In random sample consensus (RANSAC), the problem of ellipsoid fitting can be
formulated as a problem of minimization of point-to-model distance, which is
realized by maximizing model score. Hence, the performance of ellipsoid fitting
is affected by distance metric. In this paper, we proposed a novel distance
metric called the axial distance, which is converted from the algebraic
distance by introducing a scaling factor to solve nongeometric problems of the
algebraic distance. There is complementarity between the axial distance and
Sampson distance because their combination is a stricter metric when
calculating the model score of sample consensus and the weight of the weighted
least squares (WLS) fitting. Subsequently, a novel sample-consensus-based
ellipsoid fitting method is proposed by using the combination between the axial
distance and Sampson distance (CAS). We compare the proposed method with
several representative fitting methods through experiments on synthetic and
real datasets. The results show that the proposed method has a higher
robustness against outliers, consistently high accuracy, and a speed close to
that of the method based on sample consensus.

本文提出了一种基于样本一致性的新型椭球拟合方法，使用轴向距离和 Sampson 距离的组合进行计算，在合成和实际数据集上的实验结果表明，所提出的方法具有更高的鲁棒性，一致的精度和接近于基于样本一致性方法的速度。

使用轴向距离和组合的稳健椭球拟合

Robust Ellipsoid Fitting Using Axial Distance and Combination

The fundamental matrix (FM) describes the geometric relations that exist
between two images of the same scene. Different error criteria are used for
estimating FMs from an input set of correspondences. In this paper, the
accuracy and efficiency aspects of the different error criteria were studied.
We mathematically and experimentally proved that the most popular error
criterion, the symmetric epipolar distance, is biased. It was also shown that
despite the similarity between the algebraic expressions of the symmetric
epipolar distance and Sampson distance, they have different accuracy
properties. In addition, a new error criterion, Kanatani distance, was proposed
and was proved to be the most effective for use during the outlier removal
phase from accuracy and efficiency perspectives. To thoroughly test the
accuracy of the different error criteria, we proposed a randomized algorithm
for Reprojection Error-based Correspondence Generation (RE-CG). As input, RE-CG
takes an FM and a desired reprojection error value $d$. As output, RE-CG
generates a random correspondence having that error value. Mathematical
analysis of this algorithm revealed that the success probability for any given
trial is 1 - (2/3)^2 at best and is 1 - (6/7)^2 at worst while experiments
demonstrated that the algorithm often succeeds after only one trial.

本研究比较了不同误差判定准则的准确度和效率，发现流行的对称极线距离存在偏差，提出了一种新的最有效的 Kanatani 距离准则，并使用随机算法为基于再投影误差的对应生成提供了一种评估方法。