Scale-Free Registrations in 3D: 7 Degrees of Freedom with Fourier Mellin SOFT Transforms

Abstract

Fourier Mellin SOFT (FMS) as a novel method for global registration of 3D data is presented. It determines the seven degrees of freedom (7-DoF) transformation, i.e., the 6-DoF rigid motion parameters plus 1-DoF scale, between two scans, i.e., two noisy, only partially overlapping views on objects or scenes. It is based on a sequence of the 3D Fourier transform, the Mellin transform and the SO(3) Fourier transform. This combination represents a non-trivial complete 3D extension of the well known Fourier-Mellin registration for 2D images. It is accordingly based on decoupling rotation and scale from translation. First, rotation—which is the main challenge for the extension to 3D data - is tackled with a SO(3) Fourier Transform (SOFT) based on Spherical Harmonics. In a second step, scale is determined via a 3D Mellin transform. Finally, translation is calculated by Phase-Matching. Experiments are presented with simulated data sets for ground truth comparisons and with real world data including object recognition and localization in Magnetic Resonance Tomography (MRT) data, registration of 2.5D RGBD scans from a Microsoft Kinect with a scale-free 3D model generated by Multi-View Vision, and 3D mapping by registration of a sequence of consecutive scans from a low-cost actuated Laser Range Finder. The results show that the method is fast and that it can robustly handle partial overlap, interfering structures, and noise. It is also shown that the method is a very interesting option for 6-DoF registration, i.e., when scale is known.