Abstract
The ridge lines on a surface can be defined either via contact of the surface with spheres, or via extrema of principal curvatures along lines of curvature. Certain subsets of ridge lines called crest lines have been singled out by some authors for medical imaging applications. There is a related concept of sub-parabolic line on a surface, also defined via extrema of principal curvatures.
In this paper we study in detail the structure of the ridge lines, crest lines and sub-parabolic lines on a generic surface, and on a surface which is evolving in a generic (one-parameter) family. The mathematical details of this study are in Bruce et al. (1994c).
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Arnold, V.I. 1984: 1986; Third edition 1992. Catastrophe Theory. Springer-Verlag: New York-Heidelberg-Berlin.
Bruce, J.W. 1984. Generic reflections and projections. Math. Scand., 54:262–278.
Bruce, J.W. and Fidal, D.L. 1989. On binary differential equations and umbilics. Proc. Royal Soc. Edinburgh, Vol. 111A:147–168.
Bruce, J.W. and Wilkinson, T.C. 1991. Folding maps and focal sets. Proceedings of Warwick Symposium on Singularities, Springer Lecture Notes in Math., Vol. 1462:63–72, Springer-Verlag: Berlin-Heidelberg-New York.
Bruce, J.W. and Giblin, P.J., Second edition 1992. Curves and Singularities. Cambridge University Press: Cambridge, U.K.
Bruce, J.W. and Tari, F. 1994. Extrema of principal curvature and symmetry. To appear in Proc. Royal Soc. Edinbrugh.
Bruce, J.W., Giblin, P.J., and Tari, F. 1995. Families of surfaces: Height functions, Gauss maps and duals. In W.L. Marar (Ed.), Real and Complex Singularities, Pitman Research Notes in Mathematics, Vol. 333, pp. 148–178.
Bruce, J.W., Giblin, P.J., and Tari, F. 1994a. Families of surfaces: Height functions and projections to planes. University of Liverpool, Preprint.
Bruce, J.W., Giblin, P.J., and Tari, F. 1994b. Families of surfaces: Focal sets, ridges and umbilics. University of Liverpool, Preprint.
Bruce, J.W., Giblin, P.J., and Tari, F. 1996. Parabolic curves of evolving surfaces. Int. J. Computer Vision, 17:291–306.
Colchester, A.C.F. 1990. Network representation of 2D and 3D images. In 3D Imaging in Medicine, K.H. Höne, H. Fuchs, and S. Pizer (Eds.), Springer-Verlag: New York-Berlin-Heidelberg, pp. 45–62.
Crowley, J.L. and Parker, A.C. 1984. A representation of shape based on peaks and ridges in the difference of low-pass transform. IEEE Trans. PAMI, 6:156–170.
Eisenhart, L.P. 1909. Differential Geometry. Ginn and Company: London.
Gordon, A.P. 1991. Face recognition from depth maps and surface curvature. In Proceedings of SPIE Conference on Geometric Methods in Computer Vision, San Diego, CA.
Guéziec, A. 1995. Surface representation with deformable splines: using decoupled variables, IEEE Comp. Sci. and Eng. Mag. 2:69–80.
Koenderink, J. 1990. Solid Shape. M.I.T. Press: Cambridge, Mass.
Morris, R.J. 1991. Symmetry of Curves and the Geometry of Surfaces: Two Explorations with the aid of Computer Graphics. Ph.D. Thesis, University of Liverpool.
Morris, R.J. 1994. The sub-parabolic lines of a surface. To appear in The Mathematics of Surfaces VI (Brunel University, 1994), Clarendon Press: Oxford.
O'Neill, B. 1966. Elementary Differential Geametry. Academic Press: New York.
Porteous, I.R. 1983. The normal singularities of surfaces in R3. Proceedings of Symposia in Pure Mathematics, Vol. 40, Part 2, pp. 379–394, Providence, R.I.: American Mathematical Society.
Porteous, I.R. 1994. Geometric Differentiation. Cambridge University Press: Cambridge, U.K.
Sander, P.T. and Zucker, S.W. 1992. Singularities of principal direction fields from 3-D images. IEEE Trans. PAMI, 14:309–317.
Sottomayer, J. and Guttierrez, C. 1982. Structurally stable configurations of lines of principal curvature. Astérisque, Vol. 98–99, pp. 195–215.
Thirion, J.-P. 1996. New feature points based on geometric Invariants for 3D image registration. INRIA Research Report No. 1901; to appear in Int. J. Computer Vision.
Thirion, J.-P. and Gourdon, A. 1995. Computing the differential characteristics of isointensity surfaces, Computer Vision and Image Understanding, 61:190–202.
Wilkinson, T.C. 1991. The Geometry of Folding Maps, Ph.D. Thesis, University of Newcastle-upon-Tyne.
Author information
Authors and Affiliations
Additional information
Supported by the Esprit grant VIVA.
Rights and permissions
About this article
Cite this article
Bruce, J.W., Giblin, P.J. & Tari, F. Ridges, crests and sub-parabolic lines of evolving surfaces. Int J Comput Vision 18, 195–210 (1996). https://doi.org/10.1007/BF00123141
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00123141