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Analysis of Two-Dimensional Non-Rigid Shapes

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Abstract

Analysis of deformable two-dimensional shapes is an important problem, encountered in numerous pattern recognition, computer vision and computer graphics applications. In this paper, we address three major problems in the analysis of non-rigid shapes: similarity, partial similarity, and correspondence. We present an axiomatic construction of similarity criteria for deformation-invariant shape comparison, based on intrinsic geometric properties of the shapes, and show that such criteria are related to the Gromov-Hausdorff distance. Next, we extend the problem of similarity computation to shapes which have similar parts but are dissimilar when considered as a whole, and present a construction of set-valued distances, based on the notion of Pareto optimality. Finally, we show that the correspondence between non-rigid shapes can be obtained as a byproduct of the non-rigid similarity problem. As a numerical framework, we use the generalized multidimensional scaling (GMDS) method, which is the numerical core of the three problems addressed in this paper.

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References

  • Bajcsy, R., & Solina, F. (1987) Three dimensional object representation revisited. In Proceedings of ICCV (pp. 231–240).

  • Belongie, S., Malik, J., & Puzicha, J. (2002). Shape matching and object recognition using shape context. IEEE Transactions on PAMI, 24, 509–522.

    Google Scholar 

  • Berg, A.C., Berg, T. L., & Malik, J. (2005). Shape matching and object recognition using low distortion correspondences. In Proceedings of CVPR.

  • Biderman, I. (1985). Human image understanding: recent research and a theory. Computer Graphics, Vision and Image Processing, 32, 29–73.

    Article  Google Scholar 

  • Binford, T. O. (1987). Visual perception by computer. In Proceedings of IEEE conference on systems and control (pp. 231–240).

  • Black, M. J., & Anandan, P. (1993). A framework for the robust estimation of optical flow. In Proceedings of ICCV (pp. 231–236).

  • Borg, I., & Groenen, P. (1997). Modern multidimensional scaling—theory and applications. Berlin: Springer.

    MATH  Google Scholar 

  • Bronstein, A. M., Bronstein, M. M., & Kimmel, R. (2003). Expression-invariant 3D face recognition. In Lecture notes on computer science : Vol. 2688. Proceedings of audio and video-based biometric person authentication (pp. 62–69). Berlin: Springer.

    Chapter  Google Scholar 

  • Bronstein, A. M., Bronstein, M. M., & Kimmel, R. (2005a). Three-dimensional face recognition. International Journal of Computer Vision, 64(1), 5–30.

    Article  Google Scholar 

  • Bronstein, A. M., Bronstein, M. M., & Kimmel, R. (2005b). Expression-invariant face recognition via spherical embedding. In Proceedings of IEEE international conference on image processing (ICIP) (Vol. 3, pp. 756–759).

  • Bronstein, A. M., Bronstein, M. M., & Kimmel, R. (2005c). On isometric embedding of facial surfaces into \(\mathbb{S}^{3}\) . In Lecture notes on computer science : Vol. 3459. Proceedings of international conference on scale space and PDE methods in computer vision. (pp. 622–631). Berlin: Springer.

    Google Scholar 

  • Bronstein, A. M., Bronstein, M. M., & Kimmel, R. (2006a). Generalized multidimensional scaling: a framework for isometry-invariant partial surface matching. Proceedings of the National Academy of Sciences, 103(5), 1168–1172.

    Article  MathSciNet  MATH  Google Scholar 

  • Bronstein, A. M., Bronstein, M. M., & Kimmel, R. (2006b). Efficient computation of isometry-invariant distances between surfaces, SIAM Journal of Scientific Computing. 28(5), 1812–1836.

    Article  MATH  MathSciNet  Google Scholar 

  • Bronstein, A. M., Bronstein, M. M., & Kimmel, R. (2006c). Robust expression-invariant face recognition from partially missing data. In Proceedings of European conference on computer vision (ECCV) (pp. 396–408).

  • Bronstein, A. M., Bronstein, M. M., & Kimmel, R. (2006d). Face2face: an isometric model for facial animation. In Proceedings of conference on articulated motion and deformable objects (AMDO) (pp. 38–47).

