Scale-Invariant Fast Functional Registration

Muchen Sun, Allison Pinosky, Ian Abraham, Todd D. Murphey

Abstract: 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 \(L^2\)-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 \(\textsf{FLS}\) (functional least-squares), on standard 3D registration benchmarks, showing \(\textsf{FLS}\) is an order of magnitude faster than state-of-the-art functional registration algorithm without compromising accuracy and robustness. \(\textsf{FLS}\) 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 \(\textsf{FLS}\) 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.