The best circle fitting algorithm

Question

I need a very precise algorithm for fitting a circle to the set of data points (actually I need to determine the center). The data comes after the binarization and segmentation

Answer 1

"Best" depends on the kind of noise in the input data. The problem is trivial if there is no noise in the source data points: Just pick 3 points and calculate the circle.

If you expect normal distributed, independent translations of each data point, then a least mean squares algorithm should be optimal. The data points should fit the equation:

(x - xm)^2 + (y - ym)^2 = r^2

where xm, ym, r are unknown, so:

x^2 - 2*x*xm + xm^2 + y^2 - 2*y*ym + ym^2 = r^2

substitute c for r^2-xm^2-ym^2 and you have an overdetermined system of linear equations:

2*x*xm + 2*y*ym = c - x^2 - y^2

Any good linear algebra library (e.g. IPP) can solve that for you.

If you expect outliers in the data, I would suggest using a RANSAC-strategy to find the set of non-outlier points, then use the algorithm above to find the exact center for that set.