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Critical Configurations for Projective Reconstruction Fredrik Kahl

Critical Configurations for Projective Reconstruction Fredrik Kahl Joint work with Richard Hartley Chalmers University of Technology Lund University Oct 2015. Outline. Problem statement Two-view critical configurations Three views and more Conclusions. Unknown camera positions.

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Critical Configurations for Projective Reconstruction Fredrik Kahl

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  1. Critical Configurations for Projective Reconstruction Fredrik Kahl Joint work with Richard Hartley Chalmers University of Technology Lund University Oct 2015

  2. Outline • Problem statement • Two-view critical configurations • Three views and more • Conclusions

  3. Unknown camera positions Structure and Motion Problem • Given images, reconstruct: • Scene geometry (structure) • Camera positions (motion)

  4. When is the solution unique? Investigated previously by: • Krames (1940) • Buchanan (1988) • Maybank (1993) • Maybank & Shashua (1998) • Hartley & Kahl (2007) This work: Complete classification of all critical configurations in two and more views • Bertolini, Besana, Turrini (2007,2009,2015) • And others...

  5. Notation

  6. cone hyperboloid

  7. Proof based on a generalization of Pascal’s Theorem

  8. Pascal’s Theorem (1639) For generalization to quadrics, see: Richard Hartley, Fredrik Kahl, Critical Configurations for Projective Reconstruction from Multiple Views, International Journal of Computer Vision, 2007.

  9. N-view critical configurations • Given N>3 cameras and a point set, then critical iff each subset of three cameras and point set critical

  10. Open problem • What are the critical configurations for the calibrated case?

  11. Carlsson duality and critical configurations • Exchange role of points and cameras via a Cremona transformation • Dual configurations: • N cameras and M+4 points • M cameras and N+4 points • Example: ”2-view ambiguity and arbitrary points on a hyperboloid” is dual to ”arbitrary cameras and 6 points on a hyperboloid”

  12. Conclusions • Critical configurations for the structure and motion problem • Main criticalities: • (i) elliptic quartics (intersection of two quadratic surfaces) • (ii) rational quartic curve on a non-degenerate quadratic surface • (iii) twisted cubic ... • Projective geometry essential tool

  13. Thank you for your attention!

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