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Uncertainty and Projective Geometry

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Handbook of Geometric Computing

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References

  1. M. Ashdown. The GA package for MAPLE release V. http://www.mrao.cam.ac.uk/~clifford/software/GA/, May 2004.

    Google Scholar 

  2. W. Baarda (1973). S-Transformations and Criterion Matrices, vol. 5 of 1. Netherlands Geodetic Commission, Delft, 1973.

    Google Scholar 

  3. L. Brand (1966). Vector and Tensor Analysis. Wiley.

    Google Scholar 

  4. S. Carlsson (1994). The double algebra: an effective tool for computing invariants in computer vision. In: J. Mundy, Zisserman A., and D. Forsyth (eds.) Applications of Invariance in Computer Vision, LNCS, vol. 825. Springer, Berlin Heidelberg New York, pp. 145–164, 1994.

    Google Scholar 

  5. W. Chojnacki, M. J. Brooks, A. van den Hengel (2001). Rationalising the renormalisation method of Kanatani. Journal of Mathematical Imaging and Vision, 14(1):21–38.

    Article  MathSciNet  MATH  Google Scholar 

  6. W. Chojnacki, M. J. Brooks, A. van den Hengel, D. Gawley (2000). On the fitting of surfaces to data with covariances. IEEE Trans. Pattern Analysis Machine Intelligence, 22(11):1294–1303.

    Article  Google Scholar 

  7. W. Chojnacki, M. J. Brooks, A. van den Hengel, D. Gawley (2003). From FNS to HEIV: A link between two vision parameter estimation methods. IEEE Transactions on Pattern Analysis of Machine Intelligence, 26(2):264–268.

    Article  Google Scholar 

  8. R. Collins (1993). Model Acquisition Using Stochastic Projective Geometry. PhD thesis, Department of Computer Science, University of Massachusetts. Also published as UMass Computer Science Technical Report TR95-70.

    Google Scholar 

  9. A. Criminisi (2001). Accurate Visual Metrology from Single and Multiple Uncalibrated Images. Springer-Verlag London Ltd.

    Google Scholar 

  10. O. Faugeras, Q. Luong, with contributions by T. Papdopoulo (2001). The geometry of multiple images. MIT Press, Cambridge, MA.

    MATH  Google Scholar 

  11. O. Faugeras, T. Papadopoulo (1998). Grassmann-Cayley algebra for modeling systems of cameras and the algebraic equations of the manifold of trifocal tensors. In Trans. of the Royal Society A, 365:1123–1152.

    Article  MathSciNet  Google Scholar 

  12. W. Förstner (1996). 10 pros and cons against performance characterisation of vision algorithms. In: Madsen C. B. Christensen H. I., Förstner W. (eds.) Proceedings of the ECCV Workshop on Performance Characteristics of Vision Algorithms, pages 13–29, Cambridge, UK.

    Google Scholar 

  13. W. Förstner (1979). Ein Verfahren zur Schätzung von Varianz-und Kovarianzkomponenten. Allgemeine Vermessung Nachrichten, 11–12:446–453.

    Google Scholar 

  14. W. Förstner (2001). Algebraic projective geometry and direct optimal estimation of geometric entities. In Stefan Scherer (ed.) Computer Vision, Computer Graphics and Photogrammetry — A common viewpoint., Proc. 25th Workshop of the Austrian Association for Pattern Recognition (ÖAGM/AAPR), Österreichische Computer Gesellschaft, pp. 67–86, 2001

    Google Scholar 

  15. W. Förstner, A. Brunn, S. Heuel (2000). Statistically testing uncertain geometric relations. In G. Sommer, N. Krüger, and C. Perwass (eds.) Mustererkennung 2000, pages 17–26. DAGM, Springer, Berlin Heidelberg New York, 2000.

    Google Scholar 

  16. J. Haddon, D. A. Forsyth (2001). Noise in bilinear problems. In Proceedings of ICCV, volume II, pages 622–627, Vancouver, IEEE Computer Society.

