Microsc. Microanal. 23, 269–278, 2017 doi:10.1017/S1431927617000320
© MICROSCOPY SOCIETY OF AMERICA 2017
Detecting Clusters in Atom Probe Data with Gaussian Mixture Models
Jennifer Zelenty,1,* Andrew Dahl,2 Jonathan Hyde,3 George D. W. Smith,1 and Michael P. Moody1
1Department of Materials, University of Oxford, Oxford OX1 3PH, UK 2Department of Statistics, University of Oxford, Oxford OX1 3LB, UK 3National Nuclear Laboratory, Culham Science Centre, Abingdon OX14 3DB, UK
Abstract: Accurately identifying and extracting clusters from atom probe tomography (APT) reconstructions is extremely challenging, yet critical to many applications. Currently, the most prevalent approach to detect clusters is the maximum separation method, a heuristic that relies heavily upon parameters manually chosen by the user. In this work, a new clustering algorithm, Gaussian mixture model Expectation Maximization Algorithm (GEMA),
was developed. GEMA utilizes a Gaussian mixture model to probabilistically distinguish clusters from random fluctuations in the matrix. This machine learning approach maximizes the data likelihood via expectation maximization: given atomic positions, the algorithm learns the position, size, and width of each cluster. A key advantage of GEMA is that atoms are probabilistically assigned to clusters, thus reflecting scientifically meaningful uncertainty regarding atoms located near precipitate/matrix interfaces. GEMA outperforms the maximum separation method in cluster detection accuracy when applied to several realistically simulated data sets. Lastly, GEMA was successfully applied to real APT data.
Key words: atom probe tomography, cluster identification, Gaussian mixture models, expectation maximization, machine learning
INTRODUCTION
Atom probe tomography (APT), with its superior combi- nation of chemical and sub-nanometer three dimensional (3D) spatial resolution, is an excellent tool with which to detect nanoscale precipitates (Gault et al., 2012). In many cases, particularly in the earliest stages of cluster formation, APT can identify features beneath the limits of most other conventional microscopy techniques. However, there exists a caveat: the user must independently extract the precipitates from the APT data using a cluster search algorithm. As the need to characterize clusters extends across many materials and users, a versatile and reliable cluster search algorithm is essential. The analysis of nanoscale precipitates within data
provided by APT is critical to many engineering applications in materials science. The nucleation and growth of Ni–Mn–Si precipitates can cause hardening and, ultimately, embrittle- ment, in reactor pressure vessel (RPV) steels (Styman et al., 2012;Wells et al., 2014). In addition, the mechanical proper- ties of Ni-based superalloys, which are used in jet engines and wind turbines, rely crucially on various solute additions and the resulting precipitation (Pollock & Tin, 2006). Further- more, key research regarding oxide dispersion strengthened (ODS) steels focuses on the chemical characterization and distribution of oxide nanoparticles; although these highly stable particles are responsible for many of the exceptional mechanical properties of ODS steels, they are not yet fully
*Corresponding author.
jzelenty@gmail.com Received July 9, 2016; accepted February 8, 2017
characterized (Hirata et al., 2011; London et al., 2015).Hence, the need for accurate cluster identification and extraction of cluster information from APT reconstructions is imperative. Numerous algorithms have been developed to identify
individual clusters within APT data including the maximum separation method (Hyde & English, 2000), the core-linkage algorithm (Stephenson et al., 2007), and Voronoi parti- tioning (Felfer et al., 2015). Currently, the most prevalent approach used to detect clusters is the maximum separation method. This heuristic relies heavily upon four parameters, each manually chosen by the user: Dmax, the maximum distance separating two neighbouring solute atoms within the same cluster; Nmin, the minimum number of atoms a cluster must contain to be considered a genuine cluster; and L and E, which are metrics related to the incorporation of solvent atoms within clusters. Although there have been commendable efforts to provide guidelines for parameter selection and sensitivity analyses, the process still remains largely arbitrary and incon- sistent (Hyde et al., 2011; Styman et al., 2013; Jägle et al., 2014). This raises the fundamental issue of scientific
objectivity. Experiments should be reproducible, and results should be independent of the researcher. The application of current APT data clustering algorithms requires selection of numerous user-defined parameters that unavoidably intro- duce subjective variation into the analysis. The outcome is a chronic lack of comparability between quantitative data obtained by different users. This represents a significant issue for APT-based research and becomes ever more apparent as the number of researchers in the field increases. Therefore, a more objective method for cluster analysis is essential.
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