Abstract: Ellipsoid fitting is of general interest in machine vision, such as object detection and shape approximation. Most existing approaches rely on the least-squares fitting of quadrics, minimizing the algebraic or geometric distances, with additional constraints to enforce the quadric as an ellipsoid. However, they are susceptible to outliers and non-ellipsoid or biased results when the axis ratio exceeds certain thresholds. To address these problems, we propose a novel and robust method for ellipsoid fitting in a noisy, outlier-contaminated 3D environment. We explicitly model the ellipsoid by emph{kernel density estimation} (KDE) of the input data. The ellipsoid fitting is cast as a emph{maximum likelihood estimation} (MLE) problem without extra constraints, where a weighting term is added to depress outliers, and then effectively solved via the emph{Expectation-Maximization (EM)} framework. Furthermore, we introduce the emph{vector $varepsilon$ technique} to accelerate the convergence of the original EM. The proposed method is compared with representative state-of-the-art approaches by extensive experiments, and results show that our method is ellipsoid-specific, parameter free, and more robust against noise, outliers, and the large axis ratio. Our implementation is available at url{https://zikai1.github.io/}.