Abstract
There is an increasing interest in applications of algebraic geometry to computer vision. There are at least three possible reasons for this: i) certain vision problems naturally involve polynomial equations; ii) with the increase in available computing power it is easier to implement algorithms which stay close to the geometry underlying vision; and iii) algebraic geometry may in future provide methods for assessing the stability of algorithms against small perturbations in the data.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Buchanan T. 1992 Critical sets for reconstruction using lines. Proc. 2nd European Conf. on Computer Vision, ECCV92.
Demazure M. 1988 Sur deux problèmes de reconstruction. Technical Report No. 882, INRIA, Rocquencourt, France.
Faugeras O.D. 1990 On the motion of 3D curves and its relationship to optical flow. Proc. 1st European Conf. on Computer Vision, ECCV90, Lecture Notes in Computer Science 427, Springer-Verlag.
Faugeras O.D. Luong Q.-T. & Maybank S.J. 1992 Camera self-calibration: theory and experiments. Proc. 2nd European Conf. on Computer Vision, ECCV92.
Faugeras O.D. & Maybank S.J. 1990 Motion from point matches: multiplicity of solutions. International J. Computer Vision 4, 225–246.
Forsyth D., Mundy J., Zisserman A., Coelho C., Heller A. & Rothwell C. 1991 Invariant descriptors for 3-D object recognition and pose. IEEE Trans. Pattern Analysis and Machine Intelligence 13, 971–991.
Hofmann W. 1950 Das Problem der “Gerfährlichen Flächen„ in Theorie und Praxis. Dissertation, Fakultät für Bauwesen der Technischen Hochschule München, München, FR Germany. Published in Reihe C, No. 3 der Deutschen Geodetischen Kommission bei der Bayerischen Akademie der Wissenschaften, München 1953.
Horn B.K.P. 1986 Robot Vision. Cambridge, Massachusetts: The MIT Press.
Huang T.S. & Faugeras O.D. 1989 Some properties of the E matrix in two view motion estimation. IEEE Trans. Pattern Analysis and Machine Intelligence 11, 1310–1312.
Kruppa E. 1913 Zur Ermittlung eines Objektes zwei Perspektiven mit innere Orientierung. Sitz-Ber. Akad. Wiss., Wien, math. naturw. Kl., Abt. IIa. 122, 1939–1948.
Liu Y. & Huang T.S. 1988 Estimation of rigid body motion using straight line correspondences. Computer Vision, Graphics, and Image Processing 44, 35–57.
Longuet-Higgins H.C. 1981 A computer algorithm for reconstructing a scene from two projections. Nature 293, 133–135.
Longuet-Higgins H.C. 1988 Multiple interpretations of a pair of images of a surface. Proc. Royal Soc. London, Series B 227, 399–410.
Maybank S.J. 1990 The projective geometry of ambiguous surfaces. Phil. Trans. Royal Soc. London, Series A 332, 1–47.
Maybank S.J. 1991 The projection of two non-coplanar conics. In Proc. First DARPA-ESPRIT Joint Workshop on Applications of Invariant Theory to Computer Vision, Reykjavik, 25–28 March 1991.
Maybank S.J. & Faugeras O.D. 1992 A theory of self-calibration of a moving camera. Accepted by International J. Computer Vision.
Maybank S.J. 1992 Theory of Reconstruction from Image Motion. Springer-Verlag, to appear.
J. Mundy & A. Zisserman (eds.) 1992 Applications of Invariance in Computer Vision. To appear, MIT Press.
Spetsakis M.E. & Aloimonos J. 1990 Structure from motion using line correspondences. International J. Computer Vision 4, 171–183.
Sturm R. 1869 Das Problem der Projectivität und seine Anwendung auf die Flachen zweiten Grades. Math. Annalen 1, 533–573.
Tsai R.Y. & Huang T.S. 1984 Uniqueness and estimation of three-dimensional motion parameters of rigid objects with curved surfaces. IEEE Trans. Pattern Analysis and Machine Intelligence 6, 13–27.
Weng J., Huang T.S. & Ahuja N. 1992 Motion and structure from line correspondences: closed-form solution, uniqueness, and optimization. IEEE Trans. Pattern Analysis and Machine Intelligence 14, 318–336.
Wilczynski E.J. 1906 Projective Differential Geometry of Curves and Surfaces. Leipzig: Teubner.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1993 Birkhäuser Boston
About this paper
Cite this paper
Maybank, S.J. (1993). Applications of Algebraic Geometry to Computer Vision. In: Eyssette, F., Galligo, A. (eds) Computational Algebraic Geometry. Progress in Mathematics, vol 109. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-2752-6_13
Download citation
DOI: https://doi.org/10.1007/978-1-4612-2752-6_13
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-7652-4
Online ISBN: 978-1-4612-2752-6
eBook Packages: Springer Book Archive