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SpheriColor: Colormaps for Spherical Geospatial Input Topographies
Julius Rauscher, Johannes Fuchs, Daniel A. Keim, Frederik L. Dennig
TL;DR
Color can encode spherical geographic locations, but such encodings have received less attention than attribute colormaps. SpheriColor maps geospatial distances into a perceptually linear colorspace and uses gamut-constrained optimization; Simplex best preserves distances, while Ray better exploits colorspace.
Problem
Method
SpheriColor projects geospatial distances into a perceptually linear colorspace and applies two gamut-constrained optimization strategies.
Source: §3.3 · p. 2, §6 · p. 4Results
Takeaways & Limitations
Simplex is preferable when distance preservation is prioritized, while Ray is preferable when colorspace exploitation is prioritized.
Source: §5 · p. 4The evaluation is purely quantitative, and a user study is needed to test whether its metrics align with human perception and task performance in real-world geospatial visualizations.
Source: §5.1 · p. 4
Abstract
from arXiv · showhide
Multivariate geospatial data visualization often relies on multiple coordinated views, where color can be used to either link views or encode data attributes. Encoding spatial locations through color can reveal patterns in non-spatial visualizations, yet most applications of colormaps focus on high-dimensional attribute encodings instead. While 2D colormaps have been studied extensively, color encodings designed for spherical geospatial data have received much less attention, even though all locations lie on a sphere. To address this, we propose SpheriColor, a colormap generation approach for mapping geospatial references guided by the design goals of distance preservation and colorspace exploitation. We project a geospatial distance function into a perceptually linear colorspace, followed by two different gamut-constrained optimization strategies. We perform a quantitative evaluation using both real-world and synthetic datasets, demonstrating superior performance over 2D colormaps and HSLuv double cone encodings. The simplex-based optimization excels at distance preservation, whereas the ray-based approach provides better colorspace exploitation.
1 INTRODUCTION
Color can reveal geographic patterns in non-spatial views, but existing color encodings mostly target attributes rather than spherical locations. SpheriColor addresses the lack of spherical-geospatial color mappings and quantitative comparisons of their quality.
1 INTRODUCTION
- Geographic color encoding has received less attention than attribute encoding, despite its ability to reveal spatial patterns in PCPs and scatterplots and free position for other attributes.Existing methods commonly map high-dimensional attributes into colorspaces.Source: §1 · p. 1
- Existing colormap guidance emphasizes distance preservation and colorspace exploitation, but has not been extended to spherical inputs.Planar map projections can create extreme antimeridian color discontinuities when applied to the whole globe.Source: §1 · p. 1
- Out-of-gamut colors can distort perceptual distance, and device-gamut-constrained optimization had not been quantitatively evaluated against both design goals.The paper contributes SpheriColor and a quantitative comparison with other methods.Source: §1 · p. 1
2 RELATED WORK
Prior work covers multivariate spatial views and color encodings, but does not address geographic position on a spherical surface with empirical quality assessment. SpheriColor targets that gap.
2 RELATED WORK
- Multivariate spatial visualization combines maps with attribute views, including PCPs and scatterplots, while glyph maps can suffer visual overload as dimensionality increases.Prior work also combines or integrates maps and attribute views to support spatial-pattern analysis.Source: §2 · p. 2
- Existing spatially referenced multivariate methods use color for non-spatial attributes, while sophisticated geographic-position colormaps are absent to the authors' knowledge.Prior approaches include SOM-, dimensionality-reduction-, and 2D-colormap-based encodings.Source: §2 · p. 2
- Colormap research spans 1D and 2D design, data-driven methods, and spherical encodings for fiber orientations, but not spherical geographic positions.The paper also identifies a lack of empirical quantification for such geographic mappings.Source: §2 · p. 2
3 SPHERICAL SPATIAL COLORMAPS
SpheriColor maps spherical geospatial references through a perceptually linear embedding and fits that embedding to a display gamut. Its two optimization strategies balance distance preservation against colorspace use differently.
3.1 Design Goals
- The design goals are to preserve proportional relationships between spatial and perceived color distances and to maximize discriminative color use within the sRGB gamut.The section frames the approach as data-driven colormap generation for a fixed set of spherical references.Source: §3.1 · p. 2, §3 · p. 2
3.2 Naïve Approaches
- The 2D colormap inherits projection distortions, including antimeridian discontinuities, while HSLuv avoids that cut but distorts distances through its double-cone geometry.Mapping longitude to hue and latitude to lightness creates a continuous cycle and uses the saturated outer hull.Source: §3.2 · p. 2
3.3 SpheriColor
- SpheriColor derives pairwise geodesic distances between centroids, embeds the distance matrix in 3D with MDS, and maps the embedding to CAM16-UCS J*, a*, b*.The distance-based input can support alternatives such as travel-time or trajectory distances, but MDS has no stable output orientation.Source: §3.3 · p. 2
- The sRGB gamut is the target for fitting perceptual embeddings because standard displays reproduce only a limited subset of perceptual color spaces.The target gamut can be customized for a known device.Source: §3.3 · p. 2
- Simplex optimization applies a rigid transformation with uniform scaling to preserve relative geometry while fitting the embedding inside the gamut.It balances distance preservation, out-of-gamut penalties, and expansion within the gamut, but uses only the hull-gamut intersection and requires tuning.Source: §3.3 · p. 3
- Ray optimization projects embedding points from the gamut centroid onto its surface, improving gamut use while distorting pairwise distances.It is parameter-free and linear in the number of points; the Figure 2 caption reports optimal preservation for Simplex but the lowest maximum ΔE.Source: §3.3 · p. 3, Figure 2 · p. 3
4 QUANTITATIVE EVALUATION
The evaluation measures distance preservation and colorspace exploitation on synthetic and real-world data. SpheriColor’s variants lead on different goals: Simplex preserves distances best, while Ray uses more of the colorspace.
