Kruppa Equation Revisited: Its Renormalization and Degeneracy

In this paper, we study general questions about the solvability of the Kruppa equations and show that, in several special cases, the Kruppa equations can be renormalized and become linear. In particular, for cases when the camera motion is such that its rotation axis is parallel or perpendicular to translation, we can obtain linear algorithms for self-calibration. A further study of these cases not only reveals generic difficulties with degeneracy in conventional self-calibration methods based on the nonlinear Kruppa equations, but also clarifies some incomplete discussion in the literature about the solutions of the Kruppa equations. We demonstrate that Kruppa equations do not provide sufficient constraints on camera calibration and give a complete account of exactly what is missing in Kruppa equations. In particular, a clear relationship between the Kruppa equations and chirality is revealed. The results then resolve the discrepancy between the Kruppa equations and the necessary and sufficient condition for a unique calibration. Simulation results are presented for evaluation of the sensitivity and robustness of the proposed linear algorithms.

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