True colours or red herrings?: colour maps for finite-element analysis in palaeontological studies to enhance interpretation and accessibility

Accessibility is a key aspect for the presentation of research data. In palaeontology, new data is routinely obtained with computational techniques, such as finite-element analysis (FEA). FEA is used to calculate stress and deformation in objects when subjected to external forces. Results are displayed using contour plots in which colour information is used to convey the underlying biomechanical data. The Rainbow colour map is nearly exclusively used for these contour plots in palaeontological studies. However, numerous studies in other disciplines have shown the Rainbow map to be problematic due to uneven colour representation and its inaccessibility for those with colour vision deficiencies. Here, different colour maps were tested for their accuracy in representing values of FEA models. Differences in stress magnitudes (ΔS) and colour values (ΔE) of subsequent points from the FEA models were compared and their correlation was used as a measure of accuracy. The results confirm that the Rainbow colour map is not well suited to represent the underlying stress distribution of FEA models with other colour maps showing a higher discriminative power. As the performance of the colour maps varied with tested scenarios/stress types, it is recommended to use different colour maps for specific purposes.


SL, 0000-0003-3472-814X
Accessibility is a key aspect for the presentation of research data. In palaeontology, new data is routinely obtained with computational techniques, such as finite-element analysis (FEA). FEA is used to calculate stress and deformation in objects when subjected to external forces. Results are displayed using contour plots in which colour information is used to convey the underlying biomechanical data. The Rainbow colour map is nearly exclusively used for these contour plots in palaeontological studies. However, numerous studies in other disciplines have shown the Rainbow map to be problematic due to uneven colour representation and its inaccessibility for those with colour vision deficiencies. Here, different colour maps were tested for their accuracy in representing values of FEA models. Differences in stress magnitudes (ΔS) and colour values (ΔE) of subsequent points from the FEA models were compared and their correlation was used as a measure of accuracy. The results confirm that the Rainbow colour map is not well suited to represent the underlying stress distribution of FEA models with other colour maps showing a higher discriminative power. As the performance of the colour maps varied with tested scenarios/stress types, it is recommended to use different colour maps for specific purposes.  Figure 1. Problems of the Rainbow colour scheme: (a) non-uniform distances between individual colours (adapted from [27]). (b) Lack of intuitive perceptual order. (c) Rainbow colour map as seen without and with colour vision deficiency (i.e. deuteranopia and protanopia type) and in greyscale.
royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 8: 211357 the Rainbow colour map creates considerable accessibility problems for those with colour vision deficiencies (CVDs). Approximately 5-10% of the population may suffer from some form of CVD, such as red-green blindness (Deuteranopia), which renders data represented by the Rainbow colour map largely unreadable [32][33][34]. Furthermore, similar issues arise when results using the Rainbow colour map are converted to a greyscale format, such as for example for printing.
Given these inherent problems with the Rainbow colour scheme, several disciplines, including oceanography [35], meteorology [36,37] and geosciences [25,38], have started to address this issue and proposed the use of alternative colour schemes. Here, different colour maps are tested and their effectiveness for the visualization of FEA results of palaeontological models is evaluated.

Material and methods
In order to evaluate their visual effect and accessibility, different colour maps were tested for a variety of FEA models of fossil specimens and different FEA stress measures. In addition to the traditional Rainbow map (see also [39]), nine further, established colour maps were selected (version numbers are provided where present): (a) The five sequential colour maps Batlow (7.0), Inferno, Parula, Viridis and YlGnBu [25,[40][41][42][43]. Sequential colour maps vary between two colours ranging from dark to light (or vice versa) and are suitable for ordered data ranging gradually from low to high values (i.e. ratio data with an absolute zero value) [27]. (b) The three diverging colour maps Cork (7.0), Polar and Roma (7.0) [25,43]. Diverging colour maps range between two contrasting colours at either end separated by a neutral colour in the middle and are suitable for interval data that can have positive and negative values [27]. (c) As a further option, a variant of the classic Rainbow colour map known as Turbo was included in the analysis. Although Turbo similarly consists of a sequence of colours in the visible light spectrum, it has been suggested to represent a perceptually improved rainbow map with a uniform luminance [44,45]. All colour maps are non-proprietary, in some cases, versioned and available/defined via the respective references above. Not all colour maps are readily and equally available by default in all software but can be added in most cases (see also below), Other colour maps, such as qualitative, categorical or cyclic colour maps, were not tested as these are not appropriate for FEA data. The colour maps tested here were selected following their use and popularity in different applications. However, not all of the colour maps are perceptually uniform (e.g. the difference between two colours as perceived by the human eye is proportional to the numerical distance within the given colour space). Batlow, Cork, Inferno, Roma, Viridis and YlGnBu are all perceptually uniform, whereas Parula, Polar, Rainbow and Turbo are not (see also [43]).
All colour maps used for this study consist of 24 individual colour values (definitions (order and HEX colour codes) are available in the electronic supplementary material), and all outputs presented here were generated in Abaqus and model views were saved as image files. Depending on the software, custom colour maps can be created. In this example, all colour maps were created in Abaqus via a commandline script detailing the colour components via HEX codes individually (see the script in electronic supplementary material). Alternatively, new colour maps (so-called spectra in Abaqus) can be created via a tools menu and selecting successive colours via a colour picker. This process will differ for individual software. However, specific pre-designed colour maps can be generated and accessed via online tools, such as https://colorbrewer2.org/#type=sequential&scheme=BuGn&n=3 [40].
Several FEA models of fossil specimens and different skeletal elements were used here to evaluate the perceptual effects of the tested colour maps with different three-dimensional morphologies: (a) a simplified, planar model of the mandible of the sabre-toothed cat Dinofelis cristata as used in [46] (figure 2). This model was chosen to represent a geometrically simple morphology as used for FEA models not derived from computed tomography (CT) or surface-based digitization methods [47,48]. For this model, contour plots displaying the distribution of von Mises stress were chosen as an example for ratio data. (b) A three-dimensional model of the mandible of Thrinaxodon liorhinus representing a geometrically more complex morphology compared to the model of Dinofelis and derived from CT scanning [49]. For the contour plots, tensile ( positive) and compressive (negative) absolute stresses were displayed as an example for interval data. (c) A model of the skull of the therizinosaurian dinosaur Erlikosaurus andrewsi as used in [50]. In addition to the different contour plots displaying von Mises stress, the models were also displayed as perceived with deuteranopia-type CVDs. For this purpose, the images were converted accordingly using Adobe Photoshop CC 2020. (d) A model of the skull of the capitosaurian temnospondyl Parotosuchus helgolandicus [51] representing a dorsoventrally flattened skull morphology. In addition to the royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 8: 211357 different contour plots displaying von Mises stress, the models were also displayed as perceived with a protanopia-type CVD. For this purpose, the images were converted accordingly using Adobe Photoshop CC 2020. (e) A model of a dorsal vertebra of the ornithischian dinosaur Stegosaurus stenops [47] representing a post-cranial skeletal element. In addition to the different contour plots displaying von Mises stress