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!