A Stability Analysis of the Fundamental Matrix

The Fundamental matrix is a key concept when working with uncalibrated images and multiple viewpoints. It contains all the available geometric information and enables to recover the epipolar geometry from uncalibrated perspective views. This paper is about a stability analysis for the Fundamental matrix. We first present a probabilistic approach which works well. This approch, however, does not give insight into the causes of unstability. Two complementary explanations for unstability are the nature of the motions, and the interaction between motion and three-dimensional structure, which is characterized by a critical surface. Practical methods to characterize the proximity to the critical surface from image measurements, by estimating a quadratic transformation, are developped. They are then used for experiments which validate our observations. It turns out that surprisingly enough, the critical surface affects the stability of the fundamental matrix in a significant number of situations.