Therefore, previous works [@olsson2006registration; @olsson2009branch; @briales2017convex] that only compute the global minimizer will fail in this case. Revealing local minimizers is essential for an algorithm to handle all the configurations.

Second, we present an efficient and accurate solution for the least-squares 3D registration problem. We use the CGR parametrization to represent the rotation, which generates a rational cost function. We derive its first-order optimality conditions. They form a high order polynomial system, and are hard to solve. Four intermediate unknowns are introduced to relax the original problem, which results in much simpler first-order optimality conditions. Gröbner basis method [@cox2006using] is applied to solve this equation system. Then we refine the solution by the Newton-Raphson method. Third, we present an unified solution for the potential minimal configurations. Previous algorithms are proposed to solve the minimal problems case by case [@chen1990pose; @ramalingam2010p2pi; @ramalingam2013theory]. We also use the CGR parametrization to represent the rotation. which generates a third order equation system for the minimal configurations. Three intermediate unknowns are introduced to simplify the equation system. We introduce a novel hidden variable method [@cox2006using] to solve the resulting second order equation system for the rotation matrix. Then the translation can be computed from a 