Color Interpolation for Non-Euclidean Color SpacesMax Zeyen*

Los Alamos NationalLaboratory

University of Kaiserslautern

Tobias Post†

University of KaiserslauternHans Hagen‡

University of KaiserslauternJames Ahrens§

Los Alamos National Laboratory

David Rogers¶

Los Alamos National LaboratoryRoxana Bujack||

Los Alamos National Laboratory

ABSTRACT

Color interpolation is critical to many applications across a varietyof domains, like color mapping or image processing. Due to thecharacteristics of the human visual system, color spaces whose dis-tance measure is designed to mimic perceptual color differences tendto be non-Euclidean. In this setting, a generalization of establishedinterpolation schemes is not trivial. This paper presents an approachto generalize linear interpolation to colors for color spaces equippedwith an arbitrary non-Euclidean distance measure. It makes useof the fact that in Euclidean spaces, a straight line coincides withthe shortest path between two points. Additionally, we provide aninteractive implementation of our method for the CIELAB colorspace using the CIEDE2000 distance measure integrated into VTKand ParaView.

Index Terms: Human-centered computing—Visualization—Visu-alization techniques—Treemaps; Human-centered computing—Visualization—Visualization design and evaluation methods

1 INTRODUCTION

Many applications require interpolation between colors, for exam-ple, color mapping, re-sampling of color images or movies, andimage manipulations, like stitching, morphing, or contrast adaption.The most popular interpolation method is linear, where values aretaken equidistantly on a straight line connecting the sampling points.However, the state of the art indicates that human color perceptionis non-Euclidean due to the principle called hue superimportance,Figure 1. Hue superimportance [9] refers to the fact that changesin hue are perceived stronger than changes in saturation, Figure 1.The circumference of a circle of constant luminance and saturationwould be estimated to measure about 4π for its radius, which can-not be embedded in a Euclidean plane. In non-Euclidean spaces,the concept of a straight line is in general undefined. For the spe-cial case of Riemannian manifolds, the generalization of a straightline is a geodesic, i.e., a shortest path between two points, whoselength coincides with the points’ distance. Furthermore, humancolor perception is also non-Riemannian, due to the principle calleddiminishing returns [9], Figure 1. In this context, diminishing returnsrefers to the phenomenon that when presented with two colors, A,Cand their perceived middle (average/mixture) B, an observer usuallyjudges the sum of the perceived differences of each half greater thanthe difference of the two outer colors ∆(A,B)+∆(B,C)> ∆(A,C).This effect is produced by a natural contrast enhancement filter

*e-mail: mzeyen@lanl.gov†e-mail: tpost@rhrk.uni-kl.de‡e-mail: hagen@cs.uni-kl.de§e-mail: ahrens@lanl.gov¶e-mail: dhr@lanl.gov||e-mail: bujack@lanl.gov

that is built into the human perceptual system to adapt to differentviewing conditions. As a result, modern color difference formulas(e.g. CIEDE1994, CIEDE2000) that were designed to match experi-mental data produce complicated spaces, in which it is difficult tocorrectly interpolate colors.

Figure 1: Visualization of hue-superimportance and diminish-ing returns.

Figure 2: From the blacknode: 1-neighborhood orange,2-neighborhood green.

In this paper, we present a novel method for interpolating colors inarbitrary color spaces equipped with a distance measure. Analogousto the concept of a geodesic in a Riemannian manifold, we use ashortest path between the colors to be interpolated with respect to thegiven distance measure, even though its length does generally notcoincide with the distance between the colors. To be mathematicallyprecise, we require a path-connected metric space. That allows thedefinition of a shortest path even though it might not be unique orfinite. Furthermore, we provide an open source implementationof the algorithm in the Visualization Toolkit (VTK) [30, 31] andParaView [1] for the special case of CIELAB color space with theCIEDE2000 distance measure. We also conduct a parameter studyto determine the best resolution and neighborhood with respect topath lengths and runtime.

The main contributions of this paper in a nutshell are:

• Presentation of a novel method for the interpolation of colorsin arbitrary path-connected metric color spaces.

• Release of open an source implementation of the algorithm inVTK and ParaView.

• Optimization of the parameters w.r.t. path length and runtimethrough experiments to achieve interactivity.

We particularly do not claim that colormaps interpolated in non-metric spaces are better than the ones interpolated in Euclidean colorspaces. This could only be evaluated through a user study, which isoutside the scope of this paper. We simply provide the functionality.

