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
Zuzana Kúkelová 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
Zuzana Kúkelová 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
Zuzana Kúkelová 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
Zuzana Kúkelová 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
Zuzana Kúkelová 6/11 Six-point solver (Stewénius et al) - elimination procedure 9 equations from trace constraint and, and.
Zuzana Kúkelová 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.
Zuzana Kúkelová 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.
Zuzana Kúkelová 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.
Zuzana Kúkelová 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
Zuzana Kúkelová 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