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The paper gives a new interpretation and a possible optimization of the well-known $k$-means algorithm for searching for a locally optimal partition of the set $𝒜=\{a_i\in {ℝ}^n:i=1,\cdots ,m\}$ which consists of $k$ disjoint nonempty subsets ${\pi }_1,\cdots ,{\pi }_k$, $1\le k\le m$. For this purpose, a new divided $k$-means algorithm was constructed as a limit case of the known smoothed $k$-means algorithm. It is shown that the algorithm constructed in this way coincides with the $k$-means algorithm if during the iterative procedure no data points appear in the Voronoi diagram....

We consider the multiple ellipses detection problem on the basis of a data points set coming from a number of ellipses in the plane not known in advance, whereby an ellipse $E$ is viewed as a Mahalanobis circle with center $S$, radius $r$, and some positive definite matrix $\Sigma$. A very efficient method for solving this problem is proposed. The method uses a modification of the $k$-means algorithm for Mahalanobis-circle centers. The initial approximation consists of the set of circles whose centers are determined...