Journal of Mathematical Imaging and Vision

, Volume 34, Issue 2, pp 200–221 | Cite as

Robust Surface Fitting from Two Views using Restricted Correspondence

Article
First Online:
Received:
Accepted:

Abstract

The restricted correspondence problem is the task of solving the classical stereo correspondence problem when the surface being observed is known to belong to a family of surfaces that vary in a known way with one or more parameters. Under this constraint the surface can be extracted far more robustly than by classical stereo applied to an arbitrary surface, since the problem is solved semi-globally, rather than locally for each epipolar line. Here, the restricted correspondence problem is solved for two examples, the first being the extraction of the parameters of an ellipsoid from a calibrated stereo pair. The second example is the estimation of the osculating paraboloid at the frontier points of a convex object.

Keywords

Stereo vision Correspondence problem Algebraic surfaces Outlines Ellipsoids Frontier points 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Birchfield, S., Tomasi, C.: A pixel dissimilarity measure that is insensitive to image sampling. Trans. Pattern Anal. Mach. Intell. 20(4), 401–406 (1998) CrossRefGoogle Scholar
  2. 2.
    Bloomenthal, J. (ed.): Introduction to Implicit Surfaces. Morgan Kaufmann, San Mateo (1996) Google Scholar
  3. 3.
    Boykov, Y., Veksler, O., Zabih, R.: Fast approximate energy minimization via graph cuts. Trans. Pattern Anal. Mach. Intell. 23(11), 1222–1239 (2001) CrossRefGoogle Scholar
  4. 4.
    Canny, J.F.: Finding edges and line in images. Master’s thesis, MIT AI Lab (1983) Google Scholar
  5. 5.
    Collings, S.: Frontier points: theory and methods for computer vision. PhD thesis, University of Western Australia Schools of Mathematics and Statistics and Computer Science and Software Engineering (2007). www.maths.uwa.edu.au/~scolling/Thesis
  6. 6.
    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) Google Scholar
  7. 7.
    Collings, S., Kozera, R., Noakes, L.: Surface properties from n-views of a strictly convex solid. Fundam. Inf. 72(4), 437–452 (2006) MATHMathSciNetGoogle Scholar
  8. 8.
    Collings, S., Noakes, L., Kozera, R.: Recognising algebraic surfaces from two outlines. J. Math. Imaging Vis. 30(2), 181–193 (2008) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Cross, G.: Surface reconstruction from image sequences. PhD thesis, University of Oxford, Department of Engineering and Science (2000) Google Scholar
  10. 10.
    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) Google Scholar
  11. 11.
    Faugeras, O., Hotz, B., Mathieu, H., Viéville, T., Zhang, Z., Fua, P., Théron, E., Moll, L., Berry, G., Vuillemin, J., Bertin, P., Proy, C.: Real time correlation based stereo: Algorithm implementations and applications. RR-2013, INRIA, (1993). http://perception.inrialpes.fr/Publications/1993/FHMVZFTMBVBP93
  12. 12.
    Faugeras, O., Kerivan, R.: Variational principals, surface evolution, pde’s, level set methods and the stereo problem. Trans. Image Process. 7(3), 336–344 (1998) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Faugeras, O.: Three-Dimensional Computer Vision. MIT Press, Cambridge (1993) Google Scholar
  14. 14.
    Fisher, R., Dawson-Howe, K., Fitzgibbon, A., Robertson, C., Trucco, E.: Dictionary of Computer Vision and Image Processing. Wiley, New York (2005) CrossRefGoogle Scholar
  15. 15.
    Fitzgibbon, A., Pilu, M., Fisher, R.B.: Direct least squares fitting of ellipses. Trans. Pattern Anal. Mach. Intell. 21(5), 476–480 (1999) CrossRefGoogle Scholar
  16. 16.
    Forsyth, D.A.: Recognizing algebraic surfaces from their outlines. Int. J. Comput. Vis. 18(1), 21–40 (1996) CrossRefMathSciNetGoogle Scholar
  17. 17.
    Geusebroek, J.M., Burghouts, G.J., Smeulders, A.W.M.: The Amsterdam library of object images. Int. J. Comput. Vis. 61(1), 103–112 (2005). http://staff.science.uva.nl/~aloi/ CrossRefGoogle Scholar
  18. 18.
    Giblin, P.J., Weiss, R.S.: Epipolar fields on surfaces. In: Proceedings of the European Conference on Computer Vision. LNCS 800, vol. 1, pp. 14–23. Springer, Berlin (1994) Google Scholar
  19. 19.
    Gonzalez, R.C., Woods, R.E.: Digital Image Processing. Prentice Hall, Englewood Cliffs (2002) Google Scholar
  20. 20.
    Himmelblav, D.M.: Applied Nonlinear Programming. McGraw-Hill, New York (1972) Google Scholar
  21. 21.
