Abstract
Photographic outlines of 3 dimensional solids are robust and rich in information useful for surface reconstruction. This paper studies algebraic surfaces viewed from 2 cameras with known intrinsic and extrinsic parameters. It has been known for some time that for a degree d=2 (quadric) algebraic surface there is a 1-parameter family of surfaces that reproduce the outlines. When the algebraic surface has degree d>2, we prove a new result: that with known camera geometry it is possible to completely reconstruct an algebraic surface from 2 outlines i.e. the coefficients of its defining polynomial can be determined in a known coordinate frame. The proof exploits the existence of frontier points, which are calculable from the outlines. Examples and experiments are presented to demonstrate the theory and possible applications.
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Brand, M., Kang, K., Cooper, D.B.: Algebraic solution for the visual hull. In: Proceedings of the International Conference on Computer Vision and Pattern Recognition, vol. I, pp. 30–35. Vancouver, Canada (2004)
Canny, J.F.: Finding edges and line in images. Master’s thesis, MIT AI Lab. (1983)
Cipolla, R.: Active Visual Inference of Surface Shape. Springer, New York (1996)
Collings, S.: Frontier points: Theory and methods for computer vision. Ph.D. thesis, University of Western Australia Schools of Mathematics and Statistics and Computer Science and Software Engineering (2007). www.maths.uwa.edu.au/~scolling/Thesis
Collings, S., Kozera, R., Noakes, L.: Shape recovery of a strictly convex solid from n-views. In: Proceedings of the International Conference Computer Vision and Graphics, pp. 57–64. Warsaw, Poland (2005)
Collings, S., Noakes, L., Kozera, R.: The restricted correspondence problem: Curvature at frontier points and ellipsoids from two frames (2007, submitted)
Cross, G.: Surface reconstruction from image sequences. Ph.D. thesis, University of Oxford, Department of Engineering and Science (2000)
Cross, G., Zisserman, A.: Quadric reconstruction from dual space geometry. In: Proceedings of the International Conference on Computer Vision, pp. 25–31. Bombay, India. IEEE (1998)
D’Almeida, J.: Courbe de ramification de la projectoin su P2 d’une surface de P3. Duke Math. J. 65(2), 229–233 (1992)
Faugeras, O.: Three-Dimensional Computer Vision. MIT Press, Cambridge (1993)
Forsyth, D.A.: Recognizing algebraic surfaces from their outlines. Int. J. Comput. Vis. 18(1), 21–40 (1996)
Fraleigh, J.B.: A First Course in Abstract Algebra. Addison-Wesley, Reading (1994)
Fulton, W.: Intersection Theory. Springer, Berlin (1984)
Giblin, P.J., Weiss, R.S.: Epipolar fields on surfaces. In: Proceedings of the European Conference on Computer Vision. LNCS, vol. 1, pp. 14–23. Springer, Berlin (1994)
Hartley, R., Zisserman, A.: Multiple View Geometry in Computer Vision. Cambridge University Press, Cambridge (2000)
Hartshorne, R.: Algebraic Geometry. Springer, Berlin (1977)
Kang, K., Tarel, J.-P., Fishman, R., Cooper, D.B.: A linear dual-space approach to 3D surface reconstruction from occluding contours using algebraic surfaces. In: Proceedings of the International Conference on Computer Vision, vol. I, pp. 198–204. Vancouver, Canada (2001)
Karl, W.C., Verghese, G.C., Willsky, A.S.: Reconstructing ellipsoids from projections. Graph. Model. Image Process. 56(2), 124–139 (1994)
Liang, C., Wong, K.K.: Complex 3D shape recovery using a dual-space approach. In: Proceedings of the International Conference on Computer Vision and Pattern Recognition, vol. 2, pp. 878–884 (2005)
Ma, S.D., Chen, X.: Reconstruction of quadric surface from occluding contour. In: Proceedings of the International Conference on Pattern Recognition, pp. 27–31. IEEE (1994)
Marr, D., Hildreth, E.C.: Theory of edge detection. In: Proceedings of the Royal Society, vol. B, pp. 187–217 (1980)
Marsden, J.E.: Elementary Classical Analysis. Freeman, San Francisco (1974)
Porill, J., Pollard, S.: Curve matching and stereo callibration. Image Vis. Comput. 9(1), 45–50 (1991)
Rieger, J.H.: Three-dimensional motion from fixed points of a deforming profile curve. Opt. Lett. 11(3), 123–125 (1986)
Shashua, A., Toelg, S.: The quadric reference surface: theory and applications. Int. J. Comput. Vis. 23(2), 185–198 (1997)
Shashuar, A.: Q-warping: direct computation of quadratic reference surfaces. Trans. Pattern Recognit. Mach. Intell. 23(8), 920–925 (2001)
Strang, G.: Introduction to Linear Algebra. Wellesley-Cambridge Press, Cambridge (1993)
Szeliski, R.: Prediction error as a quality metric for motion and stereo. Int. J. Comput. Vis., vol. 2, pp. 781–788 (1999)
Taubin, G., Cukierman, F., Sullivan, S., Ponce, J., Kriegman, D.J.: Parameterized families of polynomials for bounded algebraic curve fitting. Trans. Pattern Anal. Mach. Intell. 16(3), 287–303 (1994)
Vogiatzis, G., Favaro, P., Cipolla, R.: Using frontier points to recover shape, reflectance and illumination. In: Proceedings of the International Conference on Computer Vision, pp. 228–235. Beijing, China. IEEE Computer Society (2005)
Wee, C.E., Goldman, R.N.: Elimination and resultants. part 1: Elimination and bivariate resultants. Comput. Graph. Appl. 15(1), 69–77 (1995)
Wee, C.E., Goldman, R.N.: Elimination and resultants, part 2: Multivariate resultants. Comput. Graph. Appl. 15(2), 60–69 (1995)
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Simon Collings is partially funded by the Interactive Virtual Environments Centre (IVEC).
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Collings, S., Kozera, R. & Noakes, L. Recognising Algebraic Surfaces from Two Outlines. J Math Imaging Vis 30, 181–193 (2008). https://doi.org/10.1007/s10851-007-0050-5
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DOI: https://doi.org/10.1007/s10851-007-0050-5