Reconstruction from Calibrated Cameras—A New Proof of the Kruppa-Demazure Theorem
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
This paper deals with the problem of reconstructing the locations of five points in space from two different images taken by calibrated cameras. Equivalently, the problem can be formulated as finding the possible relative locations and orientations, in three-dimensional Euclidean space, of two labeled stars, of five lines each, such that corresponding lines intersect.
The problem was first treated by Kruppa more than 50 years ago. He found that there were at most eleven solutions. Later Demazure and also Maybank showed that there were actually ten solutions. In this article will be given another proof of this theorem based on a different parameterisation of the problem neither using the epipoles nor the essential matrix. This is within the same point of view as direct structure recovery in the uncalibrated case. Instead of the essential matrix we use the kinetic depth vectors, which has shown to be were useful in the uncalibrated case. We will also present an algorithm that in most cases calculates the ten different solutions, although some may be complex and some may not be physically realisable. The algorithm is based on a homotopy method and tracks solutions on the so called Chasles' manifold. One of the major contributions of this paper is to bridge the gap between reconstruction methods for calibrated and uncalibrated cameras. Furthermore, we show that the twisted pair solutions are natural in this context because the kinetic depths are the same for both components.
- J.W. Bruce and P.J. Giblin, Curves and Singularities, Cambridge University Press, 1984.
- M. Demazure, “Sur deux problemes de reconstruction,” Technical Report 882, INRIA, Rocquencourt, France, 1988.
- O.D. Faugeras and S.J. Maybank, “Motion from point matches: multiplicity of solutions,” Technical Report 1157, INRIA, Rocquencourt, France, 1990.
- M.J. Greenberg and J.R. Harper, Algebraic Topology, A First Course, Addison-Wesley Publishing Company, Inc., 1981.
- A. Heyden, “Reconstruction and prediction from three images of uncalibrated cameras,” in Proc. 9th Scandinavian Conference on Image Analysis, 1995, pp. 57–66. Also in Theory & Applications of Image Processing II,World Scientific Publishing Co, Machine Perception Artificial Intelligence, 1996.
- A. Heyden, “Reconstruction from image sequences by means of relative depths,” in Proc. ICCV'95, W.L. Grimson (Ed.), IEEE Computer Society Press, 1995, pp. 1058–1063.
- E. Kruppa, “Zur ermittlung eines objektes zwei perspektiven mit innerer orientierung,” Sitz-Ber. Akad.Wiss.,Wien, Math. Naturw. Kl. Abt., Vol. IIa, No. 122, 1939–1948.
- S.J. Maybank, “The projective geometry of ambigous surfaces,” Philosophical Transactions of the Royal Society, 1990.
- S.J. Maybank, Theory of Reconstruction from Image Motion, Springer-Verlag: Berlin, Heidelberg, 1993.
- I.R. Schafarevich, Basic Algebraic Geometry I-Varieties in Projective Space, Springer Verlag, 1988.
- G. Sparr, Depth, Shape and Invariancy, in preparation for Kluwer Verlag.
- G. Sparr, “An algebraic-analytic method for affine shapes of point configurations,” in Proc. 7th Scandinavian Conference on Image Analysis, 1991, pp. 274–281. Also in Theory & Applications of Image Analysis I, World Scientific Publishing Co, Machine Perception Artificial Intelligence, 1992, pp. 87–98.
- G. Sparr, “Reconstruction and motion for deformable objects,” in Computer Vision-ECCV'94, J.-O. Eklund (Ed.), Lecture notes in Computer Science 801, Springer-Verlag, 1994, pp. 471–482.
- Reconstruction from Calibrated Cameras—A New Proof of the Kruppa-Demazure Theorem
Journal of Mathematical Imaging and Vision
Volume 10, Issue 2 , pp 123-142
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
- Additional Links
- reconstruction from calibrated cameras
- direct shape recovery
- kinetic depth
- the Kruppa-Demazure theorem
- Industry Sectors