Skip to main content
SpringerLink
Log in

Differential and Integral Geometry of Linear Scale-Spaces

  • Published:
Journal of Mathematical Imaging and Vision Aims and scope Submit manuscript

Abstract

Linear scale-space theory provides a useful framework to quantify the differential and integral geometry of spatio-temporal input images. In this paper that geometry comes about by constructing connections on the basis of the similarity jets of the linear scale-spaces and by deriving related systems of Cartan structure equations. A linear scale-space is generated by convolving an input image with Green's functions that are consistent with an appropriate Cauchy problem. The similarity jet consists of those geometric objects of the linear scale-space that are invariant under the similarity group. The constructed connection is assumed to be invariant under the group of Euclidean movements as well as under the similarity group. This connection subsequently determines a system of Cartan structure equations specifying a torsion two-form, a curvature two-form and Bianchi identities. The connection and the covariant derivatives of the curvature and torsion tensor then completely describe a particular local differential geometry of a similarity jet. The integral geometry obtained on the basis of the chosen connection is quantified by the affine translation vector and the affine rotation vectors, which are intimately related to the torsion two-form and the curvature two-form, respectively. Furthermore, conservation laws for these vectors form integral versions of the Bianchi identities. Close relations between these differential geometric identities and integral geometric conservation laws encountered in defect theory and gauge field theories are pointed out. Examples of differential and integral geometries of similarity jets of spatio-temporal input images are treated extensively.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. P. Balachandran, G. Bimonte, G. Landi, F. Lizzi, and P. Teotonio-Sobrinho, “Lattice gauge fields and noncommutative geometry,” Universidade de Sao Paulo, Instituto de Fisica-DFMA, Sao Paulo, SP, Brazil, 1996.

    Google Scholar 

  2. G. Bimonte, E. Ercolessi, F. Lizzi, G. Landi, G. Sparano, and P. Teotonio-Sobrinho, “Noncommutative lattices and their continuum limits,” Departamento de Fysica Teorica, Facultade de Ciencias, Universitad de Zaragoza, Zaragoza, Spain, 1995.

    Google Scholar 

  3. J. Blom, “Topological and geometrical aspects of image structure,” Ph. D. thesis, Utrecht University, 1992.

  4. E. Cartan, Sur les Variétés à Connexion Affine et la Théorie de la Relativité Généralisée, Gauthiers-Villars, 1955.

  5. R. Cipolla and A. Blake, “Surface orientation and time to contact from image divergence and deformation,” in Proc. ECCV'92, Santa Margherita Ligure, Italy, May 1992, pp. 187–202.

    Google Scholar 

  6. J. Damon, “Local Morse theory for solutions to the heat equation and Gaussian blurring,” Jour. Diff. Eqns., 1993.

  7. D. H. Eberly, “Geometric methods for analysis of ridges in n-dimensional images,” Ph. D. thesis, Department of Computer Vision, The University of North Carolina, Chapel Hill, North Carolina, 1994.

    Google Scholar 

  8. A. V. Evako, R. Kopperman, and Y. V. Mukhin, “Dimensional properties of graphs and digital spaces,” Journal of Mathematical Imaging and Vision, Vol. 6, No.2/3, pp. 109–119, 1996.

    Google Scholar 

  9. O. Faugeras, “On the motion of 3D curves and its relationship to optical flow,” in Proc. ECCV'90, Antibes, France, April 1990, pp. 107–117.

    Google Scholar 

  10. L. M. J. Florack and M. Nielsen, “The intrinsic structure of the optic flow field,” Technical Report ERCIM-07/94-R033 or INRIARR-2350, ERCIM, July 1994.

  11. L. M. J. Florack, B. M. ter Haar Romeny, J. J. Koenderink, and M. A. Viergever, “Cartesian differential invariants in scalespace,” Journal of Mathematical Imaging and Vision, Vol. 3, pp. 327–348, 1993.

    Google Scholar 

  12. F. W. Hehl, J. Dermott McCrea, and E. W. Mielke, Weyl Spacetimes, The Dilation Current, and Creation of Gravitating Mass by Symmetry Breaking, Verlag Peter Lang: Frankfurt, 1988, pp. 241–311.