2 RELATED WORK

Guild [6] describes the setup and the results of the colorimetricexperiments that led to the definition of the CIE standard observer.His seminal paper describes the birth of the first modern color space:

(a) Naive VTK implementation. (b) Corrected implementation.

Figure 3: VTK’s implementation of adapting midpoint and sharpnessdoes not trivially expand to our method because it is applied sep-arately between all intermediate nodes on a shortest path. Thisexample shows a sharpness of s = 1, which corresponds to constantinterpolation.

CIERGB. It is based on the three primaries from Wright’s experi-ments [37] and all currently used modern color spaces refer to it.The goal of the CIE back in those days was to embed all visiblecolors into one space to allow unambiguous reproduction of everypossible color sensation. A transformation of CIERGB to CIEXYZuses three imaginary primaries, which span the complete visiblespectrum [3, 5]. Later, the CIE’s efforts extended to the search ofa color space that would not only hold all colors but also providea metric that represents the perceived distances between all colors.This goal resulted in spaces like CIELAB, CIELUV, and CIECAM,which are transformations of CIEXYZ that approximate perceivedhuman color differences [13, 14, 16, 22, 28].

Judd [9, 10] defines the concept of an ideal color space as ”athree-dimensional array of points, each representing a color, so lo-cated that the length of the straight line between any two pointsis proportional to the perceived size of the difference between thecolors represented by the points.” In his papers, he collects evidencethat such an ideal color space cannot exist. He refers to the experi-ments of MacAdam [17–20], and Helm [7] regarding the principleof diminishing returns, and the Nickerson index of fading [23] re-garding the principle of hue-superimportance. Their experimentsall suggest that the perceived color distances cannot be embeddedin a three-dimensional Euclidean space, but that more complicatedmathematical models are needed [26, 29, 33, 34, 36].

In order to encompass the non-Euclidean behavior of human colorperception, CIELAB was equipped with other distance measures,like ∆E1994 and ∆E2000 [8, 15, 21]. Many authors agree that a per-ceptually uniform color space should be used to asses the qualityof colormaps [11, 12, 24, 27, 39, 41] but there are some problemswith the non-Euclidean distance functions. Due to their construction,they produce singularities between colors whose hues differ by 180degrees and sometimes do not fulfill the triangle inequality, whichmathematically disqualifies them as a metric [32]. On top of this,it is not easy to imagine a non-Euclidean space. It, and the pathsof colormaps in it, can’t be visualized easily. Further, concepts likelinearity, or smoothness of a colormap cannot be defined in thesespaces in a straightforward way [4]. To overcome some of thesedifficulties, Pant et al. provide a space that is close to the originaldistance measure, but is Riemannian [25]. Zerai and Triki [40] ex-tend the definition of image distortion measures (mean, variance,mean square error) to color images that are defined in a color spacethat satisfies the properties of a Riemannian manifold. These spacesare often used to describe color in a mathematical setting, as by vonHelmholtz [36], Schroedinger [29] and Stiles [38]. On the otherhand, Urban et al. [35] take one step back looking for the Euclideanspace that most closely approximates its non-Euclidean counterpart.

We believe that future color spaces will continue to better ap-proximate human color perception and embrace its complicatednon-Euclidean structure because our computational capacities willenable us to work with them despite those difficulties. In this paper,we describe a contribution to that goal that allows interpolation innon-Euclidean color spaces.

Figure 4: Average of both path distances and independent pathdistances with respect to the resolution used for computing the paths

3 ALGORITHM

In Euclidean color spaces, linear interpolation between two colorsforms a straight line, which corresponds to the shortest path betweenthose two colors. Our method generalizes linear color interpolationto non-Euclidean color spaces using this concept of interpolationalong a shortest path connecting two colors. We only assume that thespace is path-connected and has a distance measure (satisfying posi-tivity, symmetry, and identity of indiscernibles) that approximatesthe perceptually correct distance between any two colors. Throughthis notion of distance, it is possible to sample the color space andset up a graph with the color distances as weights on the edges. Thisenables us to interpolate between colors by searching a shortest pathon the graph.

