Análise topológica em imagens 3D de otólitos de peixes: explorando padrões de densidade e de morfologia

Abstract

In this thesis, we present a comparative study of otolith density variations using Topological Data Analysis (TDA). Otoliths are calcium carbonate structures found in the inner ears of fish and are commonly used to study age and growth patterns in fish populations. Traditionally, the analysis of otolith density variations has been a computationally intensive task due to the high-dimensional nature of the data. However, TDA offers a promising approach to reduce the data dimensionality and extract meaningful topological information from otolith images. We applied the Ball Mapper algorithm to a dataset of 3D otolith images from different fish species and ages. The algorithm allowed us to construct topological graphs representing the density variations in otoliths. We also explored the use of probabilistic sampling techniques to reduce the data and found that a sample size of 5% provided accurate representations of otolith density variations compared to the full dataset, after a Sample Topological Validation procedure developed here to ensure the efficiency and reliability of the sampling process. Topological invariants of the graphs, such as average clustering, node connectivity, assortativity, shortest path length, efficiency, and others, were used to comparizon between graphs. The comparizon of the topological properties of the full dataset with those of the 5% sample found a high degree of similarity, indicating that TDA with a reduced dataset can capture essential density information. Ball Mapper further allowed us to identify and eliminate dirt or anomalies present in otolith images, further enhancing the accuracy of our analysis. Overall, our study demonstrates the efficacy of TDA in studying otolith density variations with significant computational gains over traditional methods. The reduced data size using probabilistic sampling and the robustness of topological invariants provide valuable insights into the density patterns of otoliths. Another TDA technique, Persistent Homology (PH), was applied to the 3D image data with the expectation of unveiling a new classifier for otolith shape. PH demonstrated prominence even in a small sample by effectively separating otolith classes and revealing accurate quantitative separation results, showcasing potential use for otolith classification based on their 3D structure. Finally, a regression analysis demonstrated the possibility of estimating age, length, and radiodensity of otoliths based on the topological features resulting from the classification.