6,666 research outputs found
Approximating Nearest Neighbor Distances
Several researchers proposed using non-Euclidean metrics on point sets in
Euclidean space for clustering noisy data. Almost always, a distance function
is desired that recognizes the closeness of the points in the same cluster,
even if the Euclidean cluster diameter is large. Therefore, it is preferred to
assign smaller costs to the paths that stay close to the input points.
In this paper, we consider the most natural metric with this property, which
we call the nearest neighbor metric. Given a point set P and a path ,
our metric charges each point of with its distance to P. The total
charge along determines its nearest neighbor length, which is formally
defined as the integral of the distance to the input points along the curve. We
describe a -approximation algorithm and a
-approximation algorithm to compute the nearest neighbor
metric. Both approximation algorithms work in near-linear time. The former uses
shortest paths on a sparse graph using only the input points. The latter uses a
sparse sample of the ambient space, to find good approximate geodesic paths.Comment: corrected author nam
Nilpotent Approximations of Sub-Riemannian Distances for Fast Perceptual Grouping of Blood Vessels in 2D and 3D
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 and . In our distance
approximations we consider homogeneous norms on nilpotent groups that locally
approximate , and which are obtained via the exponential and logarithmic
map on . 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 .
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) that grouping via the
fast analytic approximations performs almost equally, or better, than
data-adaptive fast marching approaches on and .Comment: 18 pages, 9 figures, 3 tables, in review at JMI
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