  • Bronstein, M. M., Bronstein, A. M., Kimmel, R., & Yavneh, I. (2006e). Multigrid multidimensional scaling. Numerical Linear Algebra with Applications (NLAA), 13, 149–171.

    Article  MathSciNet  MATH  Google Scholar 

  • Bronstein, A. M., Bronstein, M. M., Bruckstein, A. M., & Kimmel, R. (2006f). Matching two-dimensional articulated shapes using generalized multidimensional scaling. In Proceedings of conference on articulated motion and deformable objects (AMDO) (pp. 48–57).

  • Bronstein, A. M., Bronstein, M. M., & Kimmel, R. (2007a). Calculus of non-rigid surfaces for geometry and texture manipulation. IEEE Transactions on Visualization and Computer Graphics, 13(5), 902–913.

    Google Scholar 

  • Bronstein, A. M., Bronstein, M. M., & Kimmel, R. (2007b). Numerical geometry of non-rigid shapes. Berlin: Springer.

    Google Scholar 

  • Bronstein, A. M., Bronstein, M. M., Devir, Y. S., Kimmel, R., & Weber, O. (2007c). Parallel algorithms for approximation of distance maps on parametric surfaces. (Technical Report CIS-2007-03). Department of Computer Science, Technion, Israel, April 2007. ACM Transactions on Graphics (in press).

  • Brooks, R. (1981). Symbolic reasoning among 3-D models and 2-D images. Artificial Intelligence, 17, 285–348.

    Article  Google Scholar 

  • Bruckstein, A. M., Holt, R. J., & Netravali, A. N. (1998). Holographic representations of images. IEEE Transactions on Image Processing, 7(11), 1583–1597.

    Article  Google Scholar 

  • Burago, D., Burago, Y., & Ivanov, S. (2001). A course in metric geometry. In Graduate studies in mathematics, (Vol. 33). Providence: American Mathematical Society.

    Google Scholar 

  • Burr, D. (1981). Elastic matching of line drawings. IEEE Transactions on PAMI, 3(6), 708–713.

    Google Scholar 

  • Charpiat, G., Maurel, P., Pons, J. P., Keriven, R., & Faugeras, O. (2007). Generalized gradients: priors on minimization flows. International Journal of Computer Vision, 73(3), 325–344.

    Article  Google Scholar 

  • Cheng, S. W., Edelsbrunner, H., Fu, P., & Lam, K. P. (2001). Design and analysis of planar shape deformation. Computational Geometry: Theory and Applications, 19(2–3), 205–218.

    MATH  MathSciNet  Google Scholar 

  • Cohen, I., Ayache, N., & Sulger, P. (1992). Tracking points on deformable objects using curvature information. In Proceedings of ECCV (pp. 458–466).

  • Connell, J. H., & Brady, M. (1987). Generating and generalizing models of visual objects. Artificial Intelligence, 31, 159–183.

    Article  Google Scholar 

  • de Rooij, S., & Vitanyi, P. (2006). Approximating rate-distortion graphs of individual data: Experiments in lossy compression and denoising. IEEE Transactions on Information Theory (submitted).

  • Elad, A., & Kimmel, R. (2001). Bending invariant representations for surfaces. In Proceedings of CVPR (pp. 168–174).

  • Elad, A., & Kimmel, R. (2002). Spherical flattening of the cortex surface. In Lecture notes in computer Science : Vol. 2191. Geometric methods in bio-medical image processing (pp. 77–89). Berlin: Springer.

    Google Scholar 

  • Eldar, Y., Lindenbaum, M., Porat, M., & Zeevi, Y. Y. (1997). The farthest point strategy for progressive image sampling. IEEE Transactions on Image Processing, 6(9), 1305–1315.

    Article  Google Scholar 

  • Everson, R. M., & Fieldsend, J. E. (2006). Multi-class ROC analysis from a multi-objective optimization perspective. Pattern Recognition Letters, 27(8), 918–927.

    Article  Google Scholar 

  • Felzenszwalb, P. F. (2005). Representation and detection of deformable shapes. IEEE Transactions on PAMI, 27(2), 208–220.

    Google Scholar 

  • Felzenszwalb, P. F., & Huttenlocher, D. P. (2005). Pictorial structures for object recognition. International Journal of Computer Vision, 61(1), 55–79.

