Point set registration based on implicit surface fitting with equivalent distance

Point set registration can be reformulated as the problem of points-to-model alignment with implicit surface fitting. This formulation avoids the correspondence search that is time-consuming. However, the minimization of the sum of the squared 'Approximate Distance' between the point set and the model under rigid transformation can be easily trapped into false positions. In this paper, we explicitly derive the detailed formulation of Levenberg-Marquadt algorithm (LMA) with the Approximate Distance for nonlinear least square optimization of registration in 3D case. Based on the analysis for the defect of the Approximate Distance, we propose a novel metric called 'Equivalent Distance' and give the full solution for the nonlinear least square optimization of the rigid transformation parameters with the Equivalent Distance. Contrary to the method with the Approximate Distance, the LMA with the Equivalent Distance can converge into optimal positions with much wider convergence range and lead to more accurate transformation parameters. Experimental results and comparisons in 3D cases demonstrate the speed, the accuracy and the convergence performance of the proposed approach.