Fitting a Model to Noisy Data Using Low-Order Implicit Curves and Surfaces

Fitting a compact model to measured data that captures the underlying relationship is a fundamental task in computer graphics and computer-aided design. Low-order implicit curves and surfaces are a practical choice in grasping this relationship since they are closed under several geometric operations (e.g. intersection, union, offset) while they offer a higher degree of smoothness than their parametric counterparts, and may be preferred especially if the object under study itself is a composition of geometric shapes. We present a method based on a blend of iterative maximum likelihood approximation of linear and quadratic curves and surfaces (with constraints), and of an alternating optimization scheme in the flavor of the standard algorithm for k-means. The algorithm alternates between two steps: (1) fitting a set of linear and quadratic curves and surfaces to previously identified groups of noisy data points, and (2) identifying new groups by assignment to the most feasible shape. Non-iterative direct methods are proposed to seed the maximum likelihood estimator with initial parameter values.