Abstract
Topological defects are singular points in vector fields, important in applications ranging from fingerprint detection to liquid crystals to biomedical imaging. In discretized vector fields, topological defects and their topological charge are identified by finite differences or finite-step paths around the tentative defect. As the topological charge is (half) integer, it cannot depend continuously on each input vector in a discrete domain. Instead, it switches discontinuously when vectors change beyond a certain amount, making the analysis of topological defects error prone in noisy data. We improve existing methods for the identification of topological defects by proposing a robustness measure for (i) the location of a defect, (ii) the existence of topological defects and the total topological charge within a given area, (iii) the annihilation of a defect pair, and (iv) the formation of a defect pair. Based on the proposed robustness measure, we show that topological defects in discrete domains can be identified with optimal trade-off between localization precision and robustness. The proposed robustness measure enables uncertainty quantification for topological defects in noisy discretized nematic fields (orientation fields) and polar fields (vector fields).
Highlights
A topological defect (TD) is a singular point in a polar or nematic vector field
We start from a definition of TDs in polar and nematic vector fields on continuous domains, from which we state the definition on discrete domains
We have proposed a robustness measure for topological defects (TDs) and their charges in discrete domains
Summary
A topological defect (TD) is a singular point in a polar or nematic vector field. Such fields are ubiquitous in science, engineering, and mathematics as coarse-grained continuous descriptors of flows, force fields, molecule and object orientation, anisotropy, etc. Previous works used vector field smoothing [e.g., 14,16,29], defect identification along larger fixed-size closed paths [17], clustering [30], filtering by temporal persistence [29], machine learning [21,25], or thresholds on the nematic order parameter, either absolute [14,15] or relative to the spatial mean [22]. While all of these methods work in practice, none of them are based on a rigorous definition of robustness of TDs, and they do not shed light onto the connection between the robustness of TDs and the geometry of the underlying vector field. We provide a data-adaptive algorithm for identifying TDs and their charges in discrete domains, which might serve as a starting point for uncertainty quantification of TDs
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