Critical Configurations for Projective Reconstruction Fredrik Kahl Joint work with Richard Hartley...
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![Page 1: Critical Configurations for Projective Reconstruction Fredrik Kahl Joint work with Richard Hartley Chalmers University of Technology Lund University Oct.](https://reader036.fdocuments.us/reader036/viewer/2022081512/5a4d1b807f8b9ab0599badea/html5/thumbnails/1.jpg)
Critical Configurations for Projective Reconstruction
Fredrik Kahl
Joint work with Richard Hartley
Chalmers University of TechnologyLund University
Oct 2015
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• Problem statement• Two-view critical configurations• Three views and more• Conclusions
Outline
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• Given images, reconstruct: – Scene geometry (structure)– Camera positions (motion)
Unknown cameraUnknown camerapositionspositions
Structure and Motion Problem
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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
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Notation
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hyperboloid cone
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Proof based on a generalization of Pascal’s Theorem
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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.
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N-view critical configurations
• Given N>3 cameras and a point set, then critical iff each subset of three cameras and point set critical
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Open problem
• What are the critical configurations for the calibrated case?
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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”
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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
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Thank you for your attention!