Abstract
AbstractThe identification of singular points or topological defects in discretized vector fields occurs in diverse areas ranging from the polarization of the cosmic microwave background to liquid crystals to fingerprint recognition and bio‐medical imaging. Due to their discrete nature, defects and their topological charge cannot depend continuously on each single vector, but they discontinuously change as soon as a vector changes by more than a threshold. Considering this threshold of admissible change at the level of vectors, we develop a robustness measure for discrete defect estimators. Here, we compare different template paths for defect estimation in discretized vector or orientation fields. Sampling prototypical vector field patterns around defects shows that the robustness increases with the length of template path, but less so in the presence of noise on the vectors. We therefore find an optimal trade‐off between resolution and robustness against noise for relatively small templates, except for the “single pixel” defect analysis, which cannot exclude zero robustness. The presented robustness measure paves the way for uncertainty quantification of defects in discretized vector fields.
Highlights
A topological defect or singular point in a two-dimensional vector field is an isolated discontinuity of an otherwise continuous vector field
To capture the role of noise in defect estimation, and to quantify the trade-off between robustness and resolution, we propose a robustness measure for singular point estimators
We define the robustness of a defect estimator as the largest orientation change along the closed path that does not alter the estimate, : min {|θj − θi − π/2 − kπ|, k ∈ Z}
Summary
A topological defect or singular point in a two-dimensional vector field is an isolated discontinuity of an otherwise continuous vector field. Such objects can be defined in both vector fields and orientation (i.e., “headless vector”) fields. To compensate for this, defect identification in discretized orientation fields often uses closed paths larger than the smallest possible loop around a single pixel. This increases robustness to noise at the expense of localization resolution [1, 4, 5]
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