Skip to main content

The Stochastic Structure of Images

  • Conference paper
Scale Space and PDE Methods in Computer Vision (Scale-Space 2005)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 3459))

Included in the following conference series:

  • 1555 Accesses

Abstract

As resolving power increases, image features evolve in an iterative fashion; large scale features will persist, and finer and finer scale features are resolved at each increase in resolution. As such, the observation process is shown to overwhelm natural image statistics. Observation by a linear imaging process imposes natural image statistics to be of random multiplicative nature, rather than additive. The scaling behavior of the random process is driven by the gradient structure in the scene irradiance. From the general structure of multiplicative processes, image statistics are proven to follow a sequential fragmentation process. From these theoretical results, analytical forms for the distributions of image derivative filter responses and gradient magnitude are derived.

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

Access this chapter

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Geusebroek, J.M., Smeulders, A.W.M.: Fragmentation in the vision of scenes. In: Proc. 9th Int. Conf. Comput. Vision., vol. 1, pp. 130–135. IEEE Computer Society, Los Alamitos (2003)

    Chapter  Google Scholar 

  2. Field, D.J.: Relations between the statistics of natural images and the response properties of cortical cells. J. Opt. Soc. Am. A 4, 2370–2393 (1987)

    Article  Google Scholar 

  3. Mallat, S.G.: A theory for multiresolution signal decomposition: The wavelet representation. IEEE Trans. Pattern Anal. Machine Intell. 11, 674–693 (1989)

    Article  MATH  Google Scholar 

  4. Ruderman, D.L., Bialek, W.: Statistics of natural images: Scaling in the woods. Phys. Rev. Let. 73, 814–817 (1994)

    Article  Google Scholar 

  5. van der Schaaf, A., van Hateren, J.H.: Modelling the power spectra of natural images: Statistics and information. Vision Res. 36, 2759–2770 (1996)

    Article  Google Scholar 

  6. Simoncelli, E.P.: Modeling the joint statistics of images in the wavelet domain. In: Proc. SPIE, vol. 3813, pp. 188–195. SPIE, Bellingham (1999)

    Chapter  Google Scholar 

  7. Sigman, M., Cecchi, G.A., Gilbert, C.D., Magnasco, M.O.: On a common circle: Natural scenes and Gestalt rules. In: Proc. Natl. Acad. Sci. USA, vol. 98, pp. 1935–1940 (2001)

    Google Scholar 

  8. Schwartz, O., Simoncelli, E.P.: Natural signal statistics and sensory gain control. Nature Neurosci. 4, 819–825 (2001)

    Article  Google Scholar 

  9. Grenander, U., Srivastava, A.: Probability models for clutter in natural images. IEEE Trans. Pattern Anal. Machine Intell. 23, 424–429 (2001)

    Article  Google Scholar 

  10. Lee, A.B., Mumford, D., Huang, J.: Occlusion models for natural images: A statistical study of a scale-invariant dead leaves model. Int. J. Comput. Vision 41, 35–59 (2001)

    Article  MATH  Google Scholar 

  11. Srivastava, A., Liu, X., Grenander, U.: Universal analytical forms for modeling image probabilities. IEEE Trans. Pattern Anal. Machine Intell. 24, 1200–1214 (2002)

    Article  Google Scholar 

  12. Geusebroek, J.M., Smeulders, A.W.M.: A physical explanation for natural image statistics. In: Chantler, M. (ed.) Proceedings of the 2nd International Workshop on Texture Analysis and Synthesis (Texture 2002), Heriot-Watt University, pp. 47–52 (2002)

    Google Scholar 

  13. Srivastava, A., Lee, A.B., Simoncelli, E.P., Zhu, S.C.: On advances in statistical modeling of natural images. J. Math. Imaging Vision 18, 17–33 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Koenderink, J.J.: The structure of images. Biol. Cybern. 50, 363–370 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  15. Havlin, S., Selinger, R.B., Schwartz, M., Stanley, H.E., Bunde, A.: Random multiplicative processes and transport in structures with correlated spatial disorder. Phys. Rev. Let. 61, 1438–1441 (1988)

    Article  Google Scholar 

  16. Castaing, B., Dubrulle, B.: Fully developed turbulence: A unifying point of view. J. Phys. II (France) 5, 895–899 (1995)

    Article  Google Scholar 

  17. Arneodo, A., Bacry, E., Manneville, S., Muzy, J.F.: Analysis of random cascades using space-scale correlation functions. Phys. Rev. Let. 80, 708–711 (1998)

    Article  Google Scholar 

  18. Brown, W.K.: A theory of sequential fragmentation and its astronomical applications. J. Astrophys. Astr. 10, 89–112 (1989)

    Article  Google Scholar 

  19. Papoulis, A., Pillai, S.U.: Probability, Random Variables, and Stochastic Processes, 4th edn. McGraw-Hill, New York (2002)

    Google Scholar 

  20. Benzi, R., Biferale, L., Crisanti, A., Paladin, G., Vergassola, M., Vulpiani, A.: A random process for the construction of multiaffine fields. Physica D 65, 352–358 (1993)

    Article  MATH  Google Scholar 

  21. Havlin, S., Schwartz, M., Selinger, R.B., Bunde, A., Stanley, H.E.: Universality classes for diffusion in the presence of correlated spatial disorder. Phys. Rev. A 40, 1717–1719 (1989)

    Article  Google Scholar 

  22. Redner, S.: Random multiplicative processes: An elementary tutorial. Am. J. Phys. 58, 267–273 (1990)

    Article  Google Scholar 

  23. Levy, M., Solomon, S.: Power laws are logaritmic boltzmann laws. Int. J. Mod. Phys. C 7, 595–601 (1996)

    Article  Google Scholar 

  24. Sornette, D., Cont, R.: Convergent multiplicative processes repelled from zero: Power-laws and truncated power-laws. J. Phys. I (France) 7, 431–444 (1997)

    Article  Google Scholar 

  25. Feller, W.: An Introduction to Probability Theory and its Applications, vol. 2. John Wiley and Sons, New York (1971)

    MATH  Google Scholar 

  26. Gnedenko, B.V., Kolmogorov, A.N.: Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Reading (1968)

    Google Scholar 

  27. Dana, K.J., van Ginneken, B., Nayar, S.K., Koenderink, J.J.: Reflectance and texture of real world surfaces. ACM Trans Graphics 18, 1–34 (1999)

    Article  Google Scholar 

  28. van Hateren, J.H., van der Schaaf, A.: Independent component filters of natural images compared with simple cells in primary visual cortex. Proc. R. Soc. Lond. B 265, 359–366 (1998)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Geusebroek, JM. (2005). The Stochastic Structure of Images. In: Kimmel, R., Sochen, N.A., Weickert, J. (eds) Scale Space and PDE Methods in Computer Vision. Scale-Space 2005. Lecture Notes in Computer Science, vol 3459. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11408031_28

Download citation

  • DOI: https://doi.org/10.1007/11408031_28

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-25547-5

  • Online ISBN: 978-3-540-32012-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics