Abstract
We define mutually consistent scale-space theories for scalar and vector images. Consistency pertains to the connection between the already established scalar theory and that for a suitably defined scalar field induced by the proposed vector scale-space. We show that one is compelled to reject the Gaussian scale-space paradigm in certain cases when scalar and vector fields are mutually dependent.
Subsequently we investigate the behaviour of critical points of a vector-valued scale-space image—i.e. points at which the vector field vanishes— as well as their singularities and unfoldings in linear scale-space.
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
R. van den Boomgaard. The morphological equivalent of the Gauss convolution. Nieuw Archief voor Wiskunde, 10(3):219–236, November 1992.
R. van den Boomgaard and A.W.M. Smeulders. The morphological structure of images, the differential equations of morphological scale-space. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16(11):1101–1113, November 1994.
J. Damon. Local Morse theory for solutions to the heat equation and Gaussian blurring. Journal of Differential Equations, 115(2):368–401, January 1995.
L.C. Evans. Partial Differential Equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, Rhode Island, 1998.
L. Florack and A. Kuijper. The topological structure of scale-space images. Journal of Mathematical Imaging and Vision, 12(1):65–79, February 2000.
L.M.J. Florack. Non-linear scale-spaces isomorphic to the linear case. In B.K. Ersb∅ll and P. Johansen, editors, Proceedings of the 11th Scandinavian Conference on Image Analysis (Kangerlussuaq, Greenland, June 7-11 1999), volume 1, pages 229–234, Lyngby, Denmark, 1999.
L.M.J. Florack, W.J. Niessen, and M. Nielsen. The intrinsic structure of optic flow incorporating measurement duality. International Journal of Computer Vision, 27(3):263–286, May 1998.
J.J. Koenderink. The structure of images. Biological Cybernetics, 50:363–370, 1984.
M. Nielsen and O.F. Olsen. The structure of the optic flow field. In H. Burkhardt and B. Neumann, editors, Proceedings of the Fifth European Conference on Computer Vision (Freiburg, Germany, June 1998), volume 1407 of Lecture Notes in Computer Science, pages 281–287. Springer-Verlag, Berlin, 1998.
A. Simmons, S.R. Arridge, P.S. Tofts, and G.J. Barker. Application of the extremum stack to neurological MRI. IEEE Transactions on Medical Imaging, 17(3):371–382, June 1998.
M. Spivak. Differential Geometry, volume 1. Publish or Perish, Berkeley, 1975.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Florack, L.M.J. (2001). Scale-Space Theories for Scalar and Vector Images. In: Kerckhove, M. (eds) Scale-Space and Morphology in Computer Vision. Scale-Space 2001. Lecture Notes in Computer Science 2106, vol 2106. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-47778-0_16
Download citation
DOI: https://doi.org/10.1007/3-540-47778-0_16
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42317-1
Online ISBN: 978-3-540-47778-5
eBook Packages: Springer Book Archive