310 research outputs found
Efficient moving point handling for incremental 3D manifold reconstruction
As incremental Structure from Motion algorithms become effective, a good
sparse point cloud representing the map of the scene becomes available
frame-by-frame. From the 3D Delaunay triangulation of these points,
state-of-the-art algorithms build a manifold rough model of the scene. These
algorithms integrate incrementally new points to the 3D reconstruction only if
their position estimate does not change. Indeed, whenever a point moves in a 3D
Delaunay triangulation, for instance because its estimation gets refined, a set
of tetrahedra have to be removed and replaced with new ones to maintain the
Delaunay property; the management of the manifold reconstruction becomes thus
complex and it entails a potentially big overhead. In this paper we investigate
different approaches and we propose an efficient policy to deal with moving
points in the manifold estimation process. We tested our approach with four
sequences of the KITTI dataset and we show the effectiveness of our proposal in
comparison with state-of-the-art approaches.Comment: Accepted in International Conference on Image Analysis and Processing
(ICIAP 2015
Demystifying Latschev's Theorem: Manifold Reconstruction from Noisy Data
For a closed Riemannian manifold and a metric space with a
small Gromov\unicode{x2013}Hausdorff distance to it, Latschev's theorem
guarantees the existence of a sufficiently small scale at which the
Vietoris\unicode{x2013}Rips complex of is homotopy equivalent to
. Despite being regarded as a stepping stone to the topological
reconstruction of Riemannian manifolds from a noisy data, the result is only a
qualitative guarantee. Until now, it had been elusive how to quantitatively
choose such a proximity scale in order to provide sampling conditions
for to be homotopy equivalent to . In this paper, we prove a
stronger and pragmatic version of Latschev's theorem, facilitating a simple
description of using the sectional curvatures and convexity radius of
as the sampling parameters. Our study also delves into the
topological recovery of a closed Euclidean submanifold from the
Vietoris\unicode{x2013}Rips complexes of a Hausdorff close Euclidean subset.
As already known for \v{C}ech complexes, we show that
Vietoris\unicode{x2013}Rips complexes also provide topologically faithful
reconstruction guarantees for submanifolds. In the Euclidean case, our sampling
conditions\unicode{x2014}using only the reach of the
submanifold\unicode{x2014}turns out to be much simpler than the previously
known reconstruction results using weak feature size and
\mu\unicode{x2013}reach.Comment: arXiv admin note: substantial text overlap with arXiv:2204.1423
A Bayesian Approach to Manifold Topology Reconstruction
In this paper, we investigate the problem of statistical reconstruction of piecewise linear manifold topology. Given a noisy, probably undersampled point cloud from a one- or two-manifold, the algorithm reconstructs an approximated most likely mesh in a Bayesian sense from which the sample might have been taken. We incorporate statistical priors on the object geometry to improve the reconstruction quality if additional knowledge about the class of original shapes is available. The priors can be formulated analytically or learned from example geometry with known manifold tessellation. The statistical objective function is approximated by a linear programming / integer programming problem, for which a globally optimal solution is found. We apply the algorithm to a set of 2D and 3D reconstruction examples, demon-strating that a statistics-based manifold reconstruction is feasible, and still yields plausible results in situations where sampling conditions are violated
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