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The use of optical flow for the analysis of non-rigid motions

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Abstract

This paper analysis the 2D motion field on the image plane produced by the 3D motion of a plane undergoing simple deformations. When the deformation can be represented by a planar linear vector field, the projected vector field, i.e., the 2D motion field of the deformation, is at most quadratic. This 2D motion field has one singular point, with eigenvalues identical to those of the singular point describing the deformation. As a consequence, the nature of the singular point of the deformation is a projective invariant. When the plane moves and experiences a linear deformation at the same time, the associated 2D motion field is at most quadratic with at most 3 singular points. In the case of a normal rototranslation, i.e., when the angular velocity is normal to the plane, and of a linear deformation, the 2D motion field has one singular point and substantial information on the rigid motion and on the deformation can be recovered from it. Experiments with image sequences of planes moving and undergoing linear deformations show that the proposed analysis can provide accurate results. In addition, experiments with deformable objects, such as water, oil, textiles and rubber show that the proposed approach can provide information on more general 3D deformations.

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References

  • Amartur, S.C. and Vesselle, H.J. 1993. A new approach to study cardiac motion: The optical flow of cine MR images. Nuclear Magnetic Resonance in Medicine, 12:59–67.

    Google Scholar 

  • Aicardi, F. and Verri, A. 1990. Limit cycles of the two-dimensional motion field. Biol. Cybern, 64:141–144.

    Google Scholar 

  • Barron, J.L., Fleet, D.J., and Beauchemin, S.S. 1994. Performance of optical flow techniques. Int. Journal of Computer Vision, 12(1):43–77.

    Google Scholar 

  • Bergholm, F. and Carlsson, S. 1991. A “Theory” of optical flow. Comput. Vis. Graph. Image Proc.: Image Understanding, 53(2):171–188.

    Google Scholar 

  • Campani, M. and Verri, A. 1992. Motion analysis from first order properties of optical flow. Comput. Vis. Graph. Image Proc.: Image Understanding, 56(1):90–107.

    Google Scholar 

  • De Micheli, E., Uras, S., and Torre, V. 1993. The accuracy of the computation of optical flow and of the recovery of motion parameters. IEEE Trans. Patt. Anal. Machine Intell., 15(5):434–447.

    Google Scholar 

  • Fennema, C.L. and Thompson, W.B. 1979. Velocity determination in scenes containing several moving objects. Comput. Vis. Graph. Image Proc., vol. 9, pp. 301–315.

    Google Scholar 

  • Ford, R.M., Strickland, R.N. and Thomas, B.A. 1994 Image models for 2-D flow visualization and compression. CVGIP: GMIP, 56(1):75–93.

    Google Scholar 

  • Francois, E. and Bouthemy, P. 1990. The derivation of qualitative information in motion analysis. Image Vis. Comput. J., 8:279–287.

    Google Scholar 

  • Gallione, A., Mc Dougull, A., Busa, W.B., Willmott, N., Cillot, I., and Whitaker, M. 1993. Redundant mechanisms of calciuminduced calcium release underlying calcium waves during fertilization of sea urchin eggs. Science, 261:348–250.

    Google Scholar 

  • Gibson, J.J. 1950. The Perception of the Visual World. Houghton Mifflin: Boston.

    Google Scholar 

  • Helmholtz, H. 1958. Uber Integrale der hydrodynamischen Gleichungen welche den Wirbelwegungen entsprechen. Crelles J., 55:25.

    Google Scholar 

  • Hildreth, E.C. 1984. The computation of the velocity field. Proc. R. Soc. London B, vol. 221, pp. 189–220.

    Google Scholar 

  • Hirsch, M.W. and Smale, S. 1974. Differential Equations, Dynamical Systems, and Linear Algebra. Academic Press: New York.

    Google Scholar 

  • Horn, B.K.P. and Schunck, B.G. 1981. Determining optical flow. Artif. Intell., 17:185–203.

    Google Scholar 

  • Jasinschi, R. and Yuille, A. 1989. Nonrigid motion and Regge calculus. J. Opt. Soc. Amer. A, 6(7):1085–1095.

    Google Scholar 

  • Koenderink, J.J. and van Doorn, A.J. 1986. Depth and shape from differential perspective in the presence of bending deformations. J. Opt. Soc. Amer. A, 3(2):242–249.

    Google Scholar 

  • Lichtenberg, A.J. and Lieberman, M.A. 1992. Regular and Chaotic Dynamics. Springer Verlag.

  • Longuet-Higgins, H.C. 1984. The visual ambiguity of a moving plane. Proc. R. Soc. London B, Vol. 223, pp. 165–175.

    Google Scholar 

  • Nagel, H.H. 1983. Displacement vectors derived from 2nd order intensity variations in image sequences. Comput. Vision Graph. Image Process, 21:85–117.

    Google Scholar 

  • Otte, M. and Nagel, H.H. 1994. Optical flow estimation: Advances and comparisons. Proc. Third Europ. Conf. on Comp. Vision Stockholm, Sweden, Vol. 1, pp. 51–60.

    Google Scholar 

  • Penna, M.A. 1992. Non-rigid motion analysis: Isometric motion. Comput. Vis. Graph. Image Proc.: Image Understanding, 56(3):366–381.

    Google Scholar 

  • Rao, A. R. and Jain, R.C. 1992. Computer flow field analysis: Oriented texture fields. IEEE Trans. Pattern Anal. Machine Intell., 14(7):693–706.

    Google Scholar 

  • Shu, C. and Jain, R.C. 1993. Direct estimation and error analysis for oriented patterns. CVGIP: Image Understanding, 58:383–398.

    Google Scholar 

  • Shulman, D. and Aloimonos, J.Y. 1988. (Non-)rigid motion interpretation: A regularized approach. Proc. Royal Soc. of London B, Vol. 233, pp. 217–234.

    Google Scholar 

  • Sommerfeld, A. 1974. Mechanics of Deformable Bodies. Academic Press: New York, USA.

    Google Scholar 

  • Subbarao, M. 1989. Interpretation of image flow: A spatio-temporal approach. IEEE Trans. Patt. Anal. Machine Intell., 3:266–278.

    Google Scholar 

  • Ullman, S. 1984. Maximizing rigidity: The incremental recovery of 3D structure from rigid and nonrigid motion. Perception, 13:255–274.

    Google Scholar 

  • Verri, A., Girosi, F., and Torre, V. 1989. Mathematical properties of the two-dimensional motion field: From singular points to motion parameters. J. Opt. Soc. Am. A, 6:698–712.

    Google Scholar 

  • Verri, A. and Poggio, T. 1989. Motion field and optical flow: Qualitative properties. IEEE Trans. Pattern Anal. Machine Intell., 11:490–498.

    Google Scholar 

  • Wohn, K. and Waxman, A.M. 1990. The analytic structure of image flows: Deformation and segmentation. CVGIP, 49:127–151.

    Google Scholar 

  • Yakamoto, M., Boulanger, P., Beraldin, J.A., and Rioux, M. 1993. Direct estimation of range flow on deformable shape from a video rate range camera. IEEE Trans. on PAMI, 15(1):82–89.

    Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Fisica, Università di Genova, Via Dodecaneso 33, 16146, Genova, Italy

    Andrea Giachetti & Vincent Torre

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  1. Andrea Giachetti
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  2. Vincent Torre
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Giachetti, A., Torre, V. The use of optical flow for the analysis of non-rigid motions. Int J Comput Vision 18, 255–279 (1996). https://doi.org/10.1007/BF00123144

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  • DOI: https://doi.org/10.1007/BF00123144

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