Robust ellipse fitting based on sparse combination of data points

Ellipse fitting is widely applied in the fields of computer vision and automatic industry control, in which the procedure of ellipse fitting often follows the preprocessing step of edge detection in the original image. Therefore, the ellipse fitting method also depends on the accuracy of edge detection besides their own performance, especially due to the introduced outliers and edge point errors from edge detection which will cause severe performance degradation. In this paper, we develop a robust ellipse fitting method to alleviate the influence of outliers. The proposed algorithm solves ellipse parameters by linearly combining a subset of ('more accurate') data points (formed from edge points) rather than all data points (which contain possible outliers). In addition, considering that squaring the fitting residuals can magnify the contributions of these extreme data points, our algorithm replaces it with the absolute residuals to reduce this influence. Moreover, the norm of data point errors is bounded, and the worst case performance optimization is formed to be robust against data point errors. The resulting mixed $l1\hbox{-}l2$ optimization problem is further derived as a second-order cone programming one and solved by the computationally efficient interior-point methods. Note that the fitting approach developed in this paper specifically deals with the overdetermined system, whereas the current sparse representation theory is only applied to underdetermined systems. Therefore, the proposed algorithm can be looked upon as an extended application and development of the sparse representation theory. Some simulated and experimental examples are presented to illustrate the effectiveness of the proposed ellipse fitting approach.