Many configurations of placing nodes and edges in different colorspaces are possible. In our implementation, we use the RGB colorspace because of its handy cubic shape and because that way, wecan easily guarantee that all graph nodes are within the display’sgamut. We place the nodes uniformly in all three dimensions, usea 26 connectivity, and allow the user to choose from different res-olutions and neighborhoods, Figure 2. In addition to its simplicity,the uniform grid allows us to compute the graph on the fly while theDijkstra algorithm runs. Therefore, we do not have to store it, whichis crucial for higher resolutions.

For a practical use in a visualization environment, the interpo-lation algorithm must run with an interactive response time. Thisrestriction makes choosing a resolution in the order of the colors inthe final colormap (typically 256) computationally prohibitive. Toovercome this problem, we make use of the fact that for small colordifferences, the paths in different color spaces do not vary much.Especially in our case, the shortest paths of CIELAB’s Euclidean∆E1976 and its non-Euclidean ∆E2000 are similar for small colordifferences. Therefore, we use the classical linear interpolation ofthe underlying Euclidean space to fill the small gaps between thenode points on the coarser shortest path in the graph. By setting theresolution, the users can choose a trade-off between accuracy andspeed depending on the computational resources at their disposal.

The calculation of a shortest color path is implemented in VTK,using a version of Dijkstra’s algorithm is used to find a shortestpath within the graph of discrete RGB colors as explained earlier.One particularity of our path finding algorithm is that the graph isgenerated on the fly to minimize the algorithm’s memory footprint,which can become quite significant for larger graph resolutions (i.e.> 500 MB).

(a) Resolution = 2 (b) Resolution = 16

(c) Resolution = 4 (d) Resolution = 32

(e) Resolution = 8 (f) Resolution = 256

Figure 5: Comparison interpolating the same two colors pink andyellow with increasing graph resolution and a neighborhood of 1.

Once the graph path n0, ...,nk of k ∈ N+ nodes is calculated, itis cached and only updated on demand in the VTK implementationto achieve a higher run-time performance. To create a colormapwith n > k entries, the path is discretized regularly according to therelative distances of the nodes on the path. Here, for each of the nfinal colormap entries, the closest left and right of the k colors onthe graph path are found and interpolated linearly in CIELAB withits Euclidean metric. This leads to almost equal distances betweenthe final colors of the colormap according to the used color distancemeasure.

In addition, VTK allows users to set midpoint m ∈ [0,1] andsharpness s ∈ [0,1] parameters for colormaps, changing the way col-ors are interpolated using a modified Hermite curve. The midpointshifts the average between two colors to the left for m < 0.5 or theright for m > 0.5. The sharpness steers the interpolation betweenlinear for s = 0 and constant for s = 1. Since VTK applies thisHermite to each pair of neighboring colors in the colormap, its naiveapplication to our color path produces false results. We correct thecomputation using a generalized Hermite curve. The effect is shownin Figure 3 using the two colors, pink (R = 180, G = 60,B = 255)and yellow (R = 255, G = 248, B = 42). These same two colors areused throughout our experiments unless mentioned otherwise.

4 EXPERIMENTS

In this section, we describe the experiments performed to select thebest parameters in terms of path length and path build time. Figure 4shows that the graph-theoretical path length, i.e., the sum over thedistances between the colors c(ni) of consecutive nodes ni

lg =k

∑i=1

∆E(c(ni),c(ni+1)), (1)

is not a good candidate because the principle of diminishing returnsgives a false advantage to coarse paths, i.e., low graph resolutionand small k. A better candidate is the path length in the color space,

l = sup0=t0,...,tn=1

k

∑i=1

∆E(c(ti),c(ti+1)) (2)

because the supremum over all possible sample points ti on the pathis independent from the resolution of the underlying graph used togenerate the path. That makes an unbiased comparison over graphsof different resolutions possible. For its practical evaluation, wechoose a discrete resolution of 2563 in RGB, includes all graph nodesof our experiments. The evaluation is performed using a version ofBresenham’s line drawing algorithm [2]. In our experiments, weconstruct all possible colormaps of maximal extension in the graph,i.e., all combinations of color pairs of the corner colors of RGB, andaverage the resulting path lengths and build times.

