I'm looking for an algorithm to find the best fit between a cloud of points and a sphere.
That is, I want to minimise
where C is the centre of the sphere, r its radius, and each P a point in my set of n points. The variables are obviously Cx, Cy, Cz, and r. In my case, I can obtain a known r beforehand, leaving only the components of C as variables.
I really don't want to have to use any kind of iterative minimisation (e.g. Newton's method, Levenberg-Marquardt, etc) - I'd prefer a set of linear equations or a solution explicitly using SVD.
There are no matrix equations forthcoming. Your choice of E is badly behaved; its partial derivatives are not even continuous, let alone linear. Even with a different objective, this optimization problem seems fundamentally non-convex; with one point P and a nonzero radius r, the set of optimal solutions is the sphere about P.
You should probably reask on an exchange with more optimization knowledge.
Short description of making matrix equation could be found here.
I've seen that WildMagic Library uses iterative method (at least in version 4)
The linked thesis contains a problem : it doesn't handle noise properly and overestimates the radius. (Centre is OK). It looks like it should be possible to come up with a correction based upon the estimate of the applied noise.
来源：https://stackoverflow.com/questions/10344119/linear-least-squares-fit-of-sphere-to-points