  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
kernel density estimation (KDE) of the input data. The ellipsoid fitting is
cast as a maximum likelihood estimation (MLE) problem without extra
constraints, where a weighting term is added to depress outliers, and then
effectively solved via the Expectation-Maximization (EM) framework.
Furthermore, we introduce the vector {\epsilon} 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 https://zikai1.github.io/.
