Recognizing Rigid Patterns of Unlabeled Point Clouds by Complete and Continuous Isometry Invariants with no False Negatives and no False Positives
This provides a robust solution for applications like object recognition in computer vision or robotics where noise and motion affect data, though it is incremental as it builds on existing concepts of isometry invariants.
The paper tackles the problem of reliably comparing rigid patterns of unlabeled point clouds by developing a complete and continuous isometry invariant that eliminates false negatives and false positives, with the invariant being computable in polynomial time for a fixed dimension.
Rigid structures such as cars or any other solid objects are often represented by finite clouds of unlabeled points. The most natural equivalence on these point clouds is rigid motion or isometry maintaining all inter-point distances. Rigid patterns of point clouds can be reliably compared only by complete isometry invariants that can also be called equivariant descriptors without false negatives (isometric clouds having different descriptions) and without false positives (non-isometric clouds with the same description). Noise and motion in data motivate a search for invariants that are continuous under perturbations of points in a suitable metric. We propose the first continuous and complete invariant of unlabeled clouds in any Euclidean space. For a fixed dimension, the new metric for this invariant is computable in a polynomial time in the number of points.