  We provide a dynamical perspective on the classical problem of 3D point cloud
registration with correspondences. A point cloud is considered as a rigid body
consisting of particles. The problem of registering two point clouds is
formulated as a dynamical system, where the dynamic model point cloud
translates and rotates in a viscous environment towards the static scene point
cloud, under forces and torques induced by virtual springs placed between each
pair of corresponding points. We first show that the potential energy of the
system recovers the objective function of the maximum likelihood estimation. We
then adopt Lyapunov analysis, particularly the invariant set theorem, to
analyze the rigid body dynamics and show that the system globally
asymptotically tends towards the set of equilibrium points, where the globally
optimal registration solution lies in. We conjecture that, besides the globally
optimal equilibrium point, the system has either three or infinite "spurious"
equilibrium points, and these spurious equilibria are all locally unstable. The
case of three spurious equilibria corresponds to generic shape of the point
cloud, while the case of infinite spurious equilibria happens when the point
cloud exhibits symmetry. Therefore, simulating the dynamics with random
perturbations guarantees to obtain the globally optimal registration solution.
Numerical experiments support our analysis and conjecture.
