Abstract
We present a new and more robust method of obtaining local projective and affine invariants. These shape descriptors are useful for object recognition because they eliminate the search for the unknown viewpoint. Being local, our invariants are much less sensitive to occlusion than the global ones used elsewhere. The basic ideas are: (i) employing an implicit curve representation without a curve parameter, to increase robustness; (ii) using a canonical coordinate system which is defined by the intrinsic properties of the shape, independently of the given coordinate system, and is thus invariant. Several configurations are treated: a general curve without any correspondence, and curves with known correspondences of one or two feature points or lines.
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References
E. Barrett, P. Payton, N. Haag and M. Brill, General methods for determining projective invariants in imagery, CVGIP:IU 53(1991)45–65.
I.N. Bronshtein and K.A. Semendyayev,Handbook of Mathematics (Van Nostrand, New York, 1985).
A.M. Bruckstein, R.J. Holt, A.N. Netravali and T.J. Richardson, Invariant signatures for planar shape recognition under partial occlusion, CVGIP:IU 58(1993)49–65.
R.O. Duda and P.E. Hart,Pattern Recognition and Scene Analysis (Wiley, New York, 1973).
D. Forsyth, J.L. Mundy, A. Zisserman, C. Coelho, A. Heller and C. Rothwell, Invariant descriptors for 3-D object recognition and pose, IEEE-PAMI 13(1991)971–991.
H. Guggenheimer,Differential Geometry (Dover, New York, 1963).
M. Halphen, Sur les invariants différentiels des courbes gauches, J. Ec. Polyt. 28(1880)1.
P. Meer and I. Weiss, Smoothed differentiation filters for images, J. Visual Commun. Image Rep. 3(1992)58–72.
J.L. Mundy and A. Zisserman (eds.),Geometric Invariance in Machine Vision (MIT Press, Cambridge, MA, 1992).
E. Rivlin and I. Weiss, Local invariants for recognition, IEEE Trans. Pattern Anal. Machine Intellig. 17(1995)226–238.
G. Salmon,Higher Plane Curves (Chelsea, New York, 1879).
G. Taubin and D.B. Cooper, Object recognition based on moment (or algebraic) invariants, in:Geometric Invariance in Machine Vision, eds. J.L. Mundy and A. Zisserman (MIT Press, Cambridge, MA, 1992).
L. Van Gool, T. Moons, E. Pauwels and A. Oosterlink, Semi-differential invariants, in:Geometric Invariance in Machine Vision, eds. J.L. Mundy and A. Zisserman (MIT Press, Cambridge, MA, 1992).
I. Weiss, Projective invariants of shapes,Proc. DARPA Image Understanding Workshop, 1988, pp. 1125–1134.
I. Weiss, Geometric invariants and object recognition, Int. J. Comp. Vision 10(1993)201–231.
I. Weiss, Noise resistant invariants of curves, IEEE Trans. Pattern Anal. Machine Intellig. 15(1993)943–948.
I. Weiss, High order differentiation filters that work, IEEE Trans. Pattern Anal. Machine Intellig. 16(1994)734–739.
E.J. Wilczynski,Projective Differential Geometry of Curves and Ruled Surfaces (Teubner, Leipzig, 1906).
E.P. Lane,A Treatise on Projective Differential Geometry (University of Chicago Press, 1942).
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The author is grateful for the support of the Air Force Office of Scientific Research under grant F49620-92-J-0332, the Defense Advanced Research Projects Agency (ARPA Order No. 8459), and the U.S. Army Topographic Engineering Center under Contract DACA76-92-C-0009.
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Weiss, I. Local projective and affine invariants. Ann Math Artif Intell 13, 203–225 (1995). https://doi.org/10.1007/BF01530828
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DOI: https://doi.org/10.1007/BF01530828