Critical Configurations for
Projective Reconstruction
Fredrik Kahl
Joint work with Richard Hartley
Chalmers University of Technology
Lund University
Oct 2015
Outline
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Problem statement
Two-view critical configurations
Three views and more
Conclusions
Structure and Motion Problem
Unknown camera
positions
• Given images, reconstruct:
– Scene geometry (structure)
– Camera positions (motion)
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...
Notation
hyperboloid
cone
Proof based on a generalization of Pascal’s Theorem
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.
N-view critical configurations
• Given N>3 cameras and a point set,
then critical iff each subset of three
cameras and point set critical
Open problem
• What are the critical configurations for
the calibrated case?
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”
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 nondegenerate quadratic surface
– (iii) twisted cubic ...
• Projective geometry essential tool
Thank you for your attention!