Feature Detection and Hypothesis Testing for Extremely Noisy Nanoparticle Images using Topological Data Analysis

ABSTRACT

We propose a flexible algorithm for feature detection and hypothesis testing in images with ultra-low signal-to-noise ratio using cubical persistent homology. Our main application is in the identification of atomic columns and other features in Transmission Electron Microscopy (TEM). Cubical persistent homology is used to identify local minima and their size in subregions in the frames of nanoparticle videos, which are hypothesized to correspond to relevant atomic features. We compare the performance of our algorithm to other employed methods for the detection of columns and their intensity. Additionally, Monte Carlo goodness-of-fit testing using real-valued summaries of persistence diagrams derived from smoothed images (generated from pixels residing in the vacuum region of an image) is developed and employed to identify whether or not the proposed atomic features generated by our algorithm are due to noise. Using these summaries derived from the generated persistence diagrams, one can produce univariate time series for the nanoparticle videos, thus, providing a means for assessing fluxional behavior. A guarantee on the false discovery rate for multiple Monte Carlo testing of identical hypotheses is also established.

KEYWORDS:

• ALPS statistic
• Catalysis
• Cubical persistent homology
• Multiple Monte Carlo testing
• Persistent entropy
• Transmission electron microscopy

Supplementary Materials

The supplementary material contains additional derivations and simulations pertaining to the main text. Every section of the supplementary material is referenced in the text above. Python code and the videos can be found in DataAndCodeForPaper.zip.

Disclosure Statement

The authors report there are no competing interests to declare.

Acknowledgments

The authors would like to thank the two reviewers and associate editors for their insightful comments which greatly aided in the presentation of the article. Furthermore, the authors would like to acknowledge Joshua Vincent, Piyush Haluai and Mai Tan at Arizona State University (ASU) for sample preparation and TEM data acquisition. We also thank Ramon Manzorro (University of Cádiz) and Advait Gilankar (ASU) for assistance in imaging simulations and data handling as well as the John M. Cowley Center for High Resolution Electron Microscopy at ASU.

Notes

1 Such a paradigm is called the elder rule and is described in detail in Edelsbrunner and Harer (2010).

2 Other filtrations could be considered here; however, besides the sublevel set filtration, all require choosing a threshold at which to binarize the image (Garin and Tauzin 2019; Turkes et al. 2021)–see also the opening paragraph of Section 4.

3 Recall that all pixels not in the image (image set) are de facto white pixels in terms of cubical homology.

4 Technically speaking these are equivalence classes of cycles, which are equivalent modulo a boundary.

5 Often, the diagonal $y=x$ is added to this diagram, but we need not consider this here.

6 For specifying polygonal regions and which pixels are contained in them, we use the Shapely Python library (Gillies 2013).

7 The locations of the blobs/atomic columns was initially calculated using the blob_log function in the Python skimage library, as in Manzorro et al. (2022). The algorithm was applied in the same fashion as Manzorro et al. (2022) to ensure optimality of parameters chosen and a fair comparison of the methods.

8 More information on this data and how it was collected can be found in Section S1 of the supplementary material.

9 Tables S3–S6, supporting this conclusion, can be seen in the supplementary material.

10 Figure 9 illustrates this for $I_{10,241}$, suggesting that $\sigma \approx 3$ is appropriate for said image.

Additional information

Funding

The authors gratefully acknowledge funding from NSF grants OAC-1940124, CCF-1934985, DMS-2114143 and specifically CBET-1604971 and OAC-1940263 for the support of sample preparation and data acquisition.