Zuzana Kukelova, Martin Bujnak, Jan Heller, Tomas Pajdla

The Art of Solving Minimal Problems Tricks: One more point?Microsoft Research CambridgeCzech Technical University in PragueCapturing Reality s.r.o.

CapturingReality

MotivationFor practical applications• sometimes more effective to use more than the minimal number of correspondences• simpler system of polynomial equations • can be solved faster, while maintaining numerical stability

Find a balance between the number of samples used in RANSAC and the speed of the solver

Usually one additional point is the best choice

Absolute pose of a camera with unknown focal length and radial distortionMinimal solution – P4Pfr[Josephson, Byröd. Pose estimation with radial distortion and unknown focal length. CVPR 2010]• 4 point correspondences, 24 solutions• LU - 1134×720 matrix, QR of a 56×56 matrix + eigenvalue computations of a 24×24

matrix

Non-minimal solution – P5Pfr[Kukelova, Bujnak, Pajdla. Real-time solution to the absolute pose problem with unknown radial distortion and focal length. ICCV 2013]• 5 point correspondences, 4 solutions• null space of a 5×8 matrix, solutions to a 4th degree polynomial +

inverse of a 3×3 matrix • 130x faster + numerically more stable than P4Pfr

Absolute pose of a camera with unknown focal length and radial distortion

The comparison of the total times of model computation and 2000 tentative matches verification in RANSAC loop and different outlier contaminations for the P4Pfr solver and the P5Pfr solver

Relative pose problem for cameras with different radial distortionsMinimal solution – F9[Kukelova, Byröd., Josephson, Pajdla, Åström. Fast and robust numerical solutions to minimal problems for cameras with radial distortion. CVIU 2010]• 9 point correspondences, 24 solutions• GJ – 9x16 + 179x203 matrix + eigenvalue computations of a 24×24 matrix

Non-minimal solution – F10• [Kukelova, Heller, Bujnak, Fitzgibbon, Pajdla. Efficient Solution to the Epipolar Geometry

for Radially Distorted Cameras. ICCV 2015]• 10 point correspondences, 10 solutions• GJ - 10×16 matrix, determinant of a 4x4 polynomial matrix, Sturm sequences• >1200x faster than F9