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Abstract

A mathematical theory for establishing correspondences between curves and for non-rigid shape comparison is developed in this paper. The proposed correspondences, called bimorphisms, are more general than those obtained from one-to-one functions. Their topology is investigated in detail.

A new criterion for non-rigid shape comparison using bimorphisms is also proposed. The criterion avoids many of the mathematical problems of previous approaches by comparing shapes non-rigidly from the bimorphism.

Geometric invariants are calculated for curves whose shapes can be exactly matched with a bimorphism. The invariants are related to the concave and convex segments of a curve and provide justification for parsing the curve into such segments.

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References

  1. V.I. Arnol'd, “Wavefront evolution and equivariant Morse lemma,” Comm. Pure Appl. Math, Vol. 29, pp. 557–582, 1976.

    Google Scholar 

  2. P. Belhumuer, “Acomputational theory for binocular stereopsis,” in Perception as Bayesian Inference, D.C. Knill and W. Richards (Eds.), Cambridge University Press, Cambridge, UK, 1996.

    Google Scholar 

  3. F.L. Bookstein, Morphometric Tools for Landmark Data, Cambridge University Press, Cambridge, UK, 1991.

    Google Scholar 

  4. F.L. Bookstein, “Landmark methods for forms without landmarks: Morphometrics of groups differences in outline shape,” Medical Image Analysis, Vol. 1, No. 3, pp. 225–243, 1996.

    Google Scholar 

  5. J.W. Bruce, “Isotopies of generic plane curves,” Glasgow Math. J., Vol. 24, pp. 195–206, 1983.

    Google Scholar 

  6. J.W. Bruce, “Seeing-The mathematical viewpoint,” The Mathematical Intelligencer, Vol. 6, pp. 18–25, 1984.

    Google Scholar 

  7. S.W. Chen, S.T. Tung, C.Y. Fang, S. Cherng, and A.K. Jain, “Extended attributed string matching for shape recognition,” Computer Vision and Image Understanding, Vol. 70, No. 1, pp. 36–50, 1998.

    Google Scholar 

  8. I. Cohen, N. Ayache, and P. Sulger, “Tracking points on deformable objects using curvature information,” in Lecture Notes in Computer Science: ECCV'92, Vol. 588, Springer-Verlag: Berlin, 1992.

    Google Scholar 

  9. M. Demi, R. Calami, G. Coppini, and G. Valli, “A visual framework for the study of cardiac motion,” Computers and Cardiology, pp. 30–34, 1990.

  10. D. Geiger, A. Gupta, L.A. Costa, and J. Vlontzos, “Dynamic programming for detecting, tracking and matching elastic contours.” IEEE Trans. on Pattern Analysis and Machine Intelligence, Vol. 17, 1995.

  11. J.W. Gorman, O.R. Mitchell, and F.P. Kuhl, “Partial shape recognition using dynamic programming,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 10, No. 2, pp. 257–266, 1988.

    Google Scholar 

  12. W.I. Grosky and R. Mehrotra, “Index-based object recognition in pictorial data management,” Computer Vision, Graphics, and Image Processing, Vol. 52, pp. 416–436, 1990.

    Google Scholar 

  13. D.D. Hoffman and W.A. Richard, “Parts of recognition,” Cognition, Vol. 18, pp. 65–96, 1985.

    Google Scholar 

  14. R.S. Ledley, “High-speed automatic analysis of biomedical pictures,” Science, Vol. 146, No. 3461, pp. 216–223, 1964.

    Google Scholar 

  15. E.T. Lee, “Shape-oriented chromosome classification,” IEEE Transactions on Systems, Man, and Cybernetics, Vol. SMC-5, pp. 629–632, 1975.

    Google Scholar 

  16. M. Leyton, “A process-grammar for shape,” Artifical Intelligence, Vol. 34, pp. 213–247, 1988.

    Google Scholar 

  17. S. Loncaric, “A survey of shape analysis techniques,” Pattern Recognition, Vol. 31, No. 8, pp. 983–1001, 1998.

    Google Scholar 

  18. J.C. McEachen II and J.S. Duncan, “Shape-based tracking of left ventricular wall-motion,” IEEE Transactions on Medical Imaging, Vol. 16, No. 3, pp. 270–283, 1997.

    Google Scholar 

  19. E.E. Milios, “Shape matching using curvature processes,” Computer Vision, Graphics, and Image Processing, Vol. 47, pp. 203–226, 1989.

    Google Scholar 

  20. D. Mumford, “Mathematical theories of shape: Do they model perception?,” in Proc. Conference 1570, 1991, Society of Photooptical and Industrial Engineers, pp. 2–10.

  21. T. Pavlidis, “A review of algorithms for shape analysis,” Computer Vision, Graphics, and Image Processing, Vol. 7, pp. 243–258, 1978.

    Google Scholar 

  22. C.G. Small, The Statistical Theory of Shape, Springer-Verlag: Berlin, 1996.

    Google Scholar 

  23. H.D. Tagare, “Non-rigid curve correspondence for estimating heart motion,” in Information Processing in Medical Imaging, Vermont, 1997.

  24. H.D. Tagare, “Shape-based nonrigid correspondence with application to heart motion analysis,” IEEE Trans. Medical Imaging, Vol. 8, No. 7, 1999.

  25. W.H. Tsai and S.S. Yu, “Attributed string matching with merging for shape recognition,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 7, No. 4, pp. 453–462, 1985.

    Google Scholar 

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Tagare, H.D., O'Shea, D. & Groisser, D. Non-Rigid Shape Comparison of Plane Curves in Images. Journal of Mathematical Imaging and Vision 16, 57–68 (2002). https://doi.org/10.1023/A:1013938519103

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