critical configurations for projective reconstruction fredrik kahl joint work with richard hartley...
DESCRIPTION
Given images, reconstruct: –Scene geometry (structure) –Camera positions (motion) Unknown camera positions Structure and Motion ProblemTRANSCRIPT
Critical Configurations for Projective Reconstruction
Fredrik Kahl
Joint work with Richard Hartley
Chalmers University of TechnologyLund University
Oct 2015
• Problem statement• Two-view critical configurations• Three views and more• Conclusions
Outline
• Given images, reconstruct: – Scene geometry (structure)– Camera positions (motion)
Unknown cameraUnknown camerapositionspositions
Structure and Motion Problem
Investigated previously by:• Krames (1940)
• Hartley & Kahl (2007)
• Buchanan (1988)• Maybank (1993)• Maybank & Shashua (1998)
When is the solution unique?
• Bertolini, Besana, Turrini (2007,2009,2015)
• And others...
This work: Complete classification of all critical configurations in two and more views
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 non-degenerate quadratic surface
– (iii) twisted cubic ...• Projective geometry essential tool
Thank you for your attention!