# On the Existence of Epipolar Matrices

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## Abstract

This paper considers the foundational question of the existence of a fundamental (resp. essential) matrix given *m* point correspondences in two views. We present a complete answer for the existence of fundamental matrices for any value of *m*. We disprove the widely held beliefs that fundamental matrices always exist whenever \(m \le 7\). At the same time, we prove that they exist unconditionally when \(m \le 5\). Under a mild genericity condition, we show that an essential matrix always exists when \(m \le 4\). We also characterize the six and seven point configurations in two views for which all matrices satisfying the epipolar constraint have rank at most one.

## Keywords

Structure from motion Epipolar geometry Algebraic geometry## References

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