    Google Scholar 

  13. G. T. Herman and Z. Enping, “Jordan surfaces in simply connected digital spaces,” Journal of Mathematical Imaging and Vision, Vol. 6, No.2/3, pp. 121–138, 1996.

    Google Scholar 

  14. B. K. P. Horn, Robot Vision, MIT Press: Cambridge MA, 1986.

    Google Scholar 

  15. P. Johansen, S. Skelboe, K. Grue, and J. D. Andersen, “Representing signals by their top points in scale-space,” in Proceedings of the 8th International Conference on Pattern Recognition, 1986, pp. 215–217.

  16. P. Johansen, “On the classification of top-points in scale-space,” Journal of Mathematical Imaging and Vision, Vol. 4, pp. 57–68, 1994.

    Google Scholar 

  17. A. Kadi and D. G. B. Edelen, A gauge theory of dislocations and disclinations, Lecture Notes in Physics, Vol. 174, Springer-Verlag: Berlin, 1983.

    Google Scholar 

  18. S. N. Kalitzin, B. M. ter Haar Romeny, A. H. Salden, P. F. M. Nacken, and M. A. Viergever, “Topological numbers and singularities in scalar images; scale-space evolution properties,” Journal of Mathematical Imaging and Vision, accepted.

  19. H. Kleinert, Gauge Fields in Condensed Matter, World Scientific Publishing Co.: Singapore, Vol. 1–2, 1989.

    Google Scholar 

  20. J. J. Koenderink, “The structure of images,” Biol. Cybern., Vol. 50, pp. 363–370, 1984.

    Google Scholar 

  21. J. J. Koenderink, “Scale-time,” Biol. Cybern., Vol. 58, pp. 159–162, 1988.

    Google Scholar 

  22. J. J. Koenderink and W. Richards, “Two-dimensional curvature operators,” Journal of the Optical Society of America-A, Vol. 5, pp. 1136–1141, 1988.

    Google Scholar 

  23. J. J. Koenderink and A. J. van Doorn, “Invariant properties of the motion parallax field due to the movement of rigid bodies relative to an observer,” Optica Acta, Vol. 22, pp. 773–791, 1975.

    Google Scholar 

  24. J. J. Koenderink and A. J. van Doorn, “A description of the structure of visual images in terms of an ordered hierarchy of light and dark blobs,” in Second Int. Visual Psychophysics and Medical Imaging Conf., IEEE Cat. No. 81 CH 1676-6, 1981.

  25. J. J. Koenderink and A. J. van Doorn, “Local features of smooth shapes: Ridges and courses,” in Proc. SPIE Geometric Methods in Computer Vision II, 1993, Vol. 2031, pp. 2–13.

    Google Scholar 

  26. J. J. Koenderink, “A hitherto unnoticed singularity of scalespace,” IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. 11, pp. 1222–1224, 1989.

    Google Scholar 

  27. T. Lindeberg, “Scale-space for discrete signals,” IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. 12, pp. 234–245, 1990.

    Google Scholar 

  28. T. Lindeberg, “Scale-space behaviour of local extrema and blobs,” Journal of Mathematical Imaging and Vision, Vol. 1, pp. 65–99, 1992.

    Google Scholar 

  29. T. Lindeberg, Scale-Space Theory in Computer Vision, The Kluwer International Series in Engineering and Computer Science., Kluwer Academic Publishers: Dordrecht, The Netherlands, 1994.

    Google Scholar 

  30. McAndrew and C. A. Osborne, “A survey of algebraic methods in digital topology,” Journal of Mathematical Imaging and Vision, Vol. 6, No.2/3, pp. 139–159, 1996.

    Google Scholar 

  31. M. Newman, “A fundamental group for gray-scale digital images,” Journal of Mathematical Imaging and Vision, Vol. 6, No.2/3, pp. 139–159, 1996.

