A Novel Error Criterion of Fundamental Matrix Based on Principal Component Analysis

Abstract

Estimating the fundamental matrix (FM) using the known corresponding points is a key step for three-dimensional (3D) scene reconstruction, and its uncertainty directly affects camera calibration and point-cloud calculation. The symmetric epipolar distance is the most popular error criterion for estimating FM error, but it depends on the accuracy, number, and distribution of known corresponding points and is biased. This study mainly focuses on the error quantitative criterion of FM itself. First, the calculated FM process is reviewed with the known corresponding points. Matrix differential theory is then used to derive the covariance equation of FMs in detail. Subsequently, the principal component analysis method is followed to construct the scalar function as a novel error criterion to measure FM error. Finally, three experiments with different types of stereo images are performed to verify the rationality of the proposed method. Experiments found that the scalar function had approximately 90% correlation degree with the Manhattan norm, and greater than 80% with the epipolar geometric distance. Consequently, the proposed method is also appropriate for estimating FM error, in which the error ellipse or normal distribution curve is the reasonable error boundary of FM. When the error criterion value of this method falls into a normal distribution curve or an error ellipse, its corresponding FM is considered to have less error and be credible. Otherwise, it may be necessary to recalculate an FM to reconstruct 3D models.

Keywords

• fundamental matrix
• error
• covariance
• matrix differential theory
• principal component analysis
• Manhattan norm