Fit sphere to planar circles

Hi all,

I'm trying to find the center and radius of a mostly spherical body in a voxel volume (femoral head in a CT scan, to be precise).

My approach so far has been to use hough circle detection in the area of interest, looped over each slice. This detection works best around the hemisphere (rather than pole) regions of the sphere, and I am using some logic to cull away any false positive circles detected by the hough transform.

My question is this: what is the best way to fit a sphere to a series of planar circles?

My data is currently in the form of:
circs = [...
240.5275 145.0197 -564.5000 20.5078;
241.1135 146.1915 -559.5000 22.8516;
241.1135 146.1915 -554.5000 23.4375;
241.6994 146.7775 -549.5000 23.4375;
241.1135 146.7775 -544.5000 22.2656;]

The columns are the X, Y, and Z locations of 5 circle origins, and their radii.

I'm confident that the average of the X and Y columns represent the X and Y sphere centre, but I don't know the best way to get the Z location.

A simple average of Z locations is inaccurate because the detected circles are not necessarily centrally detected along the Z-direction.

Any thoughts? Perhaps I could try to fit a curve to the radii values and find its peak, but I'm not sure how to fit anything robustly to so few data points.

Any help would be very welcome.
Thanks,
Sven.
.

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