  Estimating the relative camera pose from $n \geq 5$ correspondences between
two calibrated views is a fundamental task in computer vision. This process
typically involves two stages: 1) estimating the essential matrix between the
views, and 2) disambiguating among the four candidate relative poses that
satisfy the epipolar geometry. In this paper, we demonstrate a novel approach
that, for the first time, bypasses the second stage. Specifically, we show that
it is possible to directly estimate the correct relative camera pose from
correspondences without needing a post-processing step to enforce the
cheirality constraint on the correspondences. Building on recent advances in
certifiable non-minimal optimization, we frame the relative pose estimation as
a Quadratically Constrained Quadratic Program (QCQP). By applying the
appropriate constraints, we ensure the estimation of a camera pose that
corresponds to a valid 3D geometry and that is globally optimal when certified.
We validate our method through exhaustive synthetic and real-world experiments,
confirming the efficacy, efficiency and accuracy of the proposed approach. Code
is available at https://github.com/javrtg/C2P.
