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Isoperimetric Normalization of Planar Curves
August 1994 (vol. 16 no. 8)
pp. 769-777

This paper presents an algorithm for transforming closed planar curves into a canonical form, independent of the viewpoint from which the original image of the contour was taken. The transformation that takes the contour to its canonical form is a member of the projective group PGL(2), chosen because PGL(2) contains all possible transformations of a plane curve under central projection onto another plane. The scheme relies on solving computationally an "isoperimetric" problem in which a transformation is sought which maximises the area of a curve given unit perimeter. In the case that the transformation is restricted to the affine subgroup there is a unique extremising transformation for any piecewise smooth closed curve. Uniqueness holds, almost always, even for curves that are not closed. In the full projective case, isoperimetric normalization is well-defined only for closed curves. We have found computational counterexamples for which there is more than one extremal transformation. Numerical algorithms are described and demonstrated both for the affine and the projective cases. Once a canonical curve is obtained, its isoperimetric area can be regarded as an invariant descriptor of shape.

[1] K. Astrom, "Fundamental difficulties with projective normalization of planar curves," Tech. Rep., Lund Inst. of Technol., 1993.
[2] A. Blake and C. Marinos, "Shape from texture: estimation, isotropy and moments,"J. Artificial Intell., vol. 45, pp. 323-380, 1990.
[3] M. Brady and A. Yuille, "An extremum principle for shape from contour,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-6, pp. 288-301, 1984.
[4] S. Carlsson, "Projectively invariant decomposition of planar shapes," inGeometric Invariance in Computer Vision, J. L. Mundy and A. Zisserman, Eds. Cambridge, MA: MIT Press, 1992, pp. 267-276.
[5] D. A. Forsyth, J. L. Mundy, A. Zisserman, and C. M. Brown, "Projectively invariant representations using implicit algebraic curves," inProc. European Conf. Computer Vision, June 1990.
[6] W.E.L. Grimson,Object Recognition by Computer: The Role of Geometric Constraints, MIT Press, Cambridge, Mass., 1990.
[7] B. K. P. Horn,Robot Vision. Cambridge, MA: M.I.T. Press, 1986.
[8] D. P. Huttenlocher, J. Arkin, and W. J. Rucklidge, "An efficiently computable metric for comparing polygonal shapes," inProc. 1st ACM-SIAM Symp. Discrete Algorithms, 1990, pp. 147-153.
[9] Y. Lamdan, J. T. Schwartz, and H. J. Wolfson, "Object recognition by affine invariant matching," inProc. CVPR 88, 1988.
[10] C. A. Rothwell, A. Zisserman, D. A. Forsyth, and J. L. Mundy, "Canonical frames for planar object recognition," inProc. 2nd European Conf. Comput. Vision, 1992, pp. 757-772.
[11] C. A. Rothwell, A. Zisserman, D. A. Forsyth, and J. L. Mundy, "Fast recognition using algebraic invariants," inGeometric Invariance in Computer VisionJ. L. Mundy and A. Zisserman, Eds. Cambridge, MA: MIT Press, 1992, pp. 398-407.
[12] J. G. Semple and G. T. Kneebone,Algebraic projective geometry. Oxford: Oxford Univ. Press, 1952.
[13] C. E. Springer,Geometry and Analysis of Projective Spaces, vol. 1. London: Freeman, 1964.
[14] J. L. Troutman,Variational calculus with elementary convexity. New York: Springer-Verlag, 1983.

Index Terms:
edge detection; image processing; computational geometry; closed planar curves; canonical form; isoperimetric normalization; projective group PGL(2); piecewise smooth closed curve; invariant shape descriptor; object recognition
D. Sinclair, A. Blake, "Isoperimetric Normalization of Planar Curves," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 16, no. 8, pp. 769-777, Aug. 1994, doi:10.1109/34.308471
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