Least-square curve and surface localization for shape conformance checking

Verification of shape conformance for free-form curves and surfaces is commonly achieved by minimizing the sum of square deviations between measured points and a nominal curve/surface, thereby solving an optimal parameter estimation (OPE) problem. Finding the optimal rigid body (ORB) transformation between the measured points and nominal surface, an important step in the OPE problem, traditionally has involved iteratively solving a nonlinear optimization problem in six variables. This paper demonstrates that the optimization problem in six variables may be reduced to solving four degree-two implicit equations in four variables, which can be regarded as an eigenvalue problem. This results in considerable savings in the number of computations. A thorough analysis of the savings in computations and several examples are presented.