That's a reasonable approach. However I also have missing data points, which means the center of gravity is not useful. I'm much more interested in finding the circle center (for a given r) than the actual "optimal" circle radius. First I set the line L as you said. Then if I iteratively move the line L in the direction of its normal and reflecting the points to the other side for each iteration, I could find the circle solution which minimizes my objective function.

I don't need to estimate the radius. Only the circle center. I would imagine that finding the circle with a fixed radius that best fits the data set is the most robust solution. However I was not able to work out the math for this problem.

The "noise" is quite predictable in its positions but is not always present for each measurement. Maybe it would help if I explain the problem a little better. I have about 50 data points in a semicircle, which have been extracted from edges in an image. Let's say the semi-circle ranges from 10-170 degrees. In the middle of the arc, i.e. at about 90 degrees there are some noisy data points in the shape of a bigger circle which has some influence on the final radius.