Object-image correspondence for curves under central and parallel projections
Pages 373 - 382
Abstract
We present a novel algorithm for deciding whether a given planar curve is an image of a given spatial curve, obtained by a central or a parallel projection with unknown parameters. A straightforward approach to this problem consists of setting up a system of conditions on the projection parameters and then checking whether or not this system has a solution. The computational advantage of the algorithm presented here, in comparison to algorithms based on the straightforward approach, lies in a significant reduction of a number of real parameters that need to be eliminated in order to establish existence or non-existence of a projection that maps a given spatial curve to a given planar curve. Our algorithm is based on projection criteria that reduce the projection problem to a certain modification of the equivalence problem of planar curves under affine and projective transformations. The latter problem is then solved using a separating set of rational differential invariants. A similar approach can be used to solve the projection problem for finite lists of points. The motivation comes from the problem of establishing a correspondence between an object and an image, taken by a camera with unknown position and parameters.
References
[1]
G. Arnold and P. F. Stiller. Mathematical aspects of shape analysis for object recognition. In proceedings of IS&T/SPIE Symposium, 11 pp, San Jose, CA, 2007.
[2]
G. Arnold, P. F. Stiller, and K. Sturtz. Object-image metrics for generalized weak perspective projection. In Statistics and analysis of shapes, Model. Simul. Sci. Eng. Technol., 253 -- 279. Birkhauser, Boston, 2006.
[3]
W. Blaschke. Vorlesungen uber Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitatstheorie, vol. II, "Affine Differentialgeometrie". Springer, Berlin, 1923.
[4]
M. Boutin. Numerically invariant signature curves. Int. J. Computer Vision, 40: 235 -- 248, 2000.
[5]
J. M. Burdis. Object-image correspondence under projections. PhD thesis, NCSU, 2010.
[6]
J. M. Burdis and I. A. Kogan. Object-image correspondence for curves under projections, 2012. http://arxiv.org/abs/1202.1303.
[7]
E. Calabi, P. J. Olver, C. Shakiban, A. Tannenbaum, and S. Haker. Differential and numerically invariant signature curves applied to object recognition. Int. J. Computer Vision, 26:107--135, 1998.
[8]
E. Cartan. Lecons sur la geometrie projective complexe. Gauthier-Villars, Paris, 1950.
[9]
B. F. Caviness and J. R. Johnson, editors. Texts and monographs in symbolic computation. 1998.
[10]
D. Cox, J. Little, and D. O'Shea. Ideals, Varieties, and Algorithms. Undergraduate Text in Mathematics. Springer-Verlag, New-York, 1997.
[11]
O. Faugeras. Cartan's moving frame method and its application to the geometry and evolution of curves in the Euclidean, affine and projective planes. Application of Invariance in Computer Vision, J.L Mundy, A. Zisserman, D. Forsyth (eds.) Springer-Verlag Lecture Notes in Computer Science, 825: 11 -- 46, 1994.
[12]
J. Feldmar, N. Ayache, and F. Betting. 3D-2D projective registration of free-form curves and surfaces. In Proceedings of the Fifth International Conference on Computer Vision, ICCV, IEEE Computer Society, 549 -- 556, Washington, DC. 1995.
[13]
S. Feng, I. A. Kogan, and H. Krim. Classification of curves in 2D and 3D via affine integrat signature. Acta Appl. Math., 109(3): 903--937, 2010.
[14]
W. Fulton. Algebraic curves. Advanced Book Classics. Addison-Wesley Publishing Company, Redwood City, CA, 1989. Notes written with the collaboration of Richard Weiss, Reprint of 1969 original.
[15]
H. W. Guggenheimer. Differential Geometry. McGraw-Hill, New York, 1963.
[16]
C. Hann and M. Hickman. Projective curvature and integral invariants. Acta Appl. Math., 74: 549 -- 556, 2002.
[17]
R. I. Hartley and A. Zisserman. Multiple View Geometry in Computer Vision. Cambridge University Press, second edition, 2004.
[18]
H. Hong, editor. Special issue on computational quantifier elimination. volume 36 of The Computer Journal. 1993.
[19]
I. A. Kogan. Two algorithms for a moving frame construction. Canad. J. Math., 55: 266--291, 2003.
[20]
P. J. Olver. Joint invariant signatures. Found. Comp. Math, 1: 3 -- 67, 2001.
[21]
J. Sato and R. Cipolla. Affine integral invariants for extracting symmetry axes. Image and Vision Computing, 15:627--635, 1997.
[22]
A. Tarski. A Decision Method for Elementary Algebra and Geometry. Univ. of California Press, Berkeley, second edition, 1951.
[23]
L. Van Gool, T. Moons, E. Pauwels, and A. Oosterlinck. Semi-differential invariants. Geometric Invariance in Computer Vision, J.L.Mundy and A. Zisserman (eds), MIT Press, pages 157--192, 1992.
[24]
http://en.wikipedia.org/wiki/File:Pinhole-camera.svg.
[25]
http://www.math.ncsu.edu/~iakogan/symbolic/projections.html.
Index Terms
- Object-image correspondence for curves under central and parallel projections
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Published: 17 June 2012
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SoCG '12: Symposium on Computational Geometry 2012
June 17 - 20, 2012
North Carolina, Chapel Hill, USA
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