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 Σ. 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 by means of a smaller number of iterations of the DIRECT global optimization algorithm. Unlike other methods known from the literature, our method recognizes well not only ellipses with clear edges, but also ellipses with noisy edges. CPU-time necessary for running the corresponding algorithm is very short and this raises hope that, with appropriate software optimization, the algorithm could be run in real time. The method is illustrated and tested on 100 randomly generated data sets.

中文翻译：

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$k$-means 算法通过使用由 DIRECT 全局优化算法获得的初始近似值来解决多重椭圆检测问题

我们基于一个数据点集来考虑多个椭圆检测问题，该数据点集来自平面中事先未知的多个椭圆，其中椭圆 E 被视为一个马氏圆，其中心为 S、半径为 r 和一些正定的矩阵Σ。提出了一种非常有效的方法来解决这个问题。该方法使用对马哈拉诺比斯圆中心的 k 均值算法的修改。初始近似值由一组圆组成，这些圆的中心是通过 DIRECT 全局优化算法的较少迭代次数确定的。与文献中已知的其他方法不同，我们的方法不仅可以很好地识别具有清晰边缘的椭圆，还可以很好地识别具有噪声边缘的椭圆。运行相应算法所需的 CPU 时间非常短，这带来了希望，通过适当的软件优化，该算法可以实时运行。该方法在 100 个随机生成的数据集上进行了说明和测试。