Zuzana Kukelova, Martin Bujnak, Tomas Pajdla

The Art of Solving Minimal Problems Tricks: Problem FormulationMicrosoft Research CambridgeCzech Technical University in PragueCapturing Reality s.r.o.

CapturingReality

How important is the problem formulation?

“The mere formulation of a problem is often far more essential than its solution, which may be merely a matter of mathematical or experimental skills. ….”

- Albert Einstein

Problem formulationThe formulation of the problem may significantly influence

• Complexity of the final “online” solver

• Sparsity of the elimination template

• Numerical stability

• Number of solutions

P4P+fAbsolute pose of a camera with unknown focal length

• Over-constrained system – 3.5 points sufficient• Different strategies for selecting equations

P4P+fDifferent formulations• Using distances

• Using ratios of distances

• Rotation matrix parametrized using quaternions

• DIAC – dual image of the absolute quadric

P4P+fPerformance of different solvers and different formulations

Problem formulationHow important is the problem formulation?

• Success = good formulation + good solver

• Finding a good solution one have to try many formulations & solvers

• Easy with automatic solver generators

P4P+fUsing an automatic generator we can quite quickly test and evaluate different, formulations, different orderings of variables or monomials

Polynomial system

Evaluation Solver source code