## Abstract

We introduce a number of new results in the context of multi-view geometry from general algebraic curves. We start with the recovery of camera geometry from matching curves. We first show how one can compute, without any knowledge on the camera, the homography induced by a single planar curve. Then we continue with the derivation of the extended Kruppa's equations which are responsible for describing the epipolar constraint of two projections of a general algebraic curve. As part of the derivation of those constraints we address the issue of dimension analysis and as a result establish the minimal number of algebraic curves required for a solution of the epipolar geometry as a function of their degree and genus. We then establish new results on the reconstruction of general algebraic curves from multiple views. We address three different representations of curves: (i) the regular point representation in which we show that the reconstruction from two views of a curve of degree d admits two solutions, one of degree d and the other of degree d(d - 1). Moreover using this representation, we address the problem of homography recovery for planar curves, (ii) dual space representation (tangents) for which we derive a lower bound for the number of views necessary for reconstruction as a function of the curve degree and genus, and (iii) a new representation (to computer vision) based on the set of lines meeting the curve which does not require any curve fitting in image space, for which we also derive lower bounds for the number of views necessary for reconstruction as a function of curve degree alone.

Original language | English |
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Pages (from-to) | 195-219 |

Number of pages | 25 |

Journal | International Journal of Computer Vision |

Volume | 56 |

Issue number | 3 |

DOIs | |

State | Published - Feb 2004 |

### Bibliographical note

Funding Information:∗This work is partially supported by the Emmy Noether Institute for Mathematics and the Minerva Foundation of Germany, by the Excellency Center of the Israel Science Foundation “Group Theoretic Methods in the Study of Algebraic Varieties” and by EAGER (European network in Algebraic Geometry).

## Keywords

- 3D reconstruction
- Algebraic curves
- Epipolar geometry
- Kruppa's equations
- Multiple-view geometry
- Structure from motion