Estimating Size and Number Density of Three-Dimensional Particles Using Truncated Cross-Sectional Data

Abstract

Abstract The need for estimating three-dimensional (3D) information based on two-dimensional (2D) images has been increasing in numerous fields. It is essential in quality assessment, quality control, and process optimization. However, all the existing methods have not considered the data truncation issue, which is commonly faced in metrology. This paper proposes a new statistical approach to infer size distribution and volume number density (VND) of 3D particles based on 2D cross-sectional images with data truncation considered. In order to estimate the size distribution, a linkage is established between 3D particles and 2D observations with the existence of data truncation. Subsequently, this paper derives the likelihood function of 2D observations and an efficient Monte Carlo expectation-maximization algorithm is developed to estimate the parameters of size distribution. In addition, an explicit relationship between the 3D and 2D particle number densities is established and leveraged to estimate the VND and volume fraction. The effectiveness of the proposed method is demonstrated through both simulation study and real case studies in metal additive manufacturing and metal-matrix nanocomposites manufacturing.