Point clouds representing real objects are natural to study under rigid motion (compositions of translations and rotations), or under isometry, which allows reflections. Any ordered points in Euclidean space are uniquely determined under isometry by the matrix of distances. However, if points are unordered, there is an exponential number of different matrices representing the same cloud of points. Hence the key challenge is to guarantee the computability of complete invariants and continuous metrics in polynomial time in the input size. One more practical condition is an explicit parameterization of all realizable invariants that can be inverted to point clouds uniquely under isometry. If this reconstruction is also continuous, the moduli space is fully parameterized like a geographic-style map, which was unknown even for four points in the plane. The new realizable invariants extend the previous distance-based invariants (CVPR 2023). All relevant papers are at https://kurlin.org/research-papers.php#Geometric-Data-Science.

Refreshments will be served preceding the talk, beginning at 2:45.