Abstract
We show that the pre-symmetry set of a smooth surface in 3-space has the structure of the graph of a function from ℝ2 to ℝ2 in many cases of interest, generalising known results for the pre-symmetry set of a curve in the plane. We explain how this function is obtained, and illustrate with examples both on and off the diagonal. There are other cases where the pre-symmetry set is singular; we mention some of these cases but leave their investigation to another occasion.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Bruce, J.W., Giblin, P.J.: Growth, motion and one-parameter families of symmetry sets. Proc. Royal Soc. Edinburgh 104A, 179–204 (1986)
Bogaevsky, I.A.: Perestroikas of shocks and singularities of minimum functions. Physica D 173, 1–28 (2002)
Diatta, A., Giblin, P.J.: Geometry of isophote curves. In: Kimmel, R., Sochen, N.A., Weickert, J. (eds.) Scale-Space 2005. LNCS, vol. 3459, pp. 50–61. Springer, Heidelberg (2005)
Giblin, P.J., Kimia, B.B., Pollitt, A.J.: Transitions of the 3D medial axis under a one-parameter family of deformations (preprint)
Hallinan, P.L., Gordon, C.G., Yuille, A.L., Giblin, P.J., Mumford, D.: Two and Three Dimensional Patterns of the Face. A.K.Peters (1999)
Koenderink, J.J.: Solid Shape. MIT Press, Cambridge (1990)
Kuijper, A., Fogh Olsen, O.: Transitions of the Pre-Symmetry Set. In: 17th International Conference on Pattern Recognition, vol. (3), pp. 190–193 (2004)
Pollitt, A.J.: Euclidean and Affine Symmetry Sets and Medial Axes, Ph.D.Thesis, University of Liverpool (2004), http://www.liv.ac.uk/~pjgiblin
Rieger, J.: Families of maps from the plane to the plane. J. London Math. Soc. 36(2), 351–369 (1987)
Yuille, A., Leyton, M.: 3D symmetry-curvature duality theorems. Comput. Vision Graphics Image Process. 52, 124–140 (1990)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Diatta, A., Giblin, P. (2005). Pre-symmetry Sets of 3D Shapes. In: Fogh Olsen, O., Florack, L., Kuijper, A. (eds) Deep Structure, Singularities, and Computer Vision. DSSCV 2005. Lecture Notes in Computer Science, vol 3753. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11577812_4
Download citation
DOI: https://doi.org/10.1007/11577812_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-29836-6
Online ISBN: 978-3-540-32097-5
eBook Packages: Computer ScienceComputer Science (R0)