our new iterative stabilizing refinement method, we are able to assign to each location a curvature and confidence value. These values are then used to define global tortuosity measures. The method is validated on synthetic and retinal images. We show that the tortuosity measures can serve as effective biomarkers for diabetes and different stages of diabetic retinopathy.
We propose an efficient approach for the grouping of local orientations (points on vessels) via nilpotent approximations of sub-Riemannian distances in the 2D and 3D roto-translation groups SE(2) and SE(3). In our distance approximations we consider homogeneous norms on nilpotent groups that locally approximate SE(n), and which are obtained via the exponential and logarithmic map on SE(n). In a qualitative validation we show that the norms provide accurate approximations of the true sub-Riemannian distances, and we discuss their relations to the fundamental solution of the sub-Laplacian on SE(n). The quantitative experiments further confirm the accuracy of the approximations. Quantitative results are obtained by evaluating perceptual grouping performance of retinal blood vessels in 2D images and curves in challenging 3D synthetic volumes. The results show that (1) sub-Riemannian geometry is essential in achieving top performance and (2) grouping via the fast analytic approximations performs almost equally, or better, than data-adaptive fast marching approaches on R n and SE(n).
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