A robustness measure for singular point and index estimation in discretized orientation and vector fields

TL;DR

This paper introduces a robustness measure for detecting singular points in discretized vector fields, balancing resolution and noise resistance, enabling better uncertainty quantification in various scientific imaging applications.

Contribution

It develops a novel robustness measure for discrete defect estimators, comparing template paths and analyzing their effectiveness amid noise.

Findings

Robustness increases with template path length.

Optimal trade-off exists between resolution and noise robustness.

Single pixel analysis cannot exclude zero robustness.

Abstract

The 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.