  We propose the first fast and certifiable algorithm for the registration of
two sets of 3D points in the presence of large amounts of outlier
correspondences. We first reformulate the registration problem using a
Truncated Least Squares (TLS) cost that is insensitive to a large fraction of
spurious correspondences. Then, we provide a general graph-theoretic framework
to decouple scale, rotation, and translation estimation, which allows solving
in cascade for the three transformations. Despite the fact that each subproblem
is still non-convex and combinatorial in nature, we show that (i) TLS scale and
(component-wise) translation estimation can be solved in polynomial time via
adaptive voting, (ii) TLS rotation estimation can be relaxed to a semidefinite
program (SDP) and the relaxation is tight, even in the presence of extreme
outlier rates, and (iii) the graph-theoretic framework allows drastic pruning
of outliers by finding the maximum clique. We name the resulting algorithm
TEASER (Truncated least squares Estimation And SEmidefinite Relaxation). While
solving large SDP relaxations is typically slow, we develop a second fast and
certifiable algorithm, named TEASER++, that uses graduated non-convexity to
solve the rotation subproblem and leverages Douglas-Rachford Splitting to
efficiently certify global optimality.
  For both algorithms, we provide theoretical bounds on the estimation errors,
which are the first of their kind for robust registration problems. Moreover,
we test their performance on standard, object detection, and the 3DMatch
benchmarks, and show that (i) both algorithms dominate the state of the art and
are robust to more than 99% outliers, (ii) TEASER++ can run in milliseconds,
and (iii) TEASER++ is so robust it can also solve problems without
correspondences, where it largely outperforms ICP and it is more accurate than
Go-ICP while being orders of magnitude faster.
