Least Squares Orthogonal Distance Fitting

Abstract

The Least Squares Orthogonal Distance Fitting method has been motivated by the desire to produce a fitted result that is independent of the location and orientation of the object being fitted. It is typically used to fit various types of spline curves to a set of discrete experimental measurements of an object’s shape, but we will use it instead to fit a smooth parametric function whose shape can be continuously changed. We begin with a definition of residual error, and the corresponding error functional, which is to be optimized. The nature of the optimized solution can be characterized using the eigenvalues of the second order response matrix. The solution method can be decomposed into two nonlinear, nested, subproblems. The first problem is to minimize the distance to the spline curve at any given point. The second problem is to calculate the response to changes in the spline’s parameters, while maintaining the minimization of this distance at all points. The interaction between these two subproblems is described. Numerical convergence issues are outlined.