Because the cost function of registration depends on finding the closest point in \mathcal{S} to every point in \mathcal{M}, it can change as the algorithm is running. As such, it is difficult to prove that ICP will in fact converge exactly to the local optimum. In fact, empirically, ICP and EM-ICP do not converge to the local minimum of the cost function. Nonetheless, because ICP is intuitive to understand and straightforward to implement, it remains the most commonly used point set registration algorithm. Many variants of ICP have been proposed, affecting all phases of the algorithm from the selection and matching of points to the minimization strategy.
