Hoffmann Karl B, Sbalzarini Ivo F
Technische Universität Dresden, Faculty of Computer Science, Dresden, Germany; Max Planck Institute of Molecular Cell Biology and Genetics, Dresden, Germany; Center for Systems Biology Dresden, Dresden, Germany; and Cluster of Excellence Physics of Life, TU Dresden, Germany.
Phys Rev E. 2021 Jan;103(1-1):012602. doi: 10.1103/PhysRevE.103.012602.
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).
拓扑缺陷是矢量场中的奇点，在从指纹检测到液晶再到生物医学成像等各种应用中都很重要。在离散化的矢量场中，拓扑缺陷及其拓扑电荷通过围绕暂定缺陷的有限差分或有限步路径来识别。由于拓扑电荷是（半）整数，它在离散域中不能连续地依赖于每个输入矢量。相反，当矢量变化超过一定量时，它会不连续地切换，这使得在噪声数据中对拓扑缺陷的分析容易出错。我们通过为以下方面提出一种稳健性度量来改进现有的拓扑缺陷识别方法：（i）缺陷的位置，（ii）给定区域内拓扑缺陷的存在和总拓扑电荷，（iii）缺陷对的湮灭，以及（iv）缺陷对的形成。基于所提出的稳健性度量，我们表明离散域中的拓扑缺陷可以在定位精度和稳健性之间实现最佳权衡来识别。所提出的稳健性度量能够对噪声离散向列场（取向场）和极场（矢量场）中的拓扑缺陷进行不确定性量化。
