Resolution of the Wicksell's equation by Minimum Distance Estimation




microstructure, minimization, probability density function, Saltikov, stereology


The estimation of the grain size in granular materials is usually performed by 2D observations. Unfolding the grain size distribution from apparent 2D sizes is commonly referred as the corpuscle problem. For spherical particles, the distribution of the apparent size can be related to that of the actual size thanks to the Wicksell’s equation. The Saltikov method, which is based on Wicksell’s equation, is the most widely used method for resolving corpuscle problems. This method is recursive and works on the finite histogram of the grain size. In this paper, we propose an algorithm based on a minimizing procedure to numerically solve the Wicksell’s equation, assuming a parametric model for the distribution (e.g. lognormal distribution). This algorithm is applied on real material and the results are compared to those found using Saltikov or Saltikov-based stereology techniques. A criterion is proposed for choosing the number of bins in the Saltikov method. The accuracy of the proposed algorithm, depending on the sample size, is studied.

Author Biographies

Dorian Depriester, Arts et Métiers ParisTech

MSMP Laboratory, Associate Professor

Régis Kubler, Arts et Métiers ParisTech

MSMP Laboratory, Associate Professor


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How to Cite

Depriester, D., & Kubler, R. (2019). Resolution of the Wicksell’s equation by Minimum Distance Estimation. Image Analysis and Stereology, 38(3), 213–226.



Original Research Paper