Core Concepts
An innovative algorithm that can efficiently reconstruct closed curves directly on Riemannian manifolds from a given sparse set of sample points, overcoming the limitations of previous state-of-the-art methods.
Abstract
This paper introduces a novel method for reconstructing closed curves on Riemannian manifolds from sparse, unordered samples. The key contributions are:
Extending state-of-the-art theory and techniques for 2D curve reconstruction to manifold domains, addressing the challenges of working in non-Euclidean spaces.
Relaxing the sampling conditions required for curve reconstruction on manifolds compared to previous work, allowing for sparser and non-uniform sampling.
Generalizing the SIGDT proximity graph to manifold domains (SIGDV) and proving that it contains the correct curve reconstruction under the new sampling conditions.
Developing an algorithm that leverages the properties of the SIGDV graph to efficiently reconstruct closed curves on manifolds, even when the sampling conditions are not fully met.
The authors demonstrate the robustness and versatility of their method through qualitative experiments on various real-world applications, including motion tracking, virtual cultural heritage processing, contour matching, and sparse data visualization. The proposed solution outperforms the previous state-of-the-art approach, which fails in many cases due to its stricter sampling requirements.
Stats
The local feature size (lfs) represents the minimum distance from a point on the curve to the medial axis.
The injectivity radius (iM(p)) of a point p on the manifold M is the infimum of the distances from p to the cut locus of p.
The injective local feature size (ilfs(p)) is defined as the minimum of the local feature size and the injectivity radius at a point p.
The injective reach (ireach(I)) of an interval I on the curve is the minimum injective local feature size among points in the interval.
Quotes
"We formally improve on the state-of-the-art requirements and introduce an innovative algorithm capable of reconstructing closed curves directly on surfaces from a given sparse set of sample points."
"We extend and adapt a state-of-the-art planar curve reconstruction method to the realm of surfaces while dealing with the challenges arising from working on non-Euclidean domains."