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A Minimal Solution for Relative Pose with Unknown Focal Length

A Minimal Solution for Relative Pose with Unknown Focal Length. Henrik Stewenius, David Nister, Fredrik Kahl, Frederik Schaffalitzky Presented by Zuzana Kukelova. Six-point solver (Stew énius et al ) – posing the problem. The linear equations from the epipolar constraint

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A Minimal Solution for Relative Pose with Unknown Focal Length

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  1. A Minimal Solution for Relative Pose with Unknown FocalLength Henrik Stewenius, David Nister, Fredrik Kahl, Frederik Schaffalitzky Presented by Zuzana Kukelova

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

  3. 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 Kúkelovákukelova@cmp.felk.cvut.cz

  4. 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 Kúkelovákukelova@cmp.felk.cvut.cz

  5. 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 Kúkelovákukelova@cmp.felk.cvut.cz

  6. Six-point solver (Stewénius et al)- elimination procedure • 9 equations from trace constraint and , and . Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

  7. 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 Kúkelovákukelova@cmp.felk.cvut.cz

  8. 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 Kúkelovákukelova@cmp.felk.cvut.cz

  9. 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 Kúkelovákukelova@cmp.felk.cvut.cz

  10. 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 Kúkelovákukelova@cmp.felk.cvut.cz

  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 Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

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