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Globally Optimal Estimates for Geometric Reconstruction Problems

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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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Authors and Affiliations

  1. Computer Science and Engineering, University of California San Diego, San Diego, USA

    Fredrik Kahl

  2. Centre for Mathematical Sciences, Lund University, Lund, Sweden

    Fredrik Kahl

  3. LAAS-CNRS, Toulouse, France

    Didier Henrion

  4. Faculty of Electrical Engineering, Czech Technical University in Prague, Prague, Czech Republic

    Didier Henrion

Authors
  1. Fredrik Kahl
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  2. Didier Henrion
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Correspondence to Fredrik Kahl.

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Kahl, F., Henrion, D. Globally Optimal Estimates for Geometric Reconstruction Problems. Int J Comput Vision 74, 3–15 (2007). https://doi.org/10.1007/s11263-006-0015-y

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  • DOI: https://doi.org/10.1007/s11263-006-0015-y

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