    Article  Google Scholar 

  • Felzenszwalb, P. F., & Schwartz, J. (2007). Hierarchical matching of deformable shapes. In Proceedings of CVPR.

  • Fischler, M. A., & Elschlager, R. A. (1973). The representation and matching of pictorial structures. IEEE Transactions on Computers, 100(22), 67–92.

    Article  Google Scholar 

  • Forsyth, D. A., & Ponce, J. (2003). Computer vision: A modern approach. Englewood Cliffs: Prentice Hall.

    Google Scholar 

  • Gdalyahu, Y., & Weinshall, D. (1999). Flexible syntactic matching of curves and its application to automatic hierarchical classification of silhouettes. IEEE Transactions on PAMI, 21, 1312–1328.

    Google Scholar 

  • Geiger, D., Liu, T. L., & Kohn, R. (2003). Representation and self-similarity of shapes. IEEE Transactions on PAMI, 25(1), 86–99.

    Google Scholar 

  • Gelfand, N., Mitra, N. J., Guibas, L., & Pottmann, H. (2005). Robust global registration. In Proceedings of symposium on geometry processing (SGP).

  • Grenander, U., Chow, Y., & Keenan, D. M. (1991). Hands: A pattern theoretic study of biological shapes. Berlin: Springer.

    Google Scholar 

  • Gromov, M. (1981). Textes mathématiques: Vol. 1. Structures métriques pour les variétés riemanniennes.

  • Jacobs, D., & Ling, H. (2005). Deformation invariant image matching. In Proceedings of ICCV.

  • Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J., & Stahel, W. A. (1986). Robust statistics: The approach based on influence functions. New York

  • Hel-Or, Y., & Werman, M. (1994). Constraint-fusion for interpretation of articulated objects. In Proceedings of CVPR (pp. 39–45).

  • Hildreth, E. (1983). The measurement of visual motion. Cambridge: MIT Press.

    Google Scholar 

  • Hoffman, D., & Richards, W. (1984). Visual cognition. In Parts of recognition. Cambridge: MIT Press.

    Google Scholar 

  • Huber, P. J. (2004). Robust statistics. New York: Wiley.

    Google Scholar 

  • Jacobs, D., Weinshall, D., & Gdalyahu, Y. (2000). Class representation and image retrieval with non-metric distances. IEEE Transactions on PAMI, 22, 583–600.

    Google Scholar 

  • Jain, A., Zhong, Y., & Lakshmanan, S. (1996). Object matching using deformable templates. IEEE Transactions on PAMI, 3(18), 267–278.

    Google Scholar 

  • Kass, M., Witkin, A., & Terzopoulos, D. (1988). Snakes: active contour models. International Journal of Computer Vision, 1(4), 321–331.

    Article  Google Scholar 

  • Kimmel, R., & Sethian, J. A. (1998). Computing geodesic on manifolds. In Proceedings of US National Academy of Science (Vol. 95, pp. 8431–8435).

  • Klir, G. J., & Yuan, B. (1994). Fuzzy sets and fuzzy logic: theory and applications. Englewood Cliffs: Prentice Hall.

    Google Scholar 

  • Koenderink, J. J., & van Doorn, A. J. (1981). The shape of smooth objects and the way contours end. Perception, 11, 129–137.

    Article  Google Scholar 

  • Kupeev, K., & Wolfson, H. (1994). On shape similarity. In Proceedings of international conference on pattern recognition (pp. 227–237).

  • Lades, M., Vorbruggen, J. C., Buhmann, J., Lange, J., von der Malsburg, C., Wurtz, R. P., & Konen, W. (1993). Distortion invariant object recognition in the dynamic link architecture. IEEE Transactions on Computers, 42(3), 300–311.

    Article  Google Scholar 

  • Latecki, L. J., & Lakamper, R. (2000). Shape similarity measure based on correspondence of visual parts. IEEE Transactions on PAMI, 22(10), 1185–1190.

    Google Scholar 

  • Latecki, L. J., Lakamper, R., & Wolter, D. (2005). Optimal partial shape similarity. Image and Vision Computing, 23, 227–236.

    Article  Google Scholar 

  • Ling, H., & Jacobs, D. (2005). Using the inner-distance for classification of articulated shapes. In: Proceedings of CVPR.

