**JOURNAL PAPER**

**Principles of an image-based algorithm for the quantification of dependencies between particle selections in sampling studies**

**D.S. Dihalu; B. Geelhoed**

Faculty of Applied Sciences, Delft University of Technology, The Netherlands

**SYNOPSIS**

*C*. This allows for modeling deviations from the ideal situation where the selection of a pair of particles is composed of two independent selections. The generalized model potentially leads to more accurate variance estimates in the case of clustering of particles, differences in densities or sizes of the particles or repulsive inter-particle forces. A straightforward and practically applicable method is needed for the determination of this parameter for miscellaneous mixtures in industrial settings.

_{ij}In this contribution, the feasibility of using digital image analysis to determine this parameter

*C*, will be demonstrated. Line transect sampling of a digital image was used to construct a transition probability matrix. A new algorithm to derive quantitative estimates for

_{ij}*C*will be presented and discussed.

_{ij}The applicability was verified by a photograph of zirconium silicate particles of sizes typical for industries dealing with pharmaceutical, food/feed, and environmental applications. Conditions affecting the practical applicability are identified and potential pitfalls will be discussed, including e.g. how a potential unrepresentative surface can affect the quality of the estimate of

*C*.

_{ij}

**“Full text available only in PDF format”**

**References**

1.** **GEELHOED, B. The construction of variance estimators for particulate material sampling, arXiv:1005.2968v1 [stat.AP], http://arxiv.org/pdf/1005.2968v1.pdf, 2008.

2.** **GY, P. *Sampling of particulate materials, theory and practice*, Elsevier, Amsterdam, 1979, 1982. [ Links ]

3.** **GEELHOED, B. A generalisation of Gy's model for the fundamental sampling error. *Second World Conference on Sampling and Blending*. The Australasian Institute of Mining and Metallurgy, ISBN 1-920806-28-8, 2005. pp. 19-25. [ Links ]

4.** **GEELHOED, B. Variable second-order inclusion probabilities as a tool to predict the sampling variance. *Third World Conference on Sampling and** Blending*. Porto Alegre, 2007. pp. 82-9. [ Links ]

5.** **KORPELAINEN, M., REINIKAINEN, S-P., LAUKKANEN, J,. and MINKKINEN, P. Estimation of Uncertainty of Concentration Estimates Obtained by Image Analysis, *Journal of Chemo metrics*, vol. 16, 2002. pp. 548-554. [ Links ]

6.** **GEELHOED, B. Variable second-order inclusion probabilities during the sampling of industrial mixtures of particles, *Applied Stochastic Models in** Business and Industry*, vol. 22, 2006. pp. 495-501. [ Links ]

7.** **KAISER, L. Unbiased Estimation in Line-Intercept Sampling, *Biometrics*, vol. 39, 1983. pp. 965-976. [ Links ]

8.** **PONTIUS, J. Estimation of the mean in line intercept sampling.* Environmental and Ecological Statistics *5, 1998. pp. 371-379. [ Links ]

9.** **BUCKLAND, S.T. *Introduction to distance sampling: estimating abundance of biological populations*, New York, Oxford University Press; 2001. [ Links ]

10.** **Carl Zeiss Axiovision User's Guide Release 4.7 2008. [ Links ]

11.** **GEELHOED, B., KOSTER-AMMERLAAN, M.J.J., KRAAIJVELD, G.J.C., BODE, P., DIHALU, D.S., and CHENG, H. An experimental comparison of Gy's sampling model with a more general model for particulate material sampling. *Fourth** World Conference on Sampling and Blending*, Cape Town, South Africa. WCSB4 Conference Proceedings, 2009. pp. 27-38. [ Links ]

12.** **APPLEGATE, D.L. *The Traveling Salesman Problem*, Princeton University Press 2006. [ Links ]