# Fundamental Matrix Estimation: A Study of Error Criteria

## TL;DR

This paper compares various error criteria for fundamental matrix estimation, revealing biases in popular methods, introducing a new effective criterion, and proposing a randomized algorithm for generating correspondences with specific reprojection errors.

## Contribution

The study provides a mathematical and experimental comparison of error criteria, introduces the Kanatani distance, and develops the RE-CG algorithm for testing accuracy in FM estimation.

## Key findings

- Symmetric epipolar distance is biased.
- Sampson distance differs in accuracy from symmetric epipolar distance.
- Kanatani distance is most effective for outlier removal.

## Abstract

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.