The performance of ellipse fitting may significantly degrade in the presence of outliers, which can be caused by occlusion of the object, mirror reflection or other objects in the process of edge detection. In this paper, we propose an ellipse fitting method that is robust against the outliers, and thus maintaining stable performance when outliers can be present. We formulate an optimization problem for ellipse fitting based on the maximum entropy criterion (MCC), having the Laplacian as the kernel function from the well-known fact that the ℓ1\ell _{1} -norm error measure is robust to outliers. The optimization problem is highly nonlinear and non-convex, and thus is very difficult to solve. To handle this difficulty, we divide it into two subproblems and solve the two subproblems in an alternate manner through iterations. The first subproblem has a closed-form solution and the second one is cast as a convex second-order cone program (SOCP) that can reach the global solution. By so doing, the alternate iterations always converge to an optimal solution, although it can be local instead of global. Furthermore, we propose a procedure to identify failed fitting of the algorithm caused by local convergence to a wrong solution, and thus, it reduces the probability of fitting failure by restarting the algorithm at a different initialization. The proposed robust ellipse fitting method is next extended to the coupled ellipses fitting problem. Both simulated and real data verify the superior performance of the proposed ellipse fitting method over the existing methods.

中文翻译：

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基于拉普拉斯核的最大熵准则的鲁棒椭圆拟合


在存在异常值的情况下，椭圆拟合的性能可能会显着下降，这可能是由于边缘检测过程中物体、镜面反射或其他物体的遮挡造成的。在本文中，我们提出了一种对异常值具有鲁棒性的椭圆拟合方法，从而在存在异常值时保持稳定的性能。我们基于最大熵准则（MCC）制定了椭圆拟合的优化问题，以拉普拉斯算子作为核函数，基于众所周知的事实，即 ℓ1\ell _{1} -范数误差测量对异常值具有鲁棒性。优化问题是高度非线性和非凸的，因此求解起来非常困难。为了解决这个困难，我们将其分为两个子问题，并通过迭代交替地解决这两个子问题。第一个子问题有一个封闭形式的解，第二个子问题被转换为可以达到全局解的凸二阶锥程序（SOCP）。通过这样做，交替迭代总是收敛到最优解，尽管它可以是局部的而不是全局的。此外，我们提出了一种过程来识别由于局部收敛到错误解而导致的算法拟合失败，因此，通过在不同的初始化时重新启动算法来降低拟合失败的概率。接下来将所提出的鲁棒椭圆拟合方法扩展到耦合椭圆拟合问题。模拟和实际数据都验证了所提出的椭圆拟合方法相对于现有方法的优越性能。