## The Annals of Applied Statistics

### Fast dynamic nonparametric distribution tracking in electron microscopic data

#### Abstract

In situ transmission electron microscope (TEM) adds a promising instrument to the exploration of the nanoscale world, allowing motion pictures to be taken while nano objects are initiating, crystalizing and morphing into different sizes and shapes. To enable in-process control of nanocrystal production, this technology innovation hinges upon a solution addressing a statistical problem, which is the capability of online tracking a dynamic, time-varying probability distribution reflecting the nanocrystal growth. Because no known parametric density functions can adequately describe the evolving distribution, a nonparametric approach is inevitable. Towards this objective, we propose to incorporate the dynamic evolution of the normalized particle size distribution into a state space model, in which the density function is represented by a linear combination of B-splines and the spline coefficients are treated as states. The closed-form algorithm runs online updates faster than the frame rate of the in situ TEM video, making it suitable for in-process control purpose. Imposing the constraints of curve smoothness and temporal continuity improves the accuracy and robustness while tracking the probability distribution. We test our method on three published TEM videos. For all of them, the proposed method is able to outperform several alternative approaches.

#### Article information

Source
Ann. Appl. Stat., Volume 13, Number 3 (2019), 1537-1563.

Dates
Revised: February 2019
First available in Project Euclid: 17 October 2019

https://projecteuclid.org/euclid.aoas/1571277763

Digital Object Identifier
doi:10.1214/19-AOAS1245

Mathematical Reviews number (MathSciNet)
MR4019149

#### Citation

Qian, Yanjun; Huang, Jianhua Z.; Park, Chiwoo; Ding, Yu. Fast dynamic nonparametric distribution tracking in electron microscopic data. Ann. Appl. Stat. 13 (2019), no. 3, 1537--1563. doi:10.1214/19-AOAS1245. https://projecteuclid.org/euclid.aoas/1571277763

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#### Supplemental materials

• Supplement A: Appendices. A pdf document including Appendices A, B and C. This document provides the derivations of the Gaussian approximation of the Poisson distribution, the detailed steps of Kalman filter and the derivation of the posterior distributions of the system parameters for the proposed model.
• Supplement B: Data and codes. A zip file including the description of the testing videos and the MATLAB codes to reproduce the results in the paper. A “Data and Codes.docx” file provides the detailed guidance to use the data and codes. The three videos have been published and are free to download, and all the codes have been tested under MATLAB 2016b.