FOE estimation: Can image measurement errors be totally "corrected" by the geometric method?

For the FOE estimation, there are basically three kinds of estimation methods in the literature: algebraic, geometric, and the maximum likelihood-based ones. In this paper, our attention is focused on the geometric method. The computational complexity of the classical geometric method is usually very high because it needs to solve a non-linear minimum problem with many variables. In this work, such a minimum problem is converted into an equivalent one with only two variables and accordingly a simplified geometric method is proposed. Based on the equivalence of the classical geometric method and the proposed simplified geometric method, we show that the measurement errors can at most be ''corrected'' only in one of the two images by geometric methods. In other words, it is impossible to correct the measurement errors in both of the two images. In addition, we show that the ''corrected'' corresponding pairs by geometric methods cannot in general meet some of the inherent constraints of corresponding pairs under pure camera translations. Hence, it is not proper to consider the ''corrected'' corresponding pairs as ''faithful'' corresponding pairs in geometric methods, and the estimated FOE from such pairs is not necessarily trustworthier. Finally, a new geometric algorithm, which automatically enforces the inherent constraints, is proposed in this work, and better FOE estimation and more faithful corresponding pairs are obtained.