The richer representation the better registration.

In this paper, the registration problem is formulated as a point to model distance minimization. Unlike most of the existing works, which are based on minimizing a point-wise correspondence term, this formulation avoids the correspondence search that is time-consuming. In the first stage, the target set is described through an implicit function by employing a linear least squares fitting. This function can be either an implicit polynomial or an implicit B-spline from a coarse to fine representation. In the second stage, we show how the obtained implicit representation is used as an interface to convert point-to-point registration into point-to-implicit problem. Furthermore, we show that this registration distance is smooth and can be minimized through the Levengberg–Marquardt algorithm. All the formulations presented for both stages are compact and easy to implement. In addition, we show that our registration method can be handled using any implicit representation though some are coarse and others provide finer representations; hence, a tradeoff between speed and accuracy can be set by employing the right implicit function. Experimental results and comparisons in 2D and 3D show the robustness and the speed of convergence of the proposed approach.