The Analysis of Ambiguous Solutions in Linear Systems and its Application to Computer Vision

Marta Wilczkowiak 1 Peter Sturm 1 Edmond Boyer 1
1 MOVI - Modeling, localization, recognition and interpretation in computer vision
GRAVIR - IMAG - Graphisme, Vision et Robotique, Inria Grenoble - Rhône-Alpes, CNRS - Centre National de la Recherche Scientifique : FR71
Abstract : The success of practical systems based on computer vision algorithms may critically depend on the correct treatment of degenerate configurations and missing data. However, even if theoretical basis exist, it appears that few efforts have been made to solve these problems in practice. In this paper, we address estimation problems based on linear constraints and for which degenerate cases are in principle easy to detect. Many algorithms, in particular in computer vision, either do not detect them or simply stop and produce no output. In many cases however, degenerate situations nevertheless allow a reliable estimation of a subset of the unknowns. We present a practical approach for splitting the variable set of a degenerate linear system into underconstrained and well defined variables. It means that even if the system as a whole is underconstrained, we can still extract useful information and give the correct solution for a subset of the unknowns. Our method is based on singular value decompositions (SVD). Using a very simple analysis of the matrix nullspace, it becomes easy and fast to split the variables into uniquely defined and ambiguous ones. To illustrate our approach, we present its applications to a novel iterative 3D reconstruction algorithm as well as to plane-based camera calibration.
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Marta Wilczkowiak, Peter Sturm, Edmond Boyer. The Analysis of Ambiguous Solutions in Linear Systems and its Application to Computer Vision. British Machine Vision Conference (BMVC '03), Sep 2003, Norwich, United Kingdom. pp.53-62. ⟨inria-00525637⟩

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