Solving systems of polynomial inequalities in subexponential time
Journal of Symbolic Computation
Gro¨bner bases: a computational approach to commutative algebra
Gro¨bner bases: a computational approach to commutative algebra
Linear N-Point Camera Pose Determination
IEEE Transactions on Pattern Analysis and Machine Intelligence
A Note on the Number of Solutions of the Noncoplanar P4P Problem
IEEE Transactions on Pattern Analysis and Machine Intelligence
A new efficient algorithm for computing Gröbner bases without reduction to zero (F5)
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
A complete symbolic-numeric linear method for camera pose determination
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
Complete Solution Classification for the Perspective-Three-Point Problem
IEEE Transactions on Pattern Analysis and Machine Intelligence
Journal of Mathematical Imaging and Vision
Complexity of the resolution of parametric systems of polynomial equations and inequations
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
A general sufficient condition of four positive solutions of the P3P problem
Journal of Computer Science and Technology
On the Probability of the Number of Solutions for the P4P Problem
Journal of Mathematical Imaging and Vision
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Solving parametric polynomial systems
Journal of Symbolic Computation
Variant real quantifier elimination: algorithm and application
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
Solution of algebraic riccati equations using the sum of roots
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
Proceedings of the 2009 conference on Symbolic numeric computation
Solving parametric piecewise polynomial systems
Journal of Computational and Applied Mathematics
An Algorithm for Finding Repeated Solutions to the General Perspective Three-Point Pose Problem
Journal of Mathematical Imaging and Vision
Variant quantifier elimination
Journal of Symbolic Computation
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
Critical points and Gröbner bases: the unmixed case
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
Computing rational solutions of linear matrix inequalities
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
A Fundamentally New View of the Perspective Three-Point Pose Problem
Journal of Mathematical Imaging and Vision
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Classifying the Perspective-Three-Point problem (abbreviated by P3P in the sequel) consists in determining the number of possible positions of a camera with respect to the apparent position of three points. In the case where the three points form an isosceles triangle, we give a full classification of the P3P. This leads to consider a polynomial system of polynomial equations and inequalities with 4 parameters which is generically zero-dimensional. In the present situation, the parameters represent the apparent position of the three points so that solving the problem means determining all the possible numbers of real solutions with respect to the parameters' values and give a sample point for each of these possible numbers. One way for solving such systems consists first in computing a discriminant variety. Then, one has to compute at least one point in each connected component of its real complementary in the parameter's space. The last step consists in specializing the parameters appearing in the initial system by these sample points. Many computational tools may be used for implementing such a general method, starting with the well known Cylindrical Algebraic Decomposition (CAD in short), which provides more information than required. In a first stage, we propose a full algorithm based on the straightforward use of some sophisticated software such as FGb (Grobner bases computations) RS (real roots of zero-dimensional systems), DV (Discriminant varieties) and RAGlib (Critical point methods for semi-algebraic systems). We then improve the global algorithm by refining the required computable mathematical objects and related algorithms and finally provide the classification. Three full days of computation were necessary to get this classification which is obtained from more than 40000 points in the parameter's space.