1. Center for Machine Perception Department of Cybernetics, Faculty of Electrical Engineering Czech Technical University in Prague A Minimal Solution for Relative Pose with Unknown Focal Length Henrik Stewenius, David Nister, Fredrik Kahl, Frederik Schaffalitzky Presented by Zuzana Kukelova

2. Zuzana Kúkelová kukelova@cmp.felk.cvut.cz 2/11 Six-point solver (Stewénius et al) – posing the problem  The linear equations from the epipolar constraint  Parameterize the fundamental matrix with three unknowns  F i – basic vectors of the null-space  Solve for F up to scale => x = 1

3. Zuzana Kúkelová kukelova@cmp.felk.cvut.cz 3/11 Six-point solver (Stewénius et al) – posing the problem  Substitute this representation of F into the rank constraint  and the trace constraint  where and

4. Zuzana Kúkelová kukelova@cmp.felk.cvut.cz 4/11 Six-point solver (Stewénius et al) – posing the problem  10 polynomial equations in 3 unknowns – y,z,w (1 cubic and 9 of degree 5)  10 equations can be written in a matrix form  where M is a 10x33 coefficient matrix and X is a vector of 33 monomials

5. Zuzana Kúkelová kukelova@cmp.felk.cvut.cz 5/11 Six-point solver (Stewénius et al) - computing the Gröbner basis  Compute the Gröbner basis using Gröbner basis elimination procedure  Generate polynomials from the ideal  Add these polynomials to the set of original polynomial equations  Perform Gauss-Jordan elimination  Repeat and stop when a complete Gröbner basis is obtained  These computations (Gröbner basis elimination procedure) can be once made in a finite prime field to speed them up - offline  The same solver (the same sequence of eliminations) can be then applied to the original problem in - online

6. Zuzana Kúkelová kukelova@cmp.felk.cvut.cz 6/11 Six-point solver (Stewénius et al) - elimination procedure  9 equations from trace constraint and, and.

7. Zuzana Kúkelová kukelova@cmp.felk.cvut.cz 7/11 Six-point solver (Stewénius et al) - elimination procedure  The previous system after a Gauss-Jordan step and adding new equations based on multiples of the previous equations.

8. Zuzana Kúkelová kukelova@cmp.felk.cvut.cz 8/11 Six-point solver (Stewénius et al) - elimination procedure  The previous system after a Gauss-Jordan step and adding new equations based on multiples of the previous equations.

9. Zuzana Kúkelová kukelova@cmp.felk.cvut.cz 9/11 Six-point solver (Stewénius et al) - elimination procedure  Gauss-Jordan eliminated version of the previous system. This set of equations is a Gröbner basis.

10. Zuzana Kúkelová kukelova@cmp.felk.cvut.cz 10/11 Six-point solver (Stewénius et al) - action matrix  Construction of the 15x15 action matrix for multiplication by one of the unknowns  extracting the correct elements from the eliminated 18x33 matrix and organizing them

11. Zuzana Kúkelová kukelova@cmp.felk.cvut.cz 11/11 Six-point solver (Stewénius et al) - extract solutions  The eigenvectors of the action matrix give solutions for  Using back-substitution we obtain solutions for F and f  We obtain 15 complex solutions