Capturing Reality The Art of Solving Minimal Problems

Capturing. Reality The Art of Solving Minimal Problems Tricks: Problem Formulation Zuzana Kukelova, Martin Bujnak, Tomas Pajdla Microsoft Research Cambridge Czech Technical University in Prague Capturing Reality s. r. o.

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

The formulation of the problem may significantly influence • Complexity of the final “online” solver

Absolute pose of a camera with unknown focal length • Over-constrained system – 3. 5 points sufficient • Different strategies for selecting equations

Different formulations • Using distances • Using ratios of distances • Rotation matrix parametrized using quaternions • DIAC – dual image of the absolute quadric

Performance of different solvers and different formulations

How important is the problem formulation? • Success = good formulation + good solver • Finding a good solution • Easy with automatic solver generators one have to try many formulations & solvers

Using an automatic generator we can quite quickly test and evaluate different, formulations, different orderings of variables or monomials Polynomial system Evaluation Solver source code