Structuring Line Ensembles with Path-Integrated Fidelity and Structural Inconsistency Fields

(a) Conventional line density plot — **Weekly maximum temperature series from 6,187 U.S. weather stations.** Density emphasizes a broad hotspot whose query remains geographically diffuse. The Structural Inconsistency Field makes two coherent corridors salient, producing clearer query results and a north–south geographic separation.

(b) Density-guided query result — **Weekly maximum temperature series from 6,187 U.S. weather stations.** Density emphasizes a broad hotspot whose query remains geographically diffuse. The Structural Inconsistency Field makes two coherent corridors salient, producing clearer query results and a north–south geographic separation.

Abstract:

When visualizing large-scale line ensembles, trajectory continuity and visual scalability are inherently antagonistic. Trajectory-centric renderings preserve path information but rapidly degenerate into clutter, while field-based density representations improve visibility at the cost of structural coherence. We complement density-based views with a path-integrated trajectory-fidelity measure that quantifies how each trajectory agrees with its surrounding tensor field. Projecting passage-centered structural support back into image space yields a Structural Inconsistency Field, localizing where dense patterns reflect coherent structure, disagreement, outliers, or connectivity-induced ambiguity. Dynamic leave-one-out normalization reduces self-bias, while fixed-grid updates and prefix-sum evaluation enable interactive analysis and iterative extraction of coherent structures. Synthetic benchmarks, scalability analyses, and real-world case studies show that the method exposes spatially localized coherence and breakdowns concealed by density-only representations.

Online Demo: https://structuring-line-ensembles.vercel.app

Core contributions

Path-Integrated Fidelity

Accumulates local tangent–field agreement along each path into a trajectory-level structural support metric.

Dynamic Leave-One-Out

Evaluates every trajectory against its surrounding ensemble without allowing the trajectory to support itself.

Inconsistency Field

Projects passage-centered support into image space to reveal where coherence persists and where it fails.

Structure-Guided Peeling

Supports interactive querying, subset extraction, removal, and re-evaluation to uncover secondary structures.

Figures:

Input trajectories — **Construction of the structure tensor field.** (a) Trajectories are rasterized onto the analysis grid. (b) With a minimal kernel (1 × 1), the tensor records only the instantaneous tangent. (c) A wider spatial kernel (3 × 3) aggregates neighborhood statistics and reveals crossings or divergences through increased isotropy.

Local tensor accumulation — **Construction of the structure tensor field.** (a) Trajectories are rasterized onto the analysis grid. (b) With a minimal kernel (1 × 1), the tensor records only the instantaneous tangent. (c) A wider spatial kernel (3 × 3) aggregates neighborhood statistics and reveals crossings or divergences through increased isotropy.

Raw trajectories — **Path integration with respect to the tensor field.** (a) Input trajectories. (b) The associated tensor field. (c) Each trajectory is evaluated via integration of tangent–field alignment along its path.

Tensor field context — **Path integration with respect to the tensor field.** (a) Input trajectories. (b) The associated tensor field. (c) Each trajectory is evaluated via integration of tangent–field alignment along its path.

D1 synthetic dataset — **Synthetic datasets.** The colors show the generated trajectory categories for cases D1, D2, D3-C, and D3-D in the benchmark below.

D2 synthetic dataset — **Synthetic datasets.** The colors show the generated trajectory categories for cases D1, D2, D3-C, and D3-D in the benchmark below.

D1 scalar density — **Benchmark for synthetic datasets.** The columns show scalar density, our structural inconsistency field, and the qualitative image-space cluster coloring by Xue *et al.* Density and inconsistency are continuous scalar fields, whereas the cluster-coloring column uses categorical colors for image-space groups rather than ground-truth trajectory classes.

D1 structural inconsistency — **Benchmark for synthetic datasets.** The columns show scalar density, our structural inconsistency field, and the qualitative image-space cluster coloring by Xue *et al.* Density and inconsistency are continuous scalar fields, whereas the cluster-coloring column uses categorical colors for image-space groups rather than ground-truth trajectory classes.

**Scalability analysis.** Binning scales linearly with the number of trajectories, tensor-field construction is nearly constant with respect to N on a fixed grid, and prefix-sum path-extent evaluation keeps path integration lightweight.

France aviation density — **France aviation case study.** The full dataset contains 5,360 trajectories. The density map emphasizes the dominant traffic corridors but leaves minority behaviors visually entangled with them. The structural inconsistency field localizes where structural support weakens. Ranking trajectories by Φ(L) then separates the dataset into a top 80% high-score subset (4,288 trajectories), which captures the main long-range, route-regular commercial traffic pattern, and a bottom 20% low-score subset (1,072 trajectories), which analysts identified as including many smaller-aircraft trajectories, together with looping or holding-like paths and trajectories affected by recording errors.

France aviation structural inconsistency field — **France aviation case study.** The full dataset contains 5,360 trajectories. The density map emphasizes the dominant traffic corridors but leaves minority behaviors visually entangled with them. The structural inconsistency field localizes where structural support weakens. Ranking trajectories by Φ(L) then separates the dataset into a top 80% high-score subset (4,288 trajectories), which captures the main long-range, route-regular commercial traffic pattern, and a bottom 20% low-score subset (1,072 trajectories), which analysts identified as including many smaller-aircraft trajectories, together with looping or holding-like paths and trajectories affected by recording errors.

DTI density — **DTI case study.** This orthographic brain projection uses a 12,059-fiber subset sampled for browser-side analysis from a full 120,593-fiber tractography dataset. Density (a) collapses multiple overlapping anatomical structures into a hairball. Line bundling (c) creates a clean tree-like structure, but the circled regions A and B show that it can also distort or suppress bundle geometry. In contrast, the highest-scoring 50% of fibers (d) from the structural inconsistency field (b) recovers a readable scaffold without deforming the trajectories.

DTI structural inconsistency field — **DTI case study.** This orthographic brain projection uses a 12,059-fiber subset sampled for browser-side analysis from a full 120,593-fiber tractography dataset. Density (a) collapses multiple overlapping anatomical structures into a hairball. Line bundling (c) creates a clean tree-like structure, but the circled regions A and B show that it can also distort or suppress bundle geometry. In contrast, the highest-scoring 50% of fibers (d) from the structural inconsistency field (b) recovers a readable scaffold without deforming the trajectories.

Acknowledgments:

This work was funded in part by Deutsche Forschungsgemeinschaft (DFG) Project 410883423 and Project 251654672 — TRR 161 “Quantitative methods for visual computing.” Yunhai Wang was supported by the grants of NSFC (No. 62132017 and No. U2436209), the Shandong Provincial Natural Science Foundation (No. ZQ2022JQ32), the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China.

From seeing density to understanding structure.

Use path-integrated evidence to identify which trajectories agree—and where that agreement breaks down.

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