A Riemannian framework for matching point clouds represented by the Schrodinger distance transform

Yan Deng, Yan Deng

Abstract:

In this paper, we cast the problem of point cloud matching as a shape matching problem by transforming each of
the given point clouds into a shape representation called
the Schrodinger distance transform (SDT) representation. ¨
This is achieved by solving a static Schrodinger equation ¨
instead of the corresponding static Hamilton-Jacobi equation in this setting. The SDT representation is an analytic expression and following the theoretical physics literature, can be normalized to have unit L2 norm—making it
a square-root density, which is identified with a point on a
unit Hilbert sphere, whose intrinsic geometry is fully known.
The Fisher-Rao metric, a natural metric for the space of
densities leads to analytic expressions for the geodesic distance between points on this sphere. In this paper, we use
the well known Riemannian framework never before used
for point cloud matching, and present a novel matching
algorithm. We pose point set matching under rigid and
non-rigid transformations in this framework and solve for
the transformations using standard nonlinear optimization
techniques. Finally, to evaluate the performance of our
algorithm—dubbed SDTM—we present several synthetic
and real data examples along with extensive comparisons
to state-of-the-art techniques. The experiments show that
our algorithm outperforms state-of-the-art point set registration algorithms on many quantitative metrics