    Google Scholar 

  17. R. Hartley (1995). In defense of the 8 point algorithm. In ICCV 95, pages 1064–1070.

    Google Scholar 

  18. R. I. Hartley, A. Zisserman (2000). Multiple View Geometry in Computer Vision. Cambridge University Press.

    Google Scholar 

  19. H.-P. Helfrich, D. Zwick (1996). A trust region algorithm for parametric curve and surface fitting. J. Comp. Appl. Math., 73:119–134.

    Article  MathSciNet  MATH  Google Scholar 

  20. F. R. Helmert (1872). Die Ausgleichungsrechnung nach der Methode der Kleinsten Quadrate. Teubner, Leipzig.

    MATH  Google Scholar 

  21. D. Hestenes, G. Sobczyk (1984). Clifford algebra to geometric calculus. D. Reidel Publishing Comp.

    Google Scholar 

  22. D. Hestenes, R. Ziegler (1991). Projective geometry with Clifford algebra. Acta Applicandae Mathematicae.

    Google Scholar 

  23. S. Heuel (2004). Uncertain Projective Geometry — Statistical Reasoning for Polyhedral Object Reconstruction. LNCS 3008. Springer, Berlin Heidelberg New York.

    MATH  Google Scholar 

  24. K. Kanatani (1994). Statistical bias of conic fitting and renormalization. IEEE Trans. Pattern Analysis and Machin Intelligence, 16(3):320–326.

    Article  MATH  Google Scholar 

  25. K. Kanatani (1996). Statistical Optimization for Geometric Computation: Theory and Practice. Elsevier Science.

    Google Scholar 

  26. K. Kanatani, D.D. Morris (2001). Gauges and gauge transformations for uncertainty description of geometric structure with indeterminacy. IEEE Transactions on Information Theory, 47(5):1–12.

    Article  MathSciNet  Google Scholar 

  27. K.-R. Koch (1988). Parameter estimation and hypothesis testing in linear models. Springer, Berlin Heidelberg New York.

    MATH  Google Scholar 

  28. B. Matei, P. Meer (2000). A general method for errors-in-variables problems in computer vision. In Computer Vision and Pattern Recognition Conference, volume II, pages 18–25. IEEE.

    Google Scholar 

  29. E. M. Mikhail, F. Ackermann (1976). Observations and Least Squares. University Press of America.

    Google Scholar 

  30. A. Papoulis (1965). Probability, Random Variables, and Stochastic Processes. McGraw-Hill.

    Google Scholar 

  31. R. C. Rao (1973). Linear Statistical Inference and Its Applications. Wiley, NY.

    MATH  Google Scholar 

  32. B. Rosenhahn, G. Sommer (2002). Pose estimation in conformal geometric algebra. Technical Report 0206, Inst. f. Informatik u. Praktische Mathematik, Universität Kiel.

    Google Scholar 

  33. J. G. Semple, G. T. Kneebone (1952). Algebraic Projective Geometry. Oxford Science.

    Google Scholar 

  34. R. Smith, M. Self, P. Cheeseman (1991). A Stochastic Map for Uncertain Spatial Relationships. In: S. S. Iyengar, A. Elfes (eds.): Autonomous Mobile Robots: Perception, Mapping, and Navigation, vol. 1. IEEE Computer Society Press, pp. 323–330.

    Google Scholar 

  35. J. Stolfi (1991). Oriented Projective Geometry: A Framework for Geometric Computations. Academic Press, San Diego.

    MATH  Google Scholar 

  36. G. Taubin (1993). An improved algorithm for algebraic curve and surface fitting. In Fourth ICCV, Berlin, pages 658–665.

    Google Scholar 

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Förstner, W. (2005). Uncertainty and Projective Geometry. In: Handbook of Geometric Computing. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-28247-5_15

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  • DOI: https://doi.org/10.1007/3-540-28247-5_15

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