Recovering an algebraic curve using its projections from different points. Applications to static and dynamic computational vision

Journal of the European Mathematical Society

Volumn 7, Issue 2, 2005, Pages 145-172

  Kaminski, Jeremy Yirmeyahu ^a   Fryers, Michael ^b   Teicher, Mina ^a  

^a BAR ILAN UNIVERSITY   (Israel)

^b NONE   (United Kingdom)

Author keywords

Machine vision; Plane and space curves; Projections; Structure from motion

Indexed keywords

EID: 16644370922     PISSN: 14359855     EISSN: None     Source Type: Journal    
DOI: 10.4171/jems/25     Document Type: Article

References (28)

1. Avidan, S., Shashua, A.: Trajectory triangulation: 3D reconstruction of moving points from a monocular image sequence. IEEE Trans. Pattern Anal. Machine Intelligence 22, 348-357 (2000)
2. Barnabei, M., Brini, A., Rota, G.-C.: On the exterior calculus of invariant theory. J. Algebra 96, 120-160 (1985)
3. Berthilsson, R., Åström, K., Heyden, A.: Reconstruction of general curves, using factorization and bundle adjustment. Int. J. Computer Vision 41, 171-182 (2001) Zbl 1012.68712
4. Blane, M., Lei, Z., Civi, H., Cooper, D. B.: The 3L algorithm for fitting implicit polynomial curves and surfaces to data. IEEE Trans. Pattern Anal. Machine Intelligence 22, 298-313 (2000)
5. Chojnacki, W., Brooks, M., van den Hengel, A., Gawley, D.: On the fitting of surfaces to data with covariances. IEEE Trans. Pattern Anal. Machine Intelligence 22, 1294-1303 (2000)
6. Cox, D., Little, J., O'Shea, D.: Using Algebraic Geometry. Springer (1998) Zbl 0920.13026 MR 99h: 13033
7. Faugeras, O. D.: Three-Dimensional Computer Vision. A Geometrie Approach. MIT Press (1993)
8. Faugeras, O. D., Luong, Q. T.: The Geometry of Multiple Images. MIT Press (2001) Zb1 1002.68183 MR 2001k:51001
9. 5). Technical report, LIP6 (1998)
10. Faugère, J.-C.: A new efficient algorithm for computing Gröbner basis (F4). J. Pure Appl. Algebra 139, 61-88 (1999) Zbl 0930.68174 MR 2000c:13038
11. Forsyth, D.: Recognizing algebraic surfaces from their outline. Int. J. Computer Vision 18, 21-40 (1996)
12. Forsyth, D., Ponce, J.: Computer Vision: a Modern Approach. Prentice-Hall (2002)
13. Cross, Zisserman, A.: Quadric reconstruction from dual-space geometry. In: Proc. Internat. Conf. Computer Vision, IEEE Computer Soc., 25-31 (1998)
14. Greuel, G. M., Pfister, G.: A Singular Introduction to Commutative Algebra. Springer (2002) Zbl 1023.13001 MR2003k:13001
15. Harris, J.: Algebraic Geometry. A First Course. Springer (1992) Zbl 0779.14001 MR 97e: 14001
16. Hartley, R., Zisserman, A.: Multiple View Geometry in Computer Vision. Cambridge Univ. Press (2000) Zbl 0956.68149 MR 2002g:68148
17. Hartshorne, R.: Algebraic Geometry. Springer (1977) Zbl 0367.14001 MR 57 #3116
18. Kahl, F., Heyden, A.: Using conic correspondence in two images to estimate the epipolar geometry. In: Proc. Internat. Conf. Computer Vision, Narosa Publ. House, 761-766 (1998)
19. Kaminski, J. Y., Shashua, A.: On calibration and reconstruction from planar curves. In: Proc. European Conf. Computer Vision, Lecture Notes in Comput. Sci. 1842, Springer, 678-694 (2000)
20. Ma, S. D., Chen, X.: Quadric reconstruction from its occluding contours. In: Proc. Internat. Conf. Pattern of Recognition Vol. 1, 27-31 (1994)
21. Matei, B., Meer, P.: A general method for errors-in-variables problems in computer vision. In: Proc. IEEE Conf. Computer Vision and Pattern Recognition Vol. 2, 2018-2025 (2000)
22. Miranda, R.: Algebraic Curves and Riemann Surfaces. Amer. Math. Soc. (1991) Zbl 0820.14022 MR 96f: 14029
23. Quan, L., Conic reconstruction and correspondence from two views. IEEE Trans. Pattern Anal. Machine Intelligence 18 (1996)
24. Schmid, C., Zisserman, A.: The geometry and matching of curves in multiple views. In: Proc. European Conf. Computer Vision, Lecture Notes in Comput. Sci. 1406, Springer, 394-409 (1998)
25. Sommese, A. J., Verscheide, J., Wampler, C. W.: Numerical irreducible decomposition using projections from points on the components. In: Symbolic Computation: Solving Equations in Algebra, Geometry, and Engineering, E. L. Green et al. (eds.), Contemp. Math. 286, Amer. Math. Soc., 37-51 (2001) Zbl pre0 1736023 MR 2002k:65089
26. Sturmfels, B.: Solving Systems of Polynomial Equations. Amer. Math. Soc. (2002) Zbl pre01827070 MR 2003i: 13037
27. Taubin, G., Cukierman, F., Sullivan, S., Ponce, J., Kriegman, D. J.: Parameterized families of polynomials for bounded algebraic curve and surface fitting. IEEE Trans. Pattern Anal. Machine Intelligence 16, 287-303 (1994) Zbl 0810.68118
28. Tasdizen, T., Tarel, J.-P., Cooper, D. B.: Improving the stability of algebraic curves for applications. IEEE Trans. Image Process. 9, 405-416 (2000) Zbl 0962.94001 MR 2001b:94001