Fast alternating fitting methods for trigonometric curves for large data sets

This paper discusses and develops new methods for fitting trigonometric curves, such as circles, ellipses, and dumbbells, to data points in the plane. Available methods for fitting circles or ellipses are very sensitive to outliers in the data, and are time consuming when the number of data points is large. The present paper focuses on curve fitting methods that are attractive to use when the number of data points is large. We propose a direct method for fitting circles, and two iterative methods for fitting ellipses and dumbbell curves based on trigonometric polynomials. These methods efficiently minimize the sum of the squared geometric distances between the given data points and the fitted curves. In particular, we are interested in detecting the general shape of an object such as a galaxy or a nebula. Certain nebulae, for instance, the one shown in the experiment section, have a dumbbell shape. Methods for fitting dumbbell curves have not been discussed in the literature. The methods developed are not very sensitive to errors in the data points. The use of random subsampling of the data points to speed up the computations also is discussed. The techniques developed in this paper can be applied to fitting other kinds of curves as well.