  We investigate uncertainty quantification of 6D pose estimation from learned
noisy measurements (e.g. keypoints and pose hypotheses). Assuming
unknown-but-bounded measurement noises, a pose uncertainty set (PURSE) is a
subset of SE(3) that contains all possible 6D poses compatible with the
measurements. Despite being simple to formulate and its ability to embed
uncertainty, the PURSE is difficult to manipulate and interpret due to the many
abstract nonconvex polynomial constraints. An appealing simplification of PURSE
is to find its minimum enclosing geodesic ball (MEGB), i.e., a point pose
estimation with minimum worst-case error bound. We contribute (i) a geometric
interpretation of the nonconvex PURSE, and (ii) a fast algorithm to inner
approximate the MEGB. Particularly, we show the PURSE corresponds to the
feasible set of a constrained dynamical system or the intersection of multiple
geodesic balls, and this perspective allows us to design an algorithm to
densely sample the boundary of the PURSE through strategic random walks. We
then use the miniball algorithm to compute the MEGB of PURSE samples, leading
to an inner approximation. Our algorithm is named CLOSURE (enClosing baLl frOm
purSe boUndaRy samplEs) and it enables computing a certificate of approximation
tightness by calculating the relative size ratio between the inner
approximation and the outer approximation. Running on a single RTX 3090 GPU,
CLOSURE achieves the relative ratio of 92.8% on the LM-O dataset, 91.4% on the
3DMatch dataset and 96.6% on the LM dataset with the average runtime less than
0.3 second. Obtaining comparable worst-case error bound but 398x 833x and 23.6x
faster than the outer approximation GRCC, CLOSURE enables uncertainty
quantification of 6D pose estimation to be implemented in real-time robot
perception applications.
