Abstract
Recognizing shapes in multiview imaging is still a challenging task, which usually relies on geometrical invariants estimations. However, very few geometric estimators that achieve projective invariance have been devised. This paper proposes a projective length and a projective curvature estimators for plane curves, when the curves are represented by points together with their tangent directions. In this context, the estimations can be performed with only three point-tangent samples for the projective length and five samples for the projective curvature. The proposed length and curvature estimator are based on projective splines built by fitting logarithmic spirals to the point-tangent samples. They are projective invariant and convergent.
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References
Boutin, M.: Numerically invariant signature curves. Int. J. Comput. Vis. 40(3), 235–248 (2000)
Boutin, M.: On invariants of Lie groups and their application to some equivalence problems. PhD thesis, University of Minnesota (2001)
Calabi, E., Olver, P.J., Shakiban, C., Tannenbaum, A., Hacker, S.: Differential and numerically invariant signature curves applied to object recognition. Int. J. Comput. Vis. 26(2), 107–135 (1998)
Calabi, E., Olver, P.J., Tannenbaum, A.: Affine geometry, curve flows, and invariant numerical approximations. Adv. Math. 124, 154–196 (1997)
Craizer, M., Lewiner, T., Morvan, J.M.: Parabolic polygons and discrete affine geometry. In: Sibgrapi, pp. 11–18. IEEE, New York (2006)
Craizer, M., Lewiner, T., Morvan, J.M.: Combining points and tangents into parabolic polygons: an affine invariant model for plane curves. J. Math. Imaging Vis. 29(2–3), 131–140 (2007)
Fabbri, R., Kimia, B.: High-order differential geometry of curves for multiview reconstruction and matching. In: Energy Minimization Methods in Computer Vision and Pattern Recognition, pp. 645–660. Springer, Berlin (2005)
Faugeras, O.: Cartan’s moving frame method and its application to the geometry and evolution of curves in the Euclidean, affine and projective planes. In: Workshop on Applications of Invariance in Computer Vision, pp. 11–46. Springer, Berlin (1994)
Hann, C.E.: Recognizing two planar objects under a projective transformation. PhD thesis, University of Canterbury (2001)
Lazebnik, S., Furukawa, Y., Ponce, J.: Projective visual hulls. Int. J. Comput. Vis. 74(2), 137–165 (2007)
Lazebnik, S., Ponce, J.: The local projective shape of smooth surfaces and their outlines. Int. J. Comput. Vis. 63(1), 65–83 (2005)
Lewiner, T., Craizer, M.: Projective estimators for point-tangent representations of planar curves. In: Sibgrapi, pp. 223–229 (2008)
Mokhtarian, F., Mackworth, A.K.: A theory for multiscale curvature based shape representation for planar curves. IEEE Trans. Pattern Anal. Mach. Intell. 14, 789–805 (1992)
Olver, P.J.: Joint invariant signatures. Found. Comput. Math. 1(1), 3–67 (2001)
Rothwell, C., Zisserman, A., Forsyth, D., Mundy, J.L.: Planar object recognition using projective shape representation. Int. J. Comput. Vis. 16(1), 57–99 (1995)
Stolfi, J.: Oriented Projective Geometry. Academic Press, Boston (1991)
Sukno, F.M., Guerrero, J.J., Frangi, A.F.: Projective active shape models for pose-variant image analysis of quasi-planar objects: Application to facial analysis. Pattern Recogn. (2009). doi:10.1016/j.patcog.2009.07.001
Triggs, B.: Camera pose and calibration from 4 or 5 known 3D points. In: International Conference on Computer Vision, pp. 278–284 (1999)
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Lewiner, T., Craizer, M. Projective Splines and Estimators for Planar Curves. J Math Imaging Vis 36, 81–89 (2010). https://doi.org/10.1007/s10851-009-0173-y
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DOI: https://doi.org/10.1007/s10851-009-0173-y