Abstract
In this paper, we extend the notion of affine shape, introduced by Sparr, from finite point sets to more general sets. It turns out to be possible to generalize most of the theory. The extension makes it possible to reconstruct, for example, 3D-curves up to projective transformations, from a number of their 2D-projections. An algorithm is presented, which is independent of choice of coordinates, is robust, does not rely on any preselected parameters and works for an arbitrary number of images. In particular this means that a solution is given to the aperture problem of finding point correspondences between curves.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
K. Åsträm, R. Cipolla, and P.J. Giblin, “Generalised Epipolar Constraints,” in Proc. 4th European Conf. on Computer Vision, Cambridge, England, 1996, pp. 97–108.
K. Åsträm, “Invariancy methods for points, curves and surfaces in computer vision,” Lund University, Lund Institute of Technology, Department of Mathematics, 1996.
S. Carlsson, “Sufficient image structure for 3D motion and shape estimation,” in Proc. 3rd European Conf. on Computer Vision, Stockholm, Sweden, 1993, pp. 83–91.
N. Dunford and J.T. Schwarz, “Linear operators,” Spectral theory, Vol. 2, Interscience, 1963.
O.D. Faugeras and T. Papadopoulo, “A theory of the motion fields of curves,” Int. Journal of Computer Vision, Vol. 10, pp. 125–156, 1993.
R. Fletcher, “Practical Methods of Optimization”, John Wiley and sons, 1987.
T. Papadopoulo and O. Faugeras, “Computing structure and motion of general 3D curves from monocular sequences of perspective images,” in Proc. 4th European Conf. on Computer Vision, Cambridge, England, 1996, pp. 696–708.
J. Porrill and S.B. Pollard, “Curve matching and stereo calibration,” Image and Vision Computing, Vol. 9, pp. 45–50, 1991.
G. Sparr, “A common framework for kinetic depth, reconstruction and motion for deformable objects,” in Proc. 4th European Conf. on Computer Vision, Cambridge, England, 1994, pp. 471–482.
G. Sparr, “Structure and motion from kinetic depth,” in Proc. of the Sophus Lie InternationalWorkshop on Computer Vision and Applied Geometry, Nordfjordeid, Norway, 1995, to appear.
G. Sparr, “Simultaneous reconstruction of scene structure and camera locations from uncalibrated image sequences,” in Proc. 13th International Conference on Pattern Recognition, Vienna, Austria, 1996, pp. 328–333.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Berthilsson, R., Åström, K. Extension of Affine Shape. Journal of Mathematical Imaging and Vision 11, 119–136 (1999). https://doi.org/10.1023/A:1008379110792
Issue Date:
DOI: https://doi.org/10.1023/A:1008379110792