Calibration by correlation using metric embedding from non-metric similarities

Abstract

This paper presents a new intrinsic calibration method that allows us to calibrate a generic single-view point camera. From the video sequence obtained while the camera undergoes random motion, we compute the pairwise time correlation of the luminance signal for the pixels. We show that the pairwise correlation of any pixels pair is a function of the distance between the pixel directions on the visual sphere. This leads to formalizing calibration as a problem of metric embedding from non-metric measurements: we want to find the disposition of pixels on the visual sphere, from similarities that are an unknown function of the distances. This problem is a generalization of multidimensional scaling (MDS) that has so far resisted a comprehensive observability analysis and a generic solution. We show that the observability depends both on the local geometric properties as well as on the global topological properties of the target manifold. It follows that, in contrast to the Euclidean case, on the sphere we can recover the scale of the points distribution. We describe an algorithm that is robust across manifolds and can recover a metrically accurate solution when the metric information is observable. We demonstrate the performance of the algorithm for several cameras (pin-hole, fish-eye, omnidirectional).