Advances in Applied Probability

The transparent dead leaves model

B. Galerne and Y. Gousseau

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this paper we introduce the transparent dead leaves (TDL) random field, a new germ-grain model in which the grains are combined according to a transparency principle. Informally, this model may be seen as the superposition of infinitely many semitransparent objects. It is therefore of interest in view of the modeling of natural images. Properties of this new model are established and a simulation algorithm is proposed. The main contribution of the paper is to establish a central limit theorem, showing that, when varying the transparency of the grain from opacity to total transparency, the TDL model ranges from the dead leaves model to a Gaussian random field.

Article information

Adv. in Appl. Probab., Volume 44, Number 1 (2012), 1-20.

First available in Project Euclid: 8 March 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60G60: Random fields

Germ-grain model dead leaves model transparency occlusion image modeling texture modeling


Galerne, B.; Gousseau, Y. The transparent dead leaves model. Adv. in Appl. Probab. 44 (2012), no. 1, 1--20. doi:10.1239/aap/1331216642.

Export citation


  • Baddeley, A. (2007). Spatial point processes and their applications. In Stochastic Geometry (Lecture Notes Math. 1892), ed. W. Weil, Springer, Berlin, pp. 1–75.
  • Barral, J. and Mandelbrot, B. B. (2002). Multifractal products of cylindrical pulses. Prob. Theory Relat. Fields 124, 409–430.
  • Baryshnikov, Y. and Yukich, J. E. (2005). Gaussian limits for random measures in geometric probability. Ann. Appl. Prob. 15, 213–253.
  • Bickel, P. J. and Doksum, K. A. (2001). Mathematical Statistics, Vol. I, 2nd edn. Prentice Hall.
  • Bordenave, C., Gousseau, Y. and Roueff, F. (2006). The dead leaves model: a general tessellation modeling occlusion. Adv. Appl. Prob. 38, 31–46.
  • Bovier, A. and Picco, P. (1993). A law of the iterated logarithm for random geometric series. Ann. Prob. 21, 168–184.
  • Cao, F., Guichard, F. and Hornung, H. (2010). Dead leaves model for measuring texture quality on a digital camera. In Digital Photography VI (Proc. SPIE 7537), eds F. H. Imai, N. Sampat and F. Xiao, 8pp.
  • Chainais, P. (2007). Infinitely divisible cascades to model the statistics of natural images. IEEE Trans. Pattern Analysis Mach. Intellig. 29, 2105–2119.
  • Enderton, E., Sintorn, E., Shirley, P. and Luebke, D. (2010). Stochastic transparency. In Proc. 2010 ACM SIGGRAPH Symp. Interactive 3D Graphics and Games, ACM, New York, pp. 157–164.
  • Galerne, B., Gousseau, Y. and Morel, J.-M. (2011). Random phase textures: theory and synthesis. IEEE Trans. Image Process. 20, 257–267.
  • Gousseau, Y. and Roueff, F. (2007). Modeling occlusion and scaling in natural images. Multiscale Model. Simul. 6, 105–134.
  • Grosjean, B. and Moisan, L. (2009). A-contrario detectability of spots in textured backgrounds. J. Math. Imaging Vision 33, 313–337.
  • Heinrich, L. and Schmidt, V. (1985). Normal convergence of multidimensional shot noise and rates of this convergence. Adv. Appl. Prob. 17, 709–730.
  • Jeulin, D. (1997). Dead leaves models: from space tesselation to random functions. In Proc. Internat. Symp. Advances in Theory and Applications of Random Sets, ed. D. Jeulin, World Scientific, River Edge, NJ, pp. 137–156.
  • Kendall, W. S. and Thönnes, E. (1999). Perfect simulation in stochastic geometry. Pattern Recognition 32, 1569–1586.
  • Kingman, J. F. C. (1993). Poisson Processes (Oxford Stud. Prob. 3). Oxford University Press.
  • Lantuéjoul, C. (2002). Geostatistical Simulation: Models and Algorithms. Springer, Berlin.
  • Matheron, G. (1968). Schéma booléen séquentiel de partition aléatoire. Tech. Rep. 89, CMM.
  • Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, New York.
  • Michalowicz, J., Nichols, J. M., Bucholtz, F. and Olson, C. C. (2009). An Isserlis' theorem for mixed Gaussian variables: application to the auto-bispectral density. J. Statist. Phys. 136, 89–102.
  • Penrose, M. D. (2007). Gaussian limits for random geometric measures. Electron. J. Prob. 12, 989–1035.
  • Rice, J. (1977). On generalized shot noise. Adv. Appl. Prob. 9, 553–565.
  • Richard, F. and Biermé, H. (2010). Statistical tests of anisotropy for fractional Brownian textures. Application to full-field digital mammography. J. Math. Imaging Vision 36, 227–240.
  • Schmitt, M. (1991). Estimation of the density in a stationary Boolean model. J. Appl. Prob. 28, 702–708.
  • Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.
  • Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.