Fitting concentric circles to measurements

Measurements for fitting a given number of concentric circles are recorded. For each concentric circle several measurements are taken. The problem is to fit the given number of circles to the data such that all circles have a common center. This is a generalization of the problem of fitting a set of points to one circle. Three objectives, to be minimized, are considered: the least squares of distances from the circles, the maximum distance from the circles, and the sum of the distances from the circles. Very efficient optimal solution procedures are constructed. Problems based on a total of 10,000 measurements are solved in about 10 s with the least squares objective, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$>$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mo>></mo> </math> </EquationSource> </InlineEquation>2 s with the maximum distance objective, and a little more than 1 min for the minisum objective. Copyright Springer-Verlag Berlin Heidelberg 2014