  • Mémoli, F., & Sapiro, G. (2005). A theoretical and computational framework for isometry invariant recognition of point cloud data. Foundations of Computational Mathematics, 5(3), 313–347.

    Article  MATH  MathSciNet  Google Scholar 

  • Michor, P. W., & Mumford, D. (2003). Riemannian geometries on spaces of plane curves. Arxiv preprint math.DG/0312384.

  • Mio, W., Srivastava, A., & Joshi, S. (2007). On shape of plane elastic curves. International Journal of Computer Vision, 73(3), 307–324.

    Article  Google Scholar 

  • Mumford, D. (1991). Mathematical theories of shape: do they model perception? In Proceedings of SPIE (pp. 2–10).

  • Pareto, V. (1906). Manuale di economia politica.

  • Pentland, A. (1987). Recognition by parts. In Proceedings of ICCV (pp. 612–620).

  • Platel, B., Balmachnova, E., Florack, L. M. J., Kanters, F. M. W., & ter Haar Romeny, B. M. (2005). Using top-points as interest points for image matchng. In Lecture notes in computer science (Vol. 3753, p. 211).

  • Geiger, D., Basri, R., Costa, L., & Jacobs, D. (1998). Determining the similarity of deformable shapes. Vision Research, 38, 2365–2385.

    Article  Google Scholar 

  • Rivlin, E., Dickenson, S., & Rosenfeld, A. (1992). Recognition by functional parts. In Proceedings of CVPR (pp. 267–275).

  • Salukwadze, M. E. (1979). Vector-valued optimization problems in control theory. Orlando: Academic Press.

    Google Scholar 

  • Sethian, J. A. (1996). A review of the theory, algorithms, and applications of level set method for propagating surfaces. Acta numerica 309–395.

  • Siddiqi, K., & Kimia, B. (1996). A shock grammar for recognition. In Proceedings of CVPR (pp. 507–513).

  • Spira, A., & Kimmel, R. (2004). An efficient solution to the eikonal equation on parametric manifolds (Vol. 6, pp. 315–327).

  • Stark, L., & Bowyer, K. (1991). Achieving generalized object recognition through reasoning about association of function to structure. IEEE Transactions on PAMI, 10(13), 992–1006.

    Google Scholar 

  • Tappert, C. (1982). Cursive script recognition by elastic matching. IBM Journal of Research and Development, 26(6), 765–771.

    Article  Google Scholar 

  • Ullman, S. (1989). Aligning pictorial descriptions: an approach to object recognition. Cognition, 3(32), 193–254.

    Article  MathSciNet  Google Scholar 

  • Walter, J., & Ritter, H. (2002). On interactive visualization of high-dimensional data using the hyperbolic plane. In Proceedings of ACM SIGKDD int. conference on knowledge discovery and data mining (pp. 123–131).

  • Yezzi, A., & Mennucci, A. (2004). Metrics in the space of curves. Arxiv preprint math.DG/0412454.

  • Yuille, A., Cohen, D., & Hallinan, P. (1989). Feature extraction from faces using deformable templates. In Proceedings of CVPR (pp. 104–109).

  • Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338–353.

    Article  MATH  MathSciNet  Google Scholar 

  • Zadeh, L. A. (1978). Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 1(1), 3–28.

    Article  MATH  MathSciNet  Google Scholar 

  • Zhang, J., Collins, R., & Liu, Y. (2004). Representation and matching of articulated shapes. In Proceedings of CVPR (Vol. 2, pp. 342–349).

  • Zimmermann, H. J. (2001). Fuzzy set theory and its applications. Dordrecht: Kluwer Academic.

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, Technion—Israel Institute of Technology, 32000, Haifa, Israel

    Alexander M. Bronstein, Michael M. Bronstein, Alfred M. Bruckstein & Ron Kimmel

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  1. Alexander M. Bronstein
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  2. Michael M. Bronstein
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  3. Alfred M. Bruckstein
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  4. Ron Kimmel
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Correspondence to Michael M. Bronstein.

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Bronstein, A.M., Bronstein, M.M., Bruckstein, A.M. et al. Analysis of Two-Dimensional Non-Rigid Shapes. Int J Comput Vis 78, 67–88 (2008). https://doi.org/10.1007/s11263-007-0078-4

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