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Orientation-Based Differential Geometric Representations for Computer Vision Applications
March 1994 (vol. 16 no. 3)
pp. 249-258

Orientation-based representations (OBR's) have many advantages. Three orientation-based differential geometric representations in computer vision literature are critically examined. The three representations are the extended Gaussian image (EGI), the support-function-based representation (SFBR), and the generalized Gaussian image (GGI). The scope of unique representation, invariant properties from matching considerations, computation and storage requirements, and relations between the three representations are analyzed. A constructive proof of the uniqueness of the SFBR for smooth surfaces is given. It is shown that an OBR using any combination of locally defined descriptors is insufficient to uniquely characterize a surface. It must contain either global descriptors or ordering information to uniquely characterize a surface. The GGI as it was originally introduced requires the recording of one principle vector. It is shown in this paper that this is unnecessary. This reduces the storage requirement of a GGI, therefore making it a more attractive representation. The key ideas of the GGI are to represent the multiple folds of a Gaussian image separately; the use of linked data structures to preserve ordering at all levels and between the folds; and the indexing of the data structures by the unit normal. It extends the EGI approach to a much wider range of applications.

[1] P. J. Besl,Surfaces in Range Image Understanding. Berlin: Springer-Verlag, 1988.
[2] W. C. Graustein,Differential Geometry. New York: MacMillan, 1935.
[3] B. K. P. Horn, "Extended Gaussian image,"Proc. IEEE, vol. 72, pp. 1671-1686, 1984.
[4] P. Liang and J. Todhunter, "Representation and recognition of surface shapes in range images: A differential geometry approach,"Comput. Vision, Graphics, Image Processing, vol. 52, pp. 78-109, 1990.
[5] D. Marr,Vision. San Francisco: Freeman, 1982.
[6] V. S. Nalwa, "Representing oriented piecewise C2surfaces," inProc. 2nd Int. Conf. Comput. Vision, Dec. 1988, pp. 40-51.
[7] B. O'Neil,Elementary Differential Geometry. New York: Academic, 1966.
[8] A. V. Pogorelov,Extrinsic Geometry of Convex Surfaces. Providence, RI: Amer. Math. Soc., vol. 35, translation ofMath. Monographs, 1978.
[9] M. Spivak,A Comprehensive Introduction to Differential Geometry, vols. I, II, III. Boston: Publish or Perish, 1979.
[10] J. J. Koenderink,Solid Shape. Cambridge, MA: MIT Press, 1989.

Index Terms:
computer vision; data structures; image sequences; vectors; orientation-based differential geometric representations; computer vision; extended Gaussian image; support-function-based representation; generalized Gaussian image; invariant properties; matching; uniqueness; smooth surfaces; global descriptors; storage requirement; linked data structures
Citation:
P. Liang, C.H. Taubes, "Orientation-Based Differential Geometric Representations for Computer Vision Applications," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 16, no. 3, pp. 249-258, March 1994, doi:10.1109/34.276124
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