A new efficient method for solving the multiple ellipse detection problem

In this paper, we consider the multiple ellipse detection problem based on data points coming from a number of ellipses in the plane not known in advance. In so doing, data points are usually contaminated with some noisy errors. In this paper, the multiple ellipse detection problem is solved as a center-based problem from cluster analysis. Therefore, an ellipse is considered a Mahalanobis circle. In this way, we easily determine a distance from a point to the ellipse and also an ellipse as the cluster center. In the case when the number of ellipses is known in advance, an optimal partition is searched for on the basis of the k-means algorithm that is modified for this case. Hence, a good initial approximation for M-circle-centers is searched for as unit circles with the application of a few iterations of the well-known DIRECT algorithm for global optimization. In the case when the number of ellipses is not known in advance, optimal partitions with 1,2,… clusters for the case when cluster-centers are ellipses are determined by using an incremental algorithm. Among them, the partition with the most appropriate number of clusters is selected. For that purpose, a new Geometrical Objects-index (GO-index) is defined. Numerous test-examples point to high efficiency of the proposed method. Many algorithms can be found in the literature that recognize ellipses with clear edges well, but that do not recognize ellipses with unclear or noisy edges. On the other hand, our algorithm is specifically used for recognition of ellipses with unclear or noisy edges.