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Topic: Least Squares fit for a sphere
Replies: 0

 Dave Rudolf Posts: 37 Registered: 10/3/05
Least Squares fit for a sphere
Posted: Apr 23, 2009 8:46 PM

Hi all,

I am looking for ways to compute the least-squares fit for a sphere
given any set of points. The points will have noise on them, and they
might be lob-sided to one part of the sphere. Of course, the quality
of the fit depends on how much noise and how lob-sided the points are,
etc.

At the moment, I have something that works using a global optimization
method (BTRK, to be precise) that is minimizing the points' distances
from the sphere's circle iteratively. This works pretty well, but is
slow.

So, I am looking into faster ways to get the job done. I came across
one method based on what looks like a standard SVD-based method:

http://nvl.nist.gov/pub/nistpubs/jres/103/6/j36sha.pdf

So, the idea is to build up a matrix M, decompose with SVD, then pick
the row of M with the smallest non-zero singular value. Correct me if
I am wrong so far.

Most of my confusion is on what exactly M is. How do I build M? He
loses me on the third-last paragraph on page 634. He states that

grad(J) = 2( M M^T) a

First of all, I don't follow how he got that. In this case, he states
that M is the set of points. That somewhat makes sense for the example
of fitting a plane to the set of points, but I don't see how he
derived that.

Further down, he states what the objective function is for a sphere,
but I can't see how to build M for this case.

Any clarification would be greatly appreciated.

Dave