Abstract
For the robust detection of lines in 2-dimensional images, it is feasible to have a general, a priori model reflecting the properties of lines. Based on three simple and generally applicable assumptions we introduce a stochastic line propagation model with its resulting Fokker-Planck equation and Green’s function. The line model implies a line diffusion scheme that is not simply another anisotropic diffusion of scalar-valued luminosity functions, but a mechanism for the anisotropic diffusion of oriented line segments in a 3-dimensional space that encodes position and orientation.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
August, J.: The curve indicator random field. PhD thesis, Yale (2001)
Bosking, W.H., Zhang, Y., Schofield, B., Fitzpatrick, D.: Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex. J. Neurosci. 17(6), 2112–2127 (1997)
Canny, J.: A computational approach to edge detection. IEEE Trans. Pattern Anal. Mach. Intell. 8, 679–714 (1986)
Duits, R., Almsick, M.A.: The explicit solutions of linear left-invariant second order stochastic evolution equations on the 2d-Euclidean motion group. Q. Appl. Math. AMS 66(1), 27–67 (2008)
Duits, R., Franken, E.M.: Left invariant parabolic evolution equations on SE(2) and contour enhancement via invertible orientation scores, part II: Nonlinear left-invariant diffusion equations on invertible orientation scores. Q. Appl. Math. 68, 293–331 (2010)
Duits, R., Franken, E.M.: Left invariant parabolic evolution equations on SE(2) and contour enhancement via invertible orientation scores, part I: Linear left-invariant diffusion equations on SE(2). Q. Appl. Math. 68, 255–292 (2010)
Duits, R., Franken, E.M.: Left-invariant diffusions on the space of positions and orientations and their application to crossing-preserving smoothing of HARDI images. Int. J. Comput. Vis. 92(3), 231–264 (2011)
Duits, R., van Almsick, M.: The explicit solutions of linear left-invariant second order stochastic evolution equations on the 2D-Euclidean motion group. Q. Appl. Math. 66, 27–67 (2008)
Floquet, G.: Sur les équations différentielles linéaires à coefficients périodiques. Ann. Éc. Norm. Super. 12, 47–88 (1883)
Franken, E., Duits, R.: Crossing-preserving coherence-enhancing diffusion on invertible orientation scores. Int. J. Comput. Vis. 85(3), 253–278 (2009)
Iverson, L.A., Zucker, S.W.: Logical/linear operators for image curves. IEEE Trans. Pattern Anal. Mach. Intell. 17(10), 982–996 (1995)
Koffka, K.: Principles of Gestalt Psychology. Harcourt, New York (1935)
Sharon, E., Brandt, A., Basri, R.: Completion energies and scale. IEEE Trans. Pattern Anal. Mach. Intell. 22(10), 1117–1131 (2000)
van Almsick, M.A.: Context models of line and contours. PhD thesis, Eindhoven University of Technology (September 2007)
Williams, L.R., Jacobs, D.: Stochastic completion fields: a neural model of illusory contour shape and salience. Neural Comput. 9(4), 837–858 (1997)
Williams, L., Zweck, J., Wang, T., Thornber, K.: Computing stochastic completion fields in linear-time using a resolution pyramid. Comput. Vis. Image Underst. 76(3), 289–297 (1999)
Zweck, J.W., Williams, L.R.: Euclidean group invariant computation of stochastic completion fields using shiftable-twistable functions. J. Math. Imaging Vision, 1–31 (2003)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag London Limited
About this chapter
Cite this chapter
van Almsick, M. (2012). An A Priori Model of Line Propagation. In: Florack, L., Duits, R., Jongbloed, G., van Lieshout, MC., Davies, L. (eds) Mathematical Methods for Signal and Image Analysis and Representation. Computational Imaging and Vision, vol 41. Springer, London. https://doi.org/10.1007/978-1-4471-2353-8_10
Download citation
DOI: https://doi.org/10.1007/978-1-4471-2353-8_10
Published:
Publisher Name: Springer, London
Print ISBN: 978-1-4471-2352-1
Online ISBN: 978-1-4471-2353-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)