    Kanade, T.: Development of a video-rate stereo machine. In: Image Understanding Workshop, pp. 549–557 (1994) Google Scholar
  22. 22.
    Karl, W.C., Verghese, G.C., Willsky, A.S.: Reconstructing ellipsoids from projections. Graph. Models Image Process. 56(2), 124–139 (1994) CrossRefGoogle Scholar
  23. 23.
    Lucas, B.D., Kanade, T.: An iterative image registration technique with an application in stereo vision. In: Proceedings of the International Joint Conference on Artificial Intelligence (1981) Google Scholar
  24. 24.
    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) Google Scholar
  25. 25.
    Marr, D., Hildreth, E.C.: Theory of edge detection. In: Proceedings of the Royal Society, vol. B, pp. 187–217 (1980) Google Scholar
  26. 26.
    Marsden, J.E.: Elementary Classical Analysis. Freeman, San Francisco (1974) MATHGoogle Scholar
  27. 27.
    Millman, R.S., Parker, G.D.: Elements of Differential Geometry. Prentice Hall, Englewood Cliffs (1977) MATHGoogle Scholar
  28. 28.
    Otten, R.H.J.M., van Ginneken, L.P.P.P.: The Annealing Algorithm. Kluwer, Boston (1989) Google Scholar
  29. 29.
    Polak, E.: Optimization: Algorithms and Consistent Approximations. Springer, New York (1997) MATHGoogle Scholar
  30. 30.
    Porill, J., Pollard, S.: Curve matching and stereo calibration. Image Vis. Comput. 9(1), 45–50 (1991) CrossRefGoogle Scholar
  31. 31.
    Rieger, J.H.: Three-dimensional motion from fixed points of a deforming profile curve. Opt. Lett. 11(3), 123–125 (1986) CrossRefGoogle Scholar
  32. 32.
    Robert, L., Deriche, R.: Dense depth map reconstruction: A minimization and regularization approach that preserves discontinuities. In: Proceedings of the European Conference on Computer Vision, pp. 439–451. Springer, Berlin (1996) Google Scholar
  33. 33.
    Roy, S.: Stereo without epipolar lines: A maximum flow formulation. Int. J. Comput. Vis. 34(2/3), 147–161 (1999) CrossRefGoogle Scholar
  34. 34.
    Scharstein, D.: Matching images by comparing their gradient fields. In: International Conference on Pattern Recognition, vol. 1, pp. 572–575 (1994) Google Scholar
  35. 35.
    Scharstein, D., Szeliski, R., Zabih, R.: A taxonomy and evaluation of dense two-frame stereo correspondence algorithms. Int. J. Comput. Vis. 47, 7–42 (2002) MATHCrossRefGoogle Scholar
  36. 36.
    Seitz, S., Dyer, C.M.: Photorealistic scene reconstruction by voxel coloring. Int. J. Comput. Vis. 35(2), 1–23 (1999) CrossRefGoogle Scholar
  37. 37.
    Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, San Diego (1982) MATHGoogle Scholar
  38. 38.
    Shashua, A., Toelg, S.: The quadric reference surface: theory and applications. Int. J. Comput. Vis. 23(2), 185–198 (1997) CrossRefGoogle Scholar
  39. 39.
    Sneddon, I.N.: Elements of Partial Differential Equations. McGraw-Hill, New York (1957) MATHGoogle Scholar
  40. 40.
    Sun, J., Zheng, N.-N., Shum, H.-Y.: Stereo matching using belief propagation. Trans. Pattern Anal. Mach. Intell. 25(7), 787–800 (2003) CrossRefGoogle Scholar
  41. 41.
    Szeliski, R.: Prediction error as a quality metric for motion and stereo. Int. J. Comput. Vis. 2, 781–788 (1999) Google Scholar
  42. 42.
    Teutsch, C., Berndt, D., Trostmann, E., Weber, M.: Real-time detection of elliptic shapes for automated object recognition and object tracking. In: Proceedings of Machine Vision Applications in Industrial Inspection XIV, Electronic Imaging 2006, 15–19 January 2006, San Jose, California, USA, pp. 171–179. SPIE/IS&T (2006) Google Scholar
  43. 43.
    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) Google Scholar
  44. 44.
    Xie, Y., Ji, Q.: A new efficient ellipse detection method. In: Proceedings of the International Conference on Pattern Recognition, vol. 2, pp. 957–960 (2002) Google Scholar
  45. 45.
    Zitnick, C.L., Kanade, T.: A cooperative algorithm for stereo matching and occlusion detection. Trans. Pattern Anal. Mach. Intell. 22(7), 675–684 (2000) CrossRefGoogle Scholar
  46. 46.
    Zwillinger, D.: CRC Standard Mathematical Tables and Formulae, 31st edn. Chapman and Hall/CRC, London (2003) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Schools of Mathematics and StatisticsUniversity of Western AustraliaCrawleyAustralia
  2. 2.Schools of Computer Science and Software EngineeringUniversity of Western AustraliaCrawleyAustralia
  3. 3.Faculty of Mathematics and Natural SciencesUniversity of Cardinal WyszynskiWarsawPoland

Personalised recommendations