1 / 11

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

maxine-mckay
Download Presentation

A Minimal Solution for Relative Pose with Unknown Focal Length

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related