  Functional registration algorithms represent point clouds as functions (e.g.
spacial occupancy field) avoiding unreliable correspondence estimation in
conventional least-squares registration algorithms. However, existing
functional registration algorithms are computationally expensive. Furthermore,
the capability of registration with unknown scale is necessary in tasks such as
CAD model-based object localization, yet no such support exists in functional
registration. In this work, we propose a scale-invariant, linear time
complexity functional registration algorithm. We achieve linear time complexity
through an efficient approximation of L2-distance between functions using
orthonormal basis functions. The use of orthonormal basis functions leads to a
formulation that is compatible with least-squares registration. Benefited from
the least-square formulation, we use the theory of
translation-rotation-invariant measurement to decouple scale estimation and
therefore achieve scale-invariant registration. We evaluate the proposed
algorithm, named FLS (functional least-squares), on standard 3D registration
benchmarks, showing FLS is an order of magnitude faster than state-of-the-art
functional registration algorithm without compromising accuracy and robustness.
FLS also outperforms state-of-the-art correspondence-based least-squares
registration algorithm on accuracy and robustness, with known and unknown
scale. Finally, we demonstrate applying FLS to register point clouds with
varying densities and partial overlaps, point clouds from different objects
within the same category, and point clouds from real world objects with noisy
RGB-D measurements.
