Globally Optimal Estimates for Geometric Reconstruction Problems |
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Authors: | Fredrik Kahl Didier Henrion |
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Affiliation: | (1) Computer Science and Engineering, University of California San Diego, San Diego, USA;(2) Centre for Mathematical Sciences, Lund University, Lund, Sweden;(3) LAAS-CNRS, Toulouse, France;(4) Faculty of Electrical Engineering, Czech Technical University in Prague, Prague, Czech Republic |
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Abstract: | We introduce a framework for computing statistically optimal estimates of geometric reconstruction problems. While traditional algorithms often suffer from either local minima or non-optimality—or a combination of both—we pursue the goal of achieving global solutions of the statistically optimal cost-function. Our approach is based on a hierarchy of convex relaxations to solve non-convex optimization problems with polynomials. These convex relaxations generate a monotone sequence of lower bounds and we show how one can detect whether the global optimum is attained at a given relaxation. The technique is applied to a number of classical vision problems: triangulation, camera pose, homography estimation and last, but not least, epipolar geometry estimation. Experimental validation on both synthetic and real data is provided. In practice, only a few relaxations are needed for attaining the global optimum. |
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Keywords: | non-convex optimization structure from motion triangulation LMI relaxations global optimization semidefinite programming |
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