4.1 Measures
- The evaluation uses Pearson correlations for geodesic-versus-color distances, including nearest-neighbor subsets, and estimates colorspace exploitation by convex-hull intersection with the sRGB gamut.It also reports minimum, median, and maximum CAM16-UCS color differences.Source: §4.1 · p. 3
4.2 Data
- The study evaluates 1,000 uniformly sampled sphere points and 212 centroids of World Bank Economic regions, using 20 initial MDS rotations and selecting the lowest-objective configuration.The two datasets represent synthetic and real-world data, respectively.Source: §4.2 · p. 3, §4 · p. 3
4.3 Results
- For nearest neighbors in Samples (k = 1), Simplex has Pearson correlation .98, while other approaches are around 0; Ray exceeds .7 at k = 9, versus HSLuv at k = 50.Figure 3 reports Pearson correlation by neighborhood size and indicates Simplex leads especially for smaller neighborhoods.Source: §4.3 · pp. 3–4, Figure 3 · p. 4
- On Countries, Ray has .82 gamut intersection, a 0.77 ΔE minimum, and a 25.35 ΔE maximum; on Samples, Ray and HSLuv nearly fill the sRGB gamut.The reported Samples gamut intersections are .99 for Ray, .97 for HSLuv, .41 for Simplex, and .21 for the 2D colormap.Source: §4.3 · p. 4
- SpheriColor-Ray provides the best stated tradeoff across distance preservation and colorspace exploitation, while Simplex preserves distances better and has lower gamut use and saturation.Ray has stronger colorspace exploitation; the methods therefore suit different priorities.Source: §4.3 · pp. 3–4
5 DISCUSSION
SpheriColor outperforms the evaluated naïve approaches on distance preservation and colorspace exploitation, but the two goals involve a trade-off. The discussion also identifies limits in evaluation, stability, and generalization.
5 DISCUSSION
- SpheriColor outperforms evaluated naïve approaches on distance preservation and colorspace exploitation, with quality assessed on real-world and synthetic datasets.Source: §5 · p. 4, Table 1 · p. 4
- When distance preservation is prioritized, SpheriColor-Simplex is preferable; when colorspace exploitation is prioritized, SpheriColor-Ray is preferable, though distance preservation may be reduced.Simplex preserves local distances most faithfully but uses less of the colorspace and produces less saturation; Ray tends to produce more distinguishable, saturated colors.Source: §5 · p. 4
- Performance depends on embedding geometry and target gamut: Ray is designed for spherical embeddings, while Simplex may better fit planar or irregular embeddings inside the gamut.Source: §5 · p. 4
- For Samples, 5,000 points (~280 km spatial resolution) remain above 1.5∆E, while tiny geodesic distances between neighboring countries yield color distances below 1∆E.SpheriColor-Simplex has the largest minimum color distance among the distance-preserving methods.Source: §5 · p. 4
5.1 Limitations and Future Work
- The evaluation is purely quantitative, and the gamut-intersection score is a proxy rather than an established quality measure; a user study is needed to assess alignment with perception and task performance.Source: §5.1 · p. 4
- MDS embeddings are sensitive to point configuration, so adding one entity can substantially change the colormap; HSLuv double cone is an alternative for dataset-independent mappings but is inferior on the design goals.Source: §5.1 · p. 4
- More complex geospatial similarities may produce a 3D MDS embedding that no longer approximates a sphere, allowing SpheriColor-Ray to collapse points sharing an angular direction.Source: §5.1 · p. 4
6 CONCLUSION
SpheriColor maps spherical geospatial references to color by projecting a geospatial distance function into a perceptually linear colorspace and applying gamut-constrained optimization. Its two variants favor different goals: Simplex best preserves geodesic distances, while Ray optimizes colorspace exploitation.
6 CONCLUSION
- SpheriColor projects geospatial entities into a perceptually linear colorspace and applies two gamut-constrained optimizations; Simplex best preserves geodesic distances, while Ray optimizes colorspace exploitation.Source: §6 · p. 4