  It is essential for a robot to be able to detect revisits or loop closures
for long-term visual navigation.A key insight explored in this work is that the
loop-closing event inherently occurs sparsely, that is, the image currently
being taken matches with only a small subset (if any) of previous images. Based
on this observation, we formulate the problem of loop-closure detection as a
sparse, convex $\ell_1$-minimization problem. By leveraging fast convex
optimization techniques, we are able to efficiently find loop closures, thus
enabling real-time robot navigation. This novel formulation requires no offline
dictionary learning, as required by most existing approaches, and thus allows
online incremental operation. Our approach ensures a unique hypothesis by
choosing only a single globally optimal match when making a loop-closure
decision. Furthermore, the proposed formulation enjoys a flexible
representation with no restriction imposed on how images should be represented,
while requiring only that the representations are "close" to each other when
the corresponding images are visually similar. The proposed algorithm is
validated extensively using real-world datasets.
