Degen Generalized Cylinders and Their Properties

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3951)


Generalized cylinder (GC) has played an important role in computer vision since it was introduced in the 1970s. While studying GC models in human visual perception of shapes from contours, Marr assumed that GC’s limbs are planar curves. Later, Koenderink and Ponce pointed out that this assumption does not hold in general by giving some examples. In this paper, we show that straight homogeneous generalized cylinders (SHGCs) and tori (a kind of curved GCs) have planar limbs when viewed from points on specific straight lines. This property leads us to the definition and investigation of a new class of GCs, with the help of the surface model proposed by Degen for geometric modeling. We call them Degen generalized cylinders (DGCs), which include SHGCs, tori, quadrics, cyclides, and more other GCs into one model. Our rigorous discussion is based on projective geometry and homogeneous coordinates. We present some invariant properties of DGCs that reveal the relations among the planar limbs, axes, and contours of DGCs. These properties are useful for recovering DGC descriptions from image contours as well as for some other tasks in computer vision.


Computer Vision Tangent Plane Machine Intelligence Invariant Property Projective Geometry 
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.


  1. 1.
    Binford, T.: Visual perception by computer. In: IEEE Conf. Systems and Control (1971)Google Scholar
  2. 2.
    Sato, H., Binford, T.: Finding and recovering SHGC objects in an edge image. CVGIP: Graphical Model and Image Processing 57(3), 346–358 (1993)CrossRefGoogle Scholar
  3. 3.
    Dhome, M., Glachet, R., Lapreste, J.: Recovering the scaling function of a SHGC from a single perspective view. In: IEEE Proc. Computer Vision and Pattern Recognition, pp. 36–41 (1992)Google Scholar
  4. 4.
    Ulupinar, F., Nevatia, R.: Shape from contour: Straight homogeneous generalized cylinders and constant cross-section generalized cylinders. IEEE Trans. Pattern Analysis and Machine Intelligence 17(2), 120–135 (1995)CrossRefGoogle Scholar
  5. 5.
    Sayd, P., Dhome, M., Lavest, J.: Recovering generalized cylinders by monocular vision. Object Representation in Computer Vision II, 25–51 (1996)Google Scholar
  6. 6.
    Ponce, J., Chelberg, D., Mann, W.: Invariant properties of straight homogeneous generalized cylinders and their contours. IEEE Trans. Pattern Analysis and Machine Intelligence 11(9), 951–966 (1989)CrossRefGoogle Scholar
  7. 7.
    O’Donnell, T., Boult, T., Fang, X., Gupta, A.: The extruded generalized cylinder: A deformable model for object recovery. In: Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 174–181 (1994)Google Scholar
  8. 8.
    O’Donnell, T., Dubuisson-Jolly, M.P., Gupta, A.: A cooperative framework for segmentation using 2-D active contours and 3-D hybrid models as applied to branching cylindrical structures. In: Proc. Int’l Conf. Computer Vision, pp. 454–459 (1998)Google Scholar
  9. 9.
    Gross, A.D.: Analyzing generalized tubes. In: Proc. SPIE Conf. Intelligent Robots and Computer Vision XIII: 3D Vision, vol. 2354 (1994)Google Scholar
  10. 10.
    Shani, U., Ballard, D.H.: Splines as embeddings for generalized cylinders. Computer Vision, Graphics, and Image Processing (CVGIP) 27(2), 129–156 (1984)CrossRefGoogle Scholar
  11. 11.
    Zerroug, M., Nevatia, R.: Segmentation and recovery of SHGCS from a real intensity image. In: European Conf. Computer Vision, pp. 319–330 (1994)Google Scholar
  12. 12.
    Gross, A.D., Boult, T.E.: Recovery of SHGCs from a single intensity view. IEEE Trans. Pattern Analysis and Machine Intelligence 18(2), 161–180 (1996)CrossRefGoogle Scholar
  13. 13.