    Google Scholar 

  32. T. Okubo, Differential Geometry, Marcel Dekker Inc.: New York, 1987.

    Google Scholar 

  33. J. F. Pommaret, Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach: Paris, 1978.

    Google Scholar 

  34. J. H. Rieger, “Generic evolutions of edges on families of diffused gray-value surfaces,” Journal of Mathematical Imaging and Vision, Vol. 5, No.3, pp. 207–217, 1995.

    Google Scholar 

  35. A. H. Salden, “Dynamic scale-space paradigms,” Ph. D. thesis, Utrecht University, The Netherlands, 1996.

    Google Scholar 

  36. A. H. Salden, “Invariant theory,” in Gaussian Scale-Space Theory, Kluwer Academic Publishers: Dordrecht, The Netherlands, 1996.

    Google Scholar 

  37. A. H. Salden, L. M. J. Florack, and B. M. ter Haar Romeny, “Differential geometric description of 3D scalar images,” 3D Computer Vision, Utrecht, The Netherlands, Technical Report 91-23, 1991.

    Google Scholar 

  38. A. H. Salden, L. M. J. Florack, B. M. ter Haar Romeny, J. J. Koenderink, and M. A. Viergever, “Multi-scale analysis and description of image structure,” in Nieuw Archief voor Wiskunde, 1992, Vol. 10, pp. 309–326.

    Google Scholar 

  39. A. H. Salden, B. M. ter Haar Romeny, L. M. J. Florack, J. J. Koenderink, and M. A. Viergever, “A complete and irreducible set of local orthogonally invariant features of 2-dimensional images,” in Proceedings 11th IAPR Internat. Conf. on Pattern Recognition, The Hague, The Netherlands, 1992, pp. 180–184.

    Google Scholar 

  40. A. H. Salden, B. M. ter Haar Romeny, and M. A. Viergever, “Local and multilocal scale-space description,” in Proc. of the NATO Advanced Research Workshop Shape in Picture—Mathematical Description of Shape in Gray-Level Images, Vol. 126 of NATO ASI Series F, Berlin, 1994, pp. 661–670.

    Google Scholar 

  41. A. H. Salden, B. M. ter Haar Romeny, and M. A. Viergever, “Modern Geometry and Dynamic scale-space theory,” in Proc. Conf. on Differential Geometry and Computer Vision: From Pure over Applicable to Applied Differential Geometry, Nordfjordeid, Norway, August 1–7, 1995.

  42. A. H. Salden, B. M. ter Haar Romeny, and M. A. Viergever, “Algebraic invariants of linear scale-spaces,” submitted to Journal of Mathematical Imaging and Vision, March 1996.

  43. A. H. Salden, B. M. ter Haar Romeny, and M. A. Viergever, “Linear scale-space theory from physical principles,” Journal of Mathematical Imaging and Vision, accepted.

  44. L. A. Santalo, “Integral geometry in general spaces,” in Proceedings International Congress of Mathematics, Cambridge, Vol. 1, 1950, pp. 483–489.

    Google Scholar 

  45. L. A. Santalo, “Integral Geometry and Geometric Probability,” Addison-Wesley Publishing Company: London, 1976, Vol. 1.

    Google Scholar 

  46. M. Spivak, Differential Geometry, Publish or Perish, Inc.: Berkeley, California, USA, Vol. 1–5, 1975.

    Google Scholar 

  47. B. M. ter Haar Romeny, Geometry-Driven Diffusion in Computer Vision, Kluwer Academic Publishers: Dordrecht, 1994.

    Google Scholar 

  48. A. L. Yuille and T. A. Poggio, “Scaling theorems for zero-crossings,” IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. 8, pp. 15–25, 1986.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Utrecht University Hospital, Image Sciences Institute, Heidelberglaan 100, 3584 CX, Utrecht, The Netherlands

    Alfons H. Salden, Bart M. Ter Haar Romeny & Max A. Viergever

Authors
  1. Alfons H. Salden
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bart M. Ter Haar Romeny
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Max A. Viergever
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Salden, A.H., Ter Haar Romeny, B.M. & Viergever, M.A. Differential and Integral Geometry of Linear Scale-Spaces. Journal of Mathematical Imaging and Vision 9, 5–27 (1998). https://doi.org/10.1023/A:1008253609285

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1008253609285

Search

Navigation