Figure 4 demonstrates the convergence of the mean path length inspace to the path length in the graph for a neighborhood of 1. It also

(a) Linear Comparison

(b) Log-log Comparison

Figure 6: Average path build time based on resolution and neighbor-hood size.

shows that larger neighborhoods generate shorter graph-theoreticalpath lengths lg despite larger actual path lengths l. That is due tothe fact that the graph chooses seemingly shorter paths by jump-ing across regions that can be reached only through large detoursby a continuous path. Note that neighborhood 2 with resolution 1and neighborhood 3 with resolutions 1 and 2 are not representedin Figure 4 because the resolutions are too small to fit a full neigh-borhood and the results are identical to the smaller neighborhoods.Resolution 256 with neighborhood 3 is missing due to high computetimes. In accordance with the results shown in Figure 4, Figure5 demonstrates through an example that the interpolation resultsfrom our method barely change at graph resolutions of 16 or higher.The corresponding timing results can be found in Figure 6. Thecolormaps with resolution 16 take 0.1s while resolution 32 takesaround 1s. Our experiments suggest that the optimal settings w.r.t.build time and interpolation quality are a resolution of 16 and aneighborhood of 1.

Figure 7 shows two colors linearly interpolated in RGB, HSV,LAB with CIEDE1976, and LAB with CIEDE2000 color spaces.The latter was generated using the shortest path from our algorithmfrom Section 3. The path lengths depicted in Figure 7 demonstratethe fact that our interpolation method achieves the shortest pathamong all tested color interpolation methods.

Figure 8 visualizes the paths from Figure 7 and the surround-ing RGB color space embedded in the CIELAB color space. TheCIEDE1976 path is as expected a straight line, whereas the RGB

(a) RGB (path length: 91.438)

(b) HSV (path length: 115.498)

(c) CIEDE1976 (path length: 92.485)

(d) CIEDE2000 (path length:90.156)

Figure 7: Linear interpolation between pink and yellow in four differentcolor spaces. Path length is measured in CIEDE2000.

path is slightly curved but still follows the CIEDE1976 line closely.The HSV and CIEDE2000 paths have more extreme deviations.Whereas the HSV path moves more along the outskirts of the RGBcolor space, the CIEDE2000 path moves closer towards its center.We were able to observe the same behavior with other colors aswell. This phenomenon is in accordance with hue-superimportancebecause the human eye perceives desaturated colors as being closertogether than flashier ones. For colormaps with multiple controlpoints, this often results in sharper transitions at these control points,Figure 9, with the path forming a flower-like pattern.

5 CONCLUSIONS

During the course of this paper, we discussed the issues regardingcolor interpolation in non-Euclidean color spaces. We presented anapproach based on shortest paths in graphs with interactive responsetime and demonstrated it applied to the example of the CIELABcolor space with the CIEDE2000 distance measure. We conductedexperiments to select the best default parameters and compared theinterpolation results to other commonly used color spaces. Eventhough there is no mathematical proof that a unique shortest pathalways exists, it did exist in each tested configuration.

In conclusion, our findings show a robust way of interpolatingnon-Euclidean path-connected metric color spaces. This makes ourmethod extremely valuable for color interpolation in future colorspaces with more complex and accurate models of human perception.

As part of our work on this paper, we provide an extension toVTK for the CIELAB color space with the CIEDE2000 distancemeasure, which is now officially available in the current VTK nightlybuild. Additionally, our method is integrated through VTK in thecolor map editor menu of ParaView and has been available sincethe release of version 5.5 (see Figure 10). As determined by ourparameter study, the ParaView version uses a 163 discretized gridwith a one-neighborhood set up in the RGB color space and mappedover to the CIELAB color space.

In the future, we would like to extend this work to avoid the sharptransitions at control points through a generalization of the conceptof smoothness to non-Euclidean color spaces.

ACKNOWLEDGMENTS

This work was funded by the National Nuclear Security Administra-tion (NNSA) Advanced Simulation and Computing (ASC) Program,for production visualization, and by U.S. Department of Energy Of-fice of Science, Advanced Scientific Computing Research, through

Figure 8: Shortest paths from four different color spaces and the RGBcolor cube embedded in the CIELAB color space

(a) CIEDE1976

(b) CIEDE2000

Figure 9: Example of our color interpolation method for multiple non-uniformly distributed control points: {[0.0, (0, 0, 0)], [0.4, (255, 0, 0)],[0.8, (255, 255, 0)], [1.0, (255, 255, 255)]}

Figure 10: Integration of our method in the ParaView nightly build. Itcan be chosen in the colormap editor drop down menu as indicatedby the red circle.

Dr. Laura Biven. We would also like to thank Greg Abram forcontributing a resampled version of the meteor impact dataset forvisualization.

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