🍩 Database of Original & Non-Theoretical Uses of Topology

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  1. Feature Detection and Hypothesis Testing for Extremely Noisy Nanoparticle Images Using Topological Data Analysis (2023)

    Andrew M. Thomas, Peter A. Crozier, Yuchen Xu, David S. Matteson
    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.

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  2. A Proof-of-Concept Investigation Into Predicting Follicular Carcinoma on Ultrasound Using Topological Data Analysis and Radiomics (2025)

    Andrew M. Thomas, Ann C. Lin, Grace Deng, Yuchen Xu, Gustavo Fernandez Ranvier, Aida Taye, David S. Matteson, Denise Lee
    Abstract Background Sonographic risk patterns identified in established risk stratification systems (RSS) may not accurately stratify follicular carcinoma from adenoma, which share many similar US characteristics. The purpose of this study is to investigate the performance of a multimodal machine learning model utilizing radiomics and topological data analysis (TDA) to predict malignancy in follicular thyroid neoplasms on ultrasound. Patients & Methods This is a retrospective study of patients who underwent thyroidectomy with pathology confirmed follicular adenoma or carcinoma at a single academic medical center between 2010 and 2022. Features derived from radiomics and TDA were calculated from processed ultrasound images and high-dimensional features in each modality were projected onto their first two principal components. Logistic regression with L2 penalty was used to predict malignancy and performance was evaluated using leave-one-out cross-validation and area under the curve (AUC). Results Patients with follicular adenomas (n = 7) and follicular carcinomas (n = 11) with available imaging were included. The best multimodal model achieved an AUC of 0.88 (95% CI: [0.85, 1]), whereas the best radiomics model achieved an AUC of 0.68 (95% CI: [0.61, 0.84]). Conclusions We demonstrate that inclusion of topological features yields strong improvement over radiomics-based features alone in the prediction of follicular carcinoma on ultrasound. Despite low volume data, the TDA features explicitly capture shape information that likely augments performance of the multimodal machine learning model. This approach suggests that a quantitative based US RSS may contribute to the preoperative prediction of follicular carcinoma.