International Journal of Computer Vision
, Volume 56, Issue 3, pp 195219
First online:
Multiple View Geometry of General Algebraic Curves
 J.Y. KaminskiAffiliated withMathematics and Computer Science Department, BarIlan University
 , Amnon ShashuaAffiliated withSchool of Computer Science and Engineering, The Hebrew University of Jerusalem
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We introduce a number of new results in the context of multiview 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.
 Title
 Multiple View Geometry of General Algebraic Curves
 Journal

International Journal of Computer Vision
Volume 56, Issue 3 , pp 195219
 Cover Date
 200402
 DOI
 10.1023/B:VISI.0000011204.89453.4d
 Print ISSN
 09205691
 Online ISSN
 15731405
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 structure from motion
 algebraic curves
 epipolar geometry
 Kruppa's equations
 multipleview geometry
 3D reconstruction
 Industry Sectors
 Authors

 J.Y. Kaminski ^{(1)}
 Amnon Shashua ^{(2)}
 Author Affiliations

 1. Mathematics and Computer Science Department, BarIlan University, RamatGan, Israel
 2. School of Computer Science and Engineering, The Hebrew University of Jerusalem, Jerusalem, Israel