The Isophotic Metric and Its Application to Feature Sensitive Morphology on Surfaces

  • Helmut Pottmann
  • Tibor Steiner
  • Michael Hofer
  • Christoph Haider
  • Allan Hanbury
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3024)


We introduce the isophotic metric, a new metric on surfaces, in which the length of a surface curve is not just dependent on the curve itself, but also on the variation of the surface normals along it. A weak variation of the normals brings the isophotic length of a curve close to its Euclidean length, whereas a strong normal variation increases the isophotic length. We actually have a whole family of metrics, with a parameter that controls the amount by which the normals influence the metric. We are interested here in surfaces with features such as smoothed edges, which are characterized by a significant deviation of the two principal curvatures. The isophotic metric is sensitive to those features: paths along features are close to geodesics in the isophotic metric, paths across features have high isophotic length. This shape effect makes the isophotic metric useful for a number of applications. We address feature sensitive image processing with mathematical morphology on surfaces, feature sensitive geometric design on surfaces, and feature sensitive local neighborhood definition and region growing as an aid in the segmentation process for reverse engineering of geometric objects.


Principal Curvature Parameter Domain Mathematical Morphology Implicit Surface Triangle Mesh 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Bertalmio, M., Mémoli, F., Cheng, L.T., Sapiro, G., Osher, S.: Variational problems and partial differential equations on implicit surfaces, CAM Report 02-17, UCLA (April 2002)Google Scholar
  2. 2.
    Beucher, S., Blosseville, J.M., Lenoir, F.: Traffic spatial measurements using video image processing. In: Proc. SPIE, Intelligent Robots and Computer Vision, vol. 848, pp. 648–655 (1987)Google Scholar
  3. 3.
    Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. Intl. J. Computer Vision 22, 61–79 (1997)zbMATHCrossRefGoogle Scholar
  4. 4.
    Cheng, L.T., Burchard, P., Merriman, B., Osher, S.: Motion of curves constrained on surfaces using a level set approach. UCLA CAM Report 00-36 (September 2000)Google Scholar
  5. 5.
    Cipolla, R., Giblin, P.: Visual Motion of Curves and Surfaces. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  6. 6.
    do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice Hall, Englewood Cliffs (1976)zbMATHGoogle Scholar
  7. 7.
    Haider, C., Hönigmann, D.: Efficient computation of distance functions on manifolds by parallelized fast sweeping, Advanced Computer Vision, Technical Report 116, Vienna (2003)Google Scholar
  8. 8.
    Hanbury, A.: Mathematical morphology applied to circular data. In: Hawkes, P.W. (ed.) Advances in Imaging and Electron Physics, vol. 128, pp. 123–205. Academic Press, London (2003)Google Scholar
  9. 9.
    Heijmans, H.J.A.M.: Morphological Image Operators. Academic Press, Boston (1994)zbMATHGoogle Scholar
  10. 10.
    Heijmans, H.J.A.M., Nacken, P., Toet, A., Vincent, L.: Graph Morphology. Journal of Visual Communication and Image Representation 3, 24–38 (1992)CrossRefGoogle Scholar
  11. 11.
    Kimmel, R., Malladi, R., Sochen, N.: Images as embedded maps and minimal surfaces: movies, color, texture and volumetric medical images. Intl. J. Computer Vision 39, 111–129 (2000)zbMATHCrossRefGoogle Scholar
  12. 12.
    Koenderink, J.J., van Doorn, A.J.: Image processing done right. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002. LNCS, vol. 2350, pp. 158–172. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  13. 13.
    Loménie, N., Gallo, L., Cambou, N., Stamon, G.: Morphological operations on Delaunay triangulations. In: Proc. Intl. Conf. on Pattern Recognition, Barcelona, vol. 3, pp. 3556–3560 (2000)Google Scholar
  14. 14.
    Memoli, F., Sapiro, G.: Fast computation of weighted distance functions and geodesics on implicit hyper-surfaces. J. Comput. Phys. 173(2), 730–764 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Springer, New York (2003)zbMATHGoogle Scholar
  16. 16.
    Patrikalakis, N.M., Maekawa, T.: Shape Interrogation for Computer Aided Design and Manufacturing. Springer, Berlin (2002)zbMATHGoogle Scholar
  17. 17.
    Roerdink, J.B.T.M.: Mathematical morphology on the sphere. In: Proc. SPIE Conf. Visual Communications and Image Processing 1990, Lausanne, pp. 263–271 (1990)Google Scholar
  18. 18.
    Reimers, M.: Computing geodesic distance functions on triangle meshes, Technical Report, Dept. of Informatics, Univ. of Oslo (2003) in preparationGoogle Scholar
  19. 19.
    Roerdink, J.B.T.M.: Manifold Shape: from Differential Geometry to Mathematical Morphology. In: Y.L. O et al. (eds.) Shape in Picture, pp. 209–223. Springer, Berlin (1994)Google Scholar
  20. 20.
    Rössl, C., Kobbelt, L., Seidel, H.-P.: Extraction of feature lines on triangulated surfaces using morphological operators. In: Proc. Smart Graphics 2000, AAAI Spring Symposium. Stanford University, Stanford (2000)Google Scholar
  21. 21.
    Sapiro, G.: Geometric Partial Differential Equations and Image Analysis. Cambridge University Press, Cambridge (2001)zbMATHCrossRefGoogle Scholar
  22. 22.
    Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, London (1982)zbMATHGoogle Scholar
  23. 23.
    Sethian, J.A.: Level Set Methods and Fast Marching Methods. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  24. 24.
    Tsai, Y.-S.R.: Rapid and accurate computation of the distance function using grids. J. Comput. Phys. 178(1), 175–195 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Tsai, Y.-S.R., Cheng, L.-T., Osher, S., Zhao, H.-K.: Fast sweeping algorithms for a class of Hamilton-Jacobi equations. SIAM J. Numerical Analysis 41(2), 673–694 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Várady, T., Martin, R.: Reverse Engineering. In: Farin, G., Hoschek, J., Kim, M.-S. (eds.) Handbook of Computer Aided Geometric Design, pp. 651–681. Elsevier, Amsterdam (2002)CrossRefGoogle Scholar
  27. 27.
    Verly, J.G., Delanoy, R.L.: Adaptive mathematical morphology for range imagery. IEEE Transactions on Image Processing 2(2), 272–275 (1993)CrossRefGoogle Scholar
  28. 28.
    Vincent, L.: Graphs and mathematical morphology. Signal Processing 16, 365–388 (1989)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Helmut Pottmann
    • 1
  • Tibor Steiner
    • 1
  • Michael Hofer
    • 1
  • Christoph Haider
    • 2
  • Allan Hanbury
    • 3
  1. 1.Geometric Modeling and Industrial Geometry GroupVienna Univ. of TechnologyWienAustria
  2. 2.Advanced Computer Vision, Tech Gate ViennaWienAustria
  3. 3.Pattern Recognition and Image Processing GroupVienna Univ. of TechnologyWienAustria

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