Abstract
The so-called essential matrix relates corresponding points of two images from the same scene in 3D, and allows to solve the relative pose problem for the two cameras up to a global scaling factor, if the camera calibrations are known. We will discuss how Hensel’s lemma from number theory can be used to find geometric approximations to solutions of the equations describing the essential matrix. Together with recent p-adic classification methods, this leads to RanSaC p , a p-adic version of the classical RANSAC in stereo vision. This approach is motivated by the observation that using p-adic numbers often leads to more efficient algorithms than their real or complex counterparts.
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In fact, this dream became true thanks to Grothendieck’s concept of scheme: The “Riemann surface” is the affine scheme \(\mathrm{Spec}\mathbb{Z}\), the space whose points are the prime ideals \(p\mathbb{Z}\) for p = 0 or a prime number. Cf. e.g. (Hartshorne 1993)
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Bradley, P.E. (2012). p-adic Methods in Stereo Vision. In: Gaul, W., Geyer-Schulz, A., Schmidt-Thieme, L., Kunze, J. (eds) Challenges at the Interface of Data Analysis, Computer Science, and Optimization. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24466-7_18
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DOI: https://doi.org/10.1007/978-3-642-24466-7_18
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