The presence of radically irregular data points (RIDPs), which are referred to as the subset of measurements that represents no or little information, can significantly degrade the performance of ellipse fitting methods. We develop an ellipse fitting method that is robust to RIDPs based on the maximum correntropy criterion with variable center (MCC-VC), where an adaptable Laplacian kernel is used. For single ellipse fitting, we formulate a non-convex optimization problem and divide it into two subproblems, one to estimate the kernel bandwidth and the other the kernel center. We design sufficiently accurate convex approximation to each subproblem that will lead to computationally efficient closed-form solutions. The two subproblems are solved in an alternate manner until convergence is reached. We also investigate coupled ellipses fitting. While there exist multiple ellipses fitting methods in the literature, we develop a coupled ellipses fitting method by exploiting the underlying special structure, where the associations between the data points and ellipses are absent in the problem. The proposed method first introduces an association vector for each data point and then formulates a non-convex mixed-integer optimization problem to establish the data associations, which is approximately solved by relaxing it into a second-order cone program. Using the estimated data associations, we then extend the proposed single ellipse fitting method to accomplish the final coupled ellipses fitting. The proposed method is shown to perform significantly better than the existing methods using both simulated data and real images.