Comparison of two sphere fitting methods

In this paper, we compare two algorithms for obtaining estimates of the center and radius of a sphere from measurements on the surface of the sphere. The first method is the maximum-likelihood solution of the problem which leads to the use of nonlinear least squares. The second method is an approximation of the first method but is computationally simpler, requiring only a linear least-squares solution. We show that the second method has problems estimating the parameters when the center of the sphere is far from the origin of the coordinate system, and the radius of the sphere is small. In many cases which would occur in practice, however, this method gives reliable estimates. The first method estimates reliably in all these cases. We also show that, although estimates of the precision of the parameter estimates can be found with each method, the second method tends to overestimate the variability of the parameters in cases that could easily occur in practice, while the first method again gives reliable estimates. We recommend that, for parameter estimation, the second (linear approximation) method should be applied first, with the parameter estimates it generates, then used as starting values for at least one iteration of more » the first (maximum likelihood) method. We further recommend that estimates of the variance of the parameters be derived using the techniques for the first (maximum likelihood) method. « less