This Article 
 Bibliographic References 
 Add to: 
Singularities of Principal Direction Fields from 3-D Images
March 1992 (vol. 14 no. 3)
pp. 309-317

Generic singularities can provide position-independent information about the qualitative shape of surfaces. The authors determine the singularities of the principal direction fields of a surface (its umbilic points) from a computation of the index of the fields. The authors present examples both for 3-D synthetic images to which noise has been added and for clinical magnetic resonance images.

[1] P. T. Sander, "On reliably inferring differential structure from three-dimensional images," Ph.D. thesis, McGill Univ., 1988; available as Tech. Rep. CIM-88-1, McGill Res. Cent. Intell. Mach., McGill Univ., Montreal, Canada.
[2] P. T. Sander and S. W. Zucker, "Inferring surface trace and differential structure from 3-D images,"IEEE Trans. Patt. Anal. Machine Intell., vol. PAMI-12, no. 9, pp. 853-854, Sep. 1990.
[3] D. Hilbert and S. Cohn-Vossen,Geometry and the Imagination. New York: Chelsea, 1952.
[4] V. Guillemin and A. Pollack,Differential Topology. Englewood Cliffs, NJ: Prentice-Hall, 1974.
[5] S. W. Zucker, "The emerging paradigm of computational vision,"Ann. Rev. Comput. Sci., pp. 69-89, 1987.
[6] R. M. Haralick, L. T. Watson, and T. J. Laffey, "The topographic primal sketch,"Int. J. Robotics Res., vol. 2, pp. 50-70, Spring 1983.
[7] L. R. Nackman, "Two-dimensional critical points configuration graphs,"IEEE Trans. Patt. Anal. Machine Intell., vol. PAMI-6, no. 4, pp. 442- 450, 1984.
[8] P. Besl and R. Jain, "Intrinsic and extrinsic surface characteristics," inProc. IEEE Comput. Vision Patt. Recogn., June 1985, pp. 226-233.
[9] J. J. Koenderink and A. J. van Doorn, "The shape of smooth objects and the way contours end,"Perception, vol. 11, pp. 129-137, 1982.
[10] O. D. Faugeras and M. Hebert, "The representation, recognition, and locating of 3-D objects,"Int. J. Robotics Res., vol. 5, no. 3, Fall 1986, pp. 27-52.
[11] M. Brady, J. Ponce, A. Yuille, and H. Asada, "Describing surfaces," inProc. Second Int. Symp. Robotics Res., 1986, pp. 5-16.
[12] J. J. Koenderink, "The internal representation of solid shape and visual exploration,"Sensory Experience, Adaptation, and Perception(Spillman and Wooten, Eds.). New York: Lawrence Erlbaum, 1984.
[13] M. Kass and A. Witkin, "Analyzing oriented patterns,"Comp. Vision Graphics Image Processing, vol. 37, pp. 362-385, 1987.
[14] M. P. do Carmo,Differential Geometry of Curves and Surfaces. Englewood Cliffs, NJ: Prentice-Hall, 1976.
[15] M. V. Berry and J. H. Hannay, "Umbilic points on Gaussian random surfaces,"J. Phys. A: Math. Gen., vol. 10, no. 11, pp. 1809-1821, 1977.
[16] J. C. Maxwell, "On hills and dales,"London Edinburgh Dublin Philo. Mag. J. Sci., pp. 421-425, Dec. 1870; reprinted inScientific Papers of James Clerk Maxwell(W. D. Niven Ed.). New York: Dover, 1965, pp. 233-240, 1965.
[17] G. Darboux,Leçons sur la Théorie Générale des Surfaces. Paris: Gauthiers-Villars, 1896.

Index Terms:
3D images; generic singularity; picture processing; pattern recognition; principal direction fields; qualitative shape; clinical magnetic resonance images; computational geometry; pattern recognition; picture processing
P.T. Sander, S.W. Zucker, "Singularities of Principal Direction Fields from 3-D Images," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 14, no. 3, pp. 309-317, March 1992, doi:10.1109/34.120326
Usage of this product signifies your acceptance of the Terms of Use.