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Geometric invariants and object recognition

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Abstract

We discuss the role of the general invariance concept in object recognition, and review the classical and recent literature on projective invariance. Invariants help solve major problems of object recognition. For instance, different images of the same object often differ from each other, because of the different viewpoint from which they were obtained. To match the two images, common methods thus need to find the correct viewpoint, a difficult problem that can involve search in a high dimensional space of all possible points of view and/or finding point correspondences. Geometric invariants are shape descriptors, computed from the geometry of the shape, that remain unchanged under geometric transformations such as changing the viewpoint. Thus they can be matched without search. Deformations of objects are another important class of geometric changes for which invariance is useful.

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References

  • Abhyankar, S.S., 1990.Algebraic Geometry for Scientists and Engineers. American Mathematical Society: Providence, RI.

    Book  MATH  Google Scholar 

  • Arbter, K., Snyder, W.E., Burkhardt, H., and Hirzinger, G., 1990. Applications of affine-invariant Fourier descriptors to recognition of 3-D objects,IEEE Trans. Patt. Anal. Mach. Intell. 12: 640–647.

    Article  Google Scholar 

  • Ballard, D., and Brown, C.M., 1982.Computer 11263on. Prentice Hall: Englewood Cliffs, NJ.

    Google Scholar 

  • Barrett, E.B., Payton, P., Haag, N., and Brill, M., 1991. General methods for determining projective invariants in imagery,Comput. Vis. Graph. Image Process. 53: 45–65.

    Google Scholar 

  • Barrett, E.B., Brill, E.H., Haag, N.N., and Payton, P.M., 1992. Invariant linear methods in photogrammetry and model-matching, In J.L. Mundy and A. Zisserman, 1992.

  • Besl, P.J., and Jain, R.C. 1985. Three-dimensional object recognition,ACM Computing Surveys 17: 75–145.

    Article  Google Scholar 

  • Bhaskaracharya, 1150.Beejaganit, Ujjain.

  • Binford, T.O., 1981. Inferring surfaces from images,Artificial Intelligence 17: 205–244.

    Article  Google Scholar 

  • Bookstein, F.L., 1979. Fitting conic sections to scattered data,Comput. Graph. Image Process. 9: 56–71.

    Article  Google Scholar 

  • Brill, M.H., Barrett, E.B., and Payton, P.M., 1992. Projective invariants in two and three dimensions. In J.L. Mundy and A. Zisserman, 1992.

  • Brown, C.M., 1991. Numerical evaluation of differential and semidifferential invariants, Tech. Rept., University of Rochester Computer Science Department.

  • Bruckstein, A., Holt, J., Netravali, A.N., and Richardson, T.J., 1991. Invariant signatures for planar shape recognition under partial occlusion, AT&T Tech. Rept., October.

  • Burkhardt, H., Fenske, A., and Schulz-Mirbach, H., 1992. Invariants for the recognition of planar contour and gray-scale images, ESPRIT Workshop: Invariants for Recognition,Proc. 2nd Europ. Conf. Comput. Vis., Italy.

  • Burns, J.B., Weiss, R., and Riseman, E.M., 1990. View variation of point set and line segment features,Proc. DARPA Image Understanding Workshop, Pittsburgh, pp. 650–659.

  • Carlsson, S., 1992. Projective invariant decomposition of planar shapes. In J.L. Mundy and A. Zisserman, 1992.

  • Cartan, E., 1955. La théorie des groupes continus et la géometrie,Oeuvres Complétes, III/2, 1727–1861, Gauthier-Villars, Paris.

    Google Scholar 

  • Chang, S., Davis, L.S., Dunn, S.M., Eklundh, J.-O., and Rosenfeld, A., 1987. Texture discrimination by projective invariants,Patt. Recog. Letts. 5: 337–342.

    Article  Google Scholar 

  • Cyganski, D., and Orr, J., 1985. Applications of tensor theory to object recognition and orientation determination,IEEE Trans. Patt. Anal. Mach. Intell. 7: 662–673.

    Article  Google Scholar 

  • Duda, R.O., and Hart, P.E., 1973.Pattern Recognition and Scene Analysis. Wiley: New York.

    Google Scholar 

  • Faugeras, O.D., and Papadopoulo, T., 1992. Disambiguating stereo matches with spatio-temporal surfaces. In J.L. Mundy and Z. Zisserman, 1992.

  • Forsyth, D., Mundy, J.L., Zisserman, A., and Brown, C.M., 1990. Projectively invariant representations using implicit algebraic curves,Image Vis. Comput. 8: 130–136.

    Article  Google Scholar 

  • Forsyth, D., Mundy, J.L., Zisserman, A., Coelho, C., Heller, C., Heller, A., and Rothwell, C., 1991. Invariant descriptors for 3-D object recognition and pose,IEEE Trans. Patt. Anal. Mach. Intell. 13: 971–991.

    Article  Google Scholar 

  • Fubini and Čech, 1927.Geometria Proiettiva Differenziale. Zanichelli: Bologna.

    MATH  Google Scholar 

  • Gordan, P. 1885.Vorlesungen Über Invariantentheorie. Leipzig.

  • Grimson, W.E.L., and Lozano-Pérez, T., 1987. Localizing overlapping parts by searching the interpretation tree,IEEE Trans. Patt. Anal. Mach. Intell. 9: 469–482.

    Article  Google Scholar 

  • Grace, J.H., and Young, A., 1903.The Algebra of Invariants. Chelsea: New York.

    MATH  Google Scholar 

  • Guggenheimer, H., 1963.Differential Geometry. Dover: New York.

    MATH  Google Scholar 

  • Halphen, M., 1880. Sur les invariants différentiels des courbes gauches,J. Ec. Polyt. 28: 1.

    Google Scholar 

  • Hopcroft, J.P., Huttenlocher, D.P., and Wayner, P.C., 1992. Affine invariants for model-based recognition. In J.L. Mundy and A. Zisserman, 1992.

  • Hilbert, D., 1890. Über die Theorie der algebraischen Formen,Mathematische Annalen 36: 473–534.

    Article  MathSciNet  MATH  Google Scholar 

  • Hilbert, D., 1893. Über die vollen Invariantensysteme,Mathematische Annalen 42: 313–373.

    Article  MathSciNet  MATH  Google Scholar 

  • Kanatani, K., 1990.Group Theoretical Methods in Image Understanding. Springer: Berlin.

    Book  MATH  Google Scholar 

  • Kapur, D., and Mundy, J.L., 1992. Fitting affine invariants to curves. In J.L. Mundy and A. Zisserman, 1992.

  • Klein, F., 1926.Entwicklung der Mathematik. Berlin.

  • Koenderink, J.J., and Van Doorn, A.J., 1991. Affine invariants from motion,J. Opt. Soc. Amer. A, 8: 377–385.

    Article  Google Scholar 

  • Kriegman, D.J., and Ponce, J. 1990. On recognizing and positioning of curved 3-D objects from image contours,IEEE Trans. Patt. Anal. Mach. Intell., 12: 1127–1137.

    Article  Google Scholar 

  • Lamdan, Y., Schwartz, J.T., and Wolfson, H.J., 1988. Object recognition by affine invariant matching,Proc. Conf. Comput. Vis. Patt. Recog., Ann Arbor, pp. 335–344.

  • Lagrange, J.L., 1773, Berlin Memoires, p. 265. Lane, E.P., 1932.Projective Differential Geometry of Curves and Surfaces. University of Chicago Press.

  • Lane, E.P., 1942.A Treatise on Projective Differential Geometry. University of Chicago Press.

  • Lowe, D., 1985.Percentual Organization and Visual Recognition. Kluwer: Boston.

    Book  Google Scholar 

  • Maybank, S.J., 1992. The projection of two noncoplanar conics. In J.L. Mundy and A. Zisserman, 1992.

  • Meer, P., and Weiss, I., 1992. Smoothed differentiation filters for images,J. Visu. Commun. Image Represent. 3: 58–72.

    Article  Google Scholar 

  • Meer, P., and Weiss, I., 1992b. Point/line correspondence under 2D projective transformation,Proc. Conf. Comput. Vis. Patt. Recog., Urbana Champaign, IL, pp. 115–121.

  • Mohr, R. and Maurin, L., 1991. Relative positioning from geometric invariants,Proc. Conf. Comp. Vis. Patt. Recog., Maui, pp. 139–144.

  • Mumford, D., 1965.Geometric Invariant Theory. Springer: New York.

    Book  MATH  Google Scholar 

  • Mundy, J.L., and Zisserman, A., 1992. Introduction—Towards a new framework for 11263on. InGeometric Invariance in Machine 11263on. J.L. Mundy and A. Zisserman, eds., MIT Press: Cambridge, MA.

    Google Scholar 

  • Mundy, J.L., Kapur, D., Maybank, S.J., and Quan, L., 1992a. Geometric interpretation of joint invariants. In J.L. Mundy and A. Zisserman, 1992.

  • Nagata, J., 1963. Complete reducibility of rational representations of a matric group,J. Math. Kyoto Univ. 3: 369–377.

    MathSciNet  Google Scholar 

  • Nielsen, L., and Sparr, G., 1991. Projective area invariants as an extension of the cross ratio,Comput. Vis. Graph. Image Process. 54: 145–159.

    MATH  Google Scholar 

  • Olver, P.J., 1986.Application of Lie Groups to Differential Equations. Springer: New York.

    Book  Google Scholar 

  • Park, K., and Hall, E., 1987. Form recognition using moment invariants for three-dimensional perspective transformation,Proc. SPIE 726, Intelligent Robots and Computer 11263on, pp. 90–108.

  • Pizlo, Z., and Rosenfeld, A., 1991. Recognition of planar shapes from perspective images using contour-based invariants, Tech. Rept. 528, Center for Automation Research, University of Maryland.

  • Rivlin, E., and Weiss, I., 1992. Local invariants for recognition, University of Maryland CS-TR 2977.

  • Rivlin, E. and Weiss, I., 1993. Recognizing objects using deformation invariants, University of Maryland CS-TR 3041.

  • Salmon, G., 1879.Higher Plane Curves. Chelsea: New York.

    Google Scholar 

  • Springer, C.E., 1964.Geometry and Analysis of Projective Spaces. Freeman: San Francisco.

    MATH  Google Scholar 

  • Stevenson, R.L., and Delp, E.J., 1989. Invariant reconstruction of visual surfaces,Proc. IEEE Workshop on Interpretation of 3-D scenes, pp. 131–137.

  • Taubin, G., and Cooper, D.B., 1992. Object recognition based on moment (or algebraic) invariants. In J.L. Mundy and A. Zisserman, 1992.

  • Turnbull, H.W., 1928.Determinants, Matrices and Invariants. Blackie and Son: Glasgow.

    MATH  Google Scholar 

  • Ullman, S., and Basri, R., 1991. Recognition by linear combination of models,IEEE Trans. Patt. Anal. Mach. Intell. 13: 992–1006.

    Article  Google Scholar 

  • Van Gool, L., Kempenaers, P., and Osterlinck, A., 1991. Recognition and semidifferential invariants,Proc. Conf. Comput. Vis. Patt. Recog., Maui, pp. 454–460.

  • Van Gool, L., Moons, T., Pauwels, E., and Oosterlinck, A., 1992. Semidifferential invariants. In J.L. Mundy and A. Zisserman, 1992.

  • Wayner, P.C., 1991. Efficiently using invariant theory for model-based matching,Proc. Conf. Comput. Vis. Patt. Recog., Maui, pp. 473–478.

  • Weinshall, D., 1990. Qualitative depth from stereo, with applications,Comput. Vis. Graph. Image Process. 49: 222–241.

    Article  MATH  Google Scholar 

  • Weiss, I., 1988. Projective invariants of shapes,Proc. DARPA Image Understanding Workshop, Cambridge, MA, pp. 1125–1134.

  • Weiss, I., 1991. High order differentiation filters that work, Tech. Rept. 545, Center for Automation Research, University of Maryland.

  • Weiss, I., 1992a. Noise resistant invariants of curves. In J.L. Mundy and A. Zisserman, 1992.

  • Weiss, I., 1992b. Local projective and affine invariants, Tech. Rept. 612, Center for Automation Research, University of Maryland.

  • Weyl, H., 1939.The Classical Groups. Princeton University Press.

  • Wilczynski, E.J., 1906.Projective Differential Geometry of Curves and Ruled Surfaces. Teubner: Leipzig.

    MATH  Google Scholar 

  • Wilczynski, E.J., 1908. Projective differential geometry of curved surfaces (Second Memoir),Amer. Math. Soc. Trans. 79.

  • Zisserman, A., Forsyth, D.A., Mundy, J.L., and Rothwell, C.A., 1992. Recognizing general curved objects efficiently. In J.L. Mundy and A. Zisserman, 1992.

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    Isaac Weiss

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Weiss, I. Geometric invariants and object recognition. Int J Comput 11263on 10, 207–231 (1993). https://doi.org/10.1007/BF01539536

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