Robustness of topological defects in discrete domains

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).

Figure 1

Estimation of topological charge and its robustness, here shown for nematic vector fields. (a),(b) On a continuous domain, the normalized vectors along a closed path (green) around (a) a singular point are lifted to (b) a continuously changing azimuth, whose total change of $+\pi$ [red half circle in (b)] here indicates a $+1/2$ defect charge. (c),(d) On a discrete domain, discretized with arbitrary offset as shown by the thin gray lines in (a), the azimuth changes along finite path steps ${\mathbf{x}}_0,...,{\mathbf{x}}_4={\mathbf{x}}_0$ in (c). The normalized vectors in the corresponding discrete lifting [Eq. (3)] sum to the same index estimate in (d). (e),(f) Upon continuous change of a vector [compare point ${\mathbf{x}}_1$ in (c) and (e)], the index estimate discontinuously changes, here to 0 when $\mathrm{V}\left({\mathbf{x}}_1\right)$ crosses the dash-dotted black line perpendicular to $\mathrm{V}\left({\mathbf{x}}_0\right)$.

Figure 2

Data-dependent iterative path adaptation. (a) Example defect from Fig. 1 discretized on a regular Cartesian grid. Edge robustness is shown by line thickness scaling as robustness to the fourth power; “$+$” marks the grid cell containing defect charge $+1/2$. (b)–(e) Requiring robustness above thresholds of ${\mathrm{R}}_{\text{thresh}}/\left(0.5\pi \right)=$ (b) 0.50, (c) 0.75, (d) 0.82, and (e) 0.91 (arrowheads on the robustness scale) adaptively increases the area (shaded red) enclosed by the path. All surrounding grid cells have index estimate 0 with high robustness, already at the finest resolution.

Figure 3

Robust identification of TDs. (a) Detection of higher-order TDs (here, $+1$) is possible. Comparing robustness levels ${\mathrm{R}}_1$ and ${\mathrm{R}}_2$, splitting into or merging from elementary charges (here, two times $+1/2$) can be detected. (b) Large unordered region with interior edges of low robustness indicates potential pair generation prior to the formation of “microscopic” charges. For visualization, the line width of the edges scales as the fourth power of robustness as in Fig. 2; edges below robustness ${\mathrm{R}}_1$ are omitted. (c) Adjustment of detection area to the data at unordered cores (${\mathrm{R}}_0$ vs ${\mathrm{R}}_1$ versus microscopic charges). Anisotropic growth of detection areas at higher robustness indicates the direction of defect motion. Zero total charge at the highest robustness ${\mathrm{R}}_2$ indicates potential pair annihilation. The legend is common to all panels. For clarity, shading of zero-charge regions without microscopic defect(s) has been omitted in all panels.

Figure 4

Robust identification of TDs in non-Cartesian data. We show an example of an irregular planar graph (thin, curved solid lines) extracted from an image of biological cells. Connecting the cell centers converts this graph to a planar triangulation (straight solid lines; line width scales as fourth power of edge robustness). Such conversion is always possible for planar graphs in two dimensions (2D). Blue sticks visualize the elongation and orientation of the cells in the original graph, computed from their tensor of inertia. “Microscopic” topological charges are visualized by symbols ($+$ for $+1/2$, o for $-1/2$). Areas found by “expansion over critical edge” for robustness threshold $0.25\pi$ are shown shaded (blue: total charge $+1/2$; red: total charge $-1/2$; gray: total charge 0 if they contain microscopic defects).