Abstract
We present a learning method that introduces explicit knowledge into the shape correspondence problem. Given two input curves to be matched, our approach establishes a dense correspondence field between them, where the characteristics of the matching field closely resemble those in an a priori learning set. We build a shape distance matrix from the values of a shape descriptor computed at every point along the curves. This matrix embeds the correspondence problem in a highly expressive and redundant construct and provides the basis for a pattern matching strategy for curve matching. We selected the previously introduced observed transport measure as a shape descriptor, as its properties make it particularly amenable to the matching problem. Synthetic and real examples are presented along with discussions of the robustness and applications of the technique.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arkin, E., Chew, P., Huttenlocher, D., Kedem, K., and Mitchel, J. 1991. An efficient computable metric for comparing polygonal shapes. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13(3):703–716.
Attneave, F. 1954. Some informational aspects of visual perception, Psychological Review, 61(3):183–193.
Attneave, F. and Arnoult, M.D. 1966. The quantitative study of shape and pattern perception. Uhr, L., editor, Pattern Recognition, 123–141.
Belongie, S., Malik, J., and Puzicha, J. 2002. Shape Matching and Object Recognition Using Shape Context. IEEE Transactions on Pattern Analysis and Machine Intelligence, 24(4):509–522.
Bookstein, F.L. 1991. Morphometric Tools for Landmark Data: Geometry and Biology. Cambridge University Press.
Chui, H. and Rangarajan, A. 2000. A new algorithm for non-rigid point matching. In Proceedings of Computer Vision and Pattern Recognition (CVPR’00), 2044–2052.
Cohen, I., Ayache, N., and Sulget, P. 1992. Tracking Points on Deformable Objects using Curvature Information. In Proceedings of European Conference on Computer Vision (ECCV’92), 458–466.
Cohen I. and Herlin, I. 1998. Curves Matching Using Geodesic Paths. In Proceedings of Computer Vision and Pattern Recognition (CVPR’98), 741–746.
Cootes, T.F., Edwards, G.J., and Taylor, C.J. 1998. Active Appearance Models. In Proc. of ECCV, 484–498.
Davatzikos C., Prince J., and Bryan N. 1996. Image Registeration Based on Boundry Mapping. IEEE Transactions on Mediacl Imaging, 15(1):212–215.
Davies, R.H., Twining, C.J., Cootes, T.F., Waterton, J.C., and Taylor, C.J. 2002. A Minimum Description Length Approach to Statistical Shape Modelling. IEEE Transactions on Medical Imaging, 21(5):525–537.
Fleuté M., Lavallée S., and Julliard R. Incorporating a Statistically Based Shape Model into a System for Computer-Assisted Anterior Cruciate Ligament Surgery. Medical Image Analysis, 3(3):209–222, 1999.
Gdalyahu, Y. and Weinshall, D. 1999. Flexible Syntactic Matching of Curves and its Application to Automatic Hierarchical Classification of Silhouettes. IEEE Transactions on Pattern Analysis and Machine Intelligence, 21(12):1312–1328.
Goldmeier, E. 1972. Similarity in visually perceived forms. Psychological Issues. 8(1)
Goshtaby, A. 1985. Description and descrimination of planar shapes using shape matrices, IEEE Transactions on Pattern Analysis and Machine Intelligence, 7:738–743.
Grenander U., Chow Y., and Keenan D. 1991 HANDS: A Pattern Therotic Study of Biological Shapes. Springer.
Hake, H.W. 1966. From discrimination and the invariance of form. In Uhr, L., editor, Pattern Recognition: Theory, Experiments, Computer Simulations, and Dynamic Models of Form, Perception, and Discovery, 142–173.
Haker, S., Angenent, S., and Tannenbaum, A. 2003. Minimizing Flows for the Monge-Kantorovich Problem. SIAM Journal of Mathematical Analysis, 35(1):61–97.
Hebb, D.O. 1949. The organization of behavior. John Wiley.
Horn, B. 1990. Height and gradient from shading. International Journal of Computer Vision, 5(1):37–75.
Kanai, T., Suzuki, H., and Kimura, F. 2000. Metamorphosis of Arbitrary Triangular Meshes. IEEE Computer Graphics and Applications, 20(2):62–75.
Kelemen, A., Szekely, G., and Gerig, G. 1999. Three-Dimensional Model-based Segmentation of Brain MRI. IEEE Transactions on Medical Imaging, 18(10):838–849.
Kendall, D. 1984. Shape Manifolds, Procrustean Metrics and Complex Projective Spaces. Bulletin of the London Mathematical Society, 16:81–121.
Koenderink, J. and Van Doorn, A. 1986. Dynamic shape. Biological Cybernetics, 53:383–396.
Koffka, K. 1935. Principles of Gestalt psychology. Harcourt Brace Jovanovic.
Latecki, L.J., Lakamper, R., and Eckhardt, U. 2000. Shape Descriptors for Non-Rigid Shapes with a Signle Closed Contour. In Proceedings of Computer Vision and Pattern Recognition (CVPR’00), 424–429.
Leventon, M., Grimson, E., and Faugeras, O. 2000. Statistical Shape Influence in Geodesic Active Contours. In Proceedings of Computer Vision and Pattern Recognition (CVPR’00), 4–11.
Leyton, M. 1987. Symmetry-curvature duality. Computer Vision, Graphics, and Image Processing, 38:327–341.
Loncaric, S. 1998. A survey of shape analysis techniques, Pattern Recognition, 31(8):983–1001.
Marr, D. 1970. A theory for cerebral neocortex. In Proceedings of the Royal Society of London, 161–234.
Marr, D. 1976. Early processing of visual information. In Proceedings of the Royal Society of London, 483–519.
Marr, D., and Poggio, T. 1979. A computational theory of human stero vision. In Proceeding of the Royal Sociaty of London, 301–328.
McInerney, T. and Terzopoulos, D. 1996. Deformable Models in Medical Image Analysis: A Survey. Medical Image Analysis, 1(2):91–108.
Persoon, E. and Fu, K. 1977. Shape Discrimination Using Fourier Descriptors. IEEE Transactions on Systems, Man and Cybernetics, 7(3):170–179.
Pitiot, A. Automated Segmentation of Cerebral Structures Incorporating Explicit Knowledge. PhD thesis, École des mines de Paris, November 2003. URL: ftp://ftp-sop.inria.fr/epidure/Publications/Pitiot/pitiot-thesis-2003.pdf.
Pitiot, A., Delingette, H., Toga, A., and Thompson, P. Learning Object Corresspondence with the Observed Transport Shape Measure. In Proceedings of Information Processing in Medical Imaging IPMI’03, 2003.
Plato. 1967. Meno (380 BC) in Lamb W. (ed.), Plato in Twelve Volumes, vol. 3, Harvard University Press.
Prokop, R.J. and Reeves, A.P. 1992. A survey of moment-based techniques for unoccluded object representation and recognition, CVGIP: Graphics Models and Image Processing, 54(5):483–460.
Sebastian, T.B., Crisco, J.J., Klein, P.N. and Kimia, B.B. 2000. Constructing 2D Curve Atlases, In Proc. of CVPR, 70–77.
Thompson, P.M. and Toga, A.W. 1997. Detection, Visulisation and Animation of Abnormal Anatomic Structure with a Deformable Probalistic Brain Atlas Based on Random Vector Field Transformations. Medical Image Analysis, 1(4):271–294.
Trouvè A. and Younes L. 2000. Diffeomorphic Matching Problems in One Dimension: Designing and Minimizing Matching Functionals. In Proceeding of European Conference on Computer Vision (ECCV’00), 573–587.
Ulupinar, F. and Nevatia, R. 1990. Inferring shape from contour for curved surfaces. In Proceedings of the International Conference on Pattern Recognition (ICPR’90), 147–154.
van Otterloo, P.J. 1992. A Contour-Oriented Approach to Shape Analysis, Prentice Hall.
Veltkamp, R.C. 1991. Shape matching: similarity measures and algorithms. In Proceedings of the International Conference on Shape Modeling and Applications, 188–199.
Veltkamp, R.C. and Hagedoorn, M. 1999. State of the Art in Shape Matching, Technical Report UU-CS-1999-27.
Wang, Y., Peterson, B.S., and Staib, L.H. 2000. Shape-Based 3D Surface Correspondence using Geodesics and Local Geometry. In Proceedings of Computer Vision and Pattern Recognition (CVPR’00), 644–651.
Zusne, L. 1970. Visual Perception of Form, Academic Press.
Author information
Authors and Affiliations
Additional information
The study was conducted while Alain Pitiot was with ASCLEPIOS and with the Laboratory of NeuroImaging.
Rights and permissions
About this article
Cite this article
Pitiot, A., Delingette, H. & Thompson, P.M. Learning Shape Correspondence for n-D curves. Int J Comput Vision 71, 71–88 (2007). https://doi.org/10.1007/s11263-006-8114-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11263-006-8114-3