Skip to main content
Log in

Adjunctions in Pyramids, Curve Evolution and Scale-Spaces

  • Published:
International Journal of Computer Vision Aims and scope Submit manuscript

Abstract

We have been witnessing lately a convergence among mathematical morphology and other nonlinear fields, such as curve evolution, PDE-based geometrical image processing, and scale-spaces. An obvious benefit of such a convergence is a cross-fertilization of concepts and techniques among these fields. The concept of adjunction however, so fundamental in mathematical morphology, is not yet shared by other disciplines. The aim of this paper is to show that other areas in image processing can possibly benefit from the use of adjunctions. In particular, a strong relationship between pyramids and adjunctions is presented. We show how this relationship may help in analyzing existing pyramids, and construct new pyramids. Moreover, it will be explained that adjunctions based on a curve evolution scheme can provide idempotent shape filters. This idea is illustrated in this paper by means of a simple affine-invariant polygonal flow. Finally, the use of adjunctions in scale-space theory is also addressed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  • Bruckstein, A.M., Sapiro, G., and Shaked, D. 1995. Evolutions of planar polygons. Intern. J. of Pattern Recognition and Artificial Intelligence, 9(6):991–1014.

    Google Scholar 

  • Goutsias, J. and Heijmans, H.J.A.M. 2000. Nonlinear multiresolution signal decomposition schemes: Part I: morphological pyramids. IEEE Transactions on Image Processing, 9:1862–1876.

    Google Scholar 

  • Heijmans, H.J.A.M. and Keshet (Kresch), R. 2002. Inf-semilattice approach to self-dual morphology. JVCIR, 13:269–301.

    Google Scholar 

  • Heijmans, H.J.A.M. and Ronse, C. 1990. The algebraic basis of mathematical morphology. I. Dilations and erosions. Comput. Vision Graphics Image Process., 50:245–295.

    Google Scholar 

  • Heijmans, H.J.A.M. and van den Boomgaard, R. 2000. Algebraic framework for linear and morphological scale-spaces. CWI Report PNA-R0003, Amsterdam. Also in JVCIR (to appear).

  • Kimmel, R. 1995. Curve evolution on surfaces. D.Sc. thesis, Technion-Israel Institute of Technology.

  • Kresch, R. 1998. Extension of morphological operations to complete semilattices and its applications to image and video processing. In Mathematical Morphology and its Applications to Image and Signal Processing (Proc. of ISMM'98), H.J.A.M. Heijmans and J.B.T.M. Roerdink (Eds.), pp. 35–42.

  • Keshet (Kresch), R. 2000.Amorphological viewon traditional signal processing. In Mathematical Morphology and its Applications to Image and Signal Processing V (Proc. of ISMM'2000), Palo Alto, California, USA.

  • Malladi, R. and Sethian, J.A. 1995. Image processing via level set curvature flow. Proc. of the National Academy of Sciences, 92(15):7046–7050.

    Google Scholar 

  • Moisan, L. 1998. Affne plane curve evolution: A fully consistent scheme. IEEE Transactions on Image Processing, 7(3):411–420.

    Google Scholar 

  • Pauwels, E.J., Van Gool, L.J., Fiddelaers, P., and Moons, T. 1995. An extended class of scale-invariant and recursive scale space filters. IEEE Trans. on PAMI, 17(7).

  • Rudin, L., Osher, S., and Fatemi, E. 1992. Nonlinear total variation based noise removal algorithms. Physica D, 60:259–268.

    Google Scholar 

  • Sapiro, G. 1995. Geometric partial differential equations in image analysis: Past, present, and future. In Proc. ICIP'95, Vol. 3, pp. 1–4.

    Google Scholar 

  • Sapiro, G. 2001. Geometric Partial Differential Equations and Image Analysis. Cambridge University Press.

  • Serra, J. (Ed.). 1988. Image Analysis and Mathematical Morphology, Volume 2: Theoretical Advances. Academic Press: New York.

    Google Scholar 

  • Steiner, A., Kimmel, R., and Bruckstein, A.M. 1996. Planar shape enhancement and exaggeration. In Proc. ICPR 96', pp. 523–527.

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

(Kresch), R.K., Heijmans, H.J. Adjunctions in Pyramids, Curve Evolution and Scale-Spaces. International Journal of Computer Vision 52, 139–151 (2003). https://doi.org/10.1023/A:1022952007509

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1022952007509

Navigation