  Using geometric landmarks like lines and planes can increase navigation
accuracy and decrease map storage requirements compared to commonly-used LiDAR
point cloud maps. However, landmark-based registration for applications like
loop closure detection is challenging because a reliable initial guess is not
available. Global landmark matching has been investigated in the literature,
but these methods typically use ad hoc representations of 3D line and plane
landmarks that are not invariant to large viewpoint changes, resulting in
incorrect matches and high registration error. To address this issue, we adopt
the affine Grassmannian manifold to represent 3D lines and planes and prove
that the distance between two landmarks is invariant to rotation and
translation if a shift operation is performed before applying the Grassmannian
metric. This invariance property enables the use of our graph-based data
association framework for identifying landmark matches that can subsequently be
used for registration in the least-squares sense. Evaluated on a challenging
landmark matching and registration task using publicly-available LiDAR
datasets, our approach yields a 1.7x and 3.5x improvement in successful
registrations compared to methods that use viewpoint-dependent centroid and
"closest point" representations, respectively.