    Bloomenthal, J.: Modeling the mighty maple. Proc. SIGGRAPH 1985 19(3) (1985)Google Scholar
  14. 14.
    Brooks, R.A., Greiner, R., Binford, T.O.: The ACRONYM model-based vision system. In: Proc. of 6th Int’l Joint Conf. Artificial Intelligence, pp. 105–113 (1979)Google Scholar
  15. 15.
    Shafer, S., Kanade, T.: The theory of straight homogeneous generalized cylinders and a taxonomy of generalized cylinders. Technical Report, Carnegie Mellon Universit (1983)Google Scholar
  16. 16.
    Ponce, J., Chelberg, D.: Finding the limbs and cusps of generalized cylinders. Int’l Journal of Computer Vision 1, 195–210 (1987)CrossRefGoogle Scholar
  17. 17.
    Ulupinar, F., Nevatia, R.: Perception of 3-D surfaces from 2-D contours. IEEE Trans. Pattern Analysis and Machine Intelligence 15(1), 3–18 (1993)CrossRefzbMATHGoogle Scholar
  18. 18.
    Ulupinar, F., Nevatia, R.: Recovery of 3-D objects with multiple curved surfaces from 2-D contours. Artificial Intelligence 67(1), 1–28 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Giblin, P.J., Kimia, B.B.: Transitions of the 3D medial axis under a one-parameter family of deformations. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002. LNCS, vol. 2351, pp. 718–734. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  20. 20.
    Zerroug, M., Nevatia, R.: Three-dimensional descriptions based on the analysis of the invariant and quasi-invariant properties of some curved-axis generalized cylinders. IEEE Trans. Pattern Analysis and Machine Intelligence 18(3), 237–253 (1996)CrossRefGoogle Scholar
  21. 21.
    Ulupinar, F., Nevatia, R.: Recovering shape from contour for constant cross section generalized cylinders. In: Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 674–676 (1991)Google Scholar
  22. 22.
    Zerroug, M., Nevatia, R.: Quasi-invariant properties and 3-D shape recovery of non-straight, non-constant generalized cylinders. In: Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 96–103 (1993)Google Scholar
  23. 23.
    Degen, W.L.F.: Nets with plane silhouettes. In: Proc. IMA Conf. the Mathematics of Surfaces V, pp. 117–133 (1994)Google Scholar
  24. 24.
    Degen, W.L.F.: Conjugate silhouette nets. Curve and surface design 99, 37–44 (2000)Google Scholar
  25. 25.
    Marr, D.: Vision: A Computational Investigation into the Human Representation and Processing of Visual Information. Henry Holt Company (1982)Google Scholar
  26. 26.
    Zhu, Q.: Virtual edges, viewing faces, and boundary traversal in line drawing representation of objects with curved surfaces. Int’l J. Computers and Graphics 15(2), 161–173 (1991)CrossRefGoogle Scholar
  27. 27.
    Stevens, K.: The visual interpretation of surface contours. Artificial Intelligence 17(1-3), 47–73 (1981)CrossRefGoogle Scholar
  28. 28.
    Stevens, K.: Implementation of a theory for inferring surface shape from contours. MIT AI Memo-676 (1982)Google Scholar
  29. 29.
    Koenderink, J.: What does the occluding contour tell us about solid shape? Perception 13, 321–330 (1984)CrossRefGoogle Scholar
  30. 30.
    Carmo, D.: Differential geometry of curves and surfaces. Prentice Hall, Englewood Cliffs (1976)zbMATHGoogle Scholar
  31. 31.
    Marsh, D.: Applied Geometry for Computer Graphics and CAD. Undergraduate Mathematics Series. Springer, Heidelberg (1999)zbMATHGoogle Scholar
  32. 32.
    Maxwell, E.A.: General Homogeneous Coordinates in Space of Three Dimensions. Cambridge University Press, Cambridge (1951)zbMATHGoogle Scholar
  33. 33.
    Ulupinar, F., Nevatia, R.: Shape from contour: Straight homogeneous generalized cones. In: Proc. Int’l Conf. Pattern Recognition, pp. 582–586 (1990)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Department of Information EngineeringThe Chinese University of Hong KongHong KongChina
  2. 2.Microsoft Research AsiaBeijingChina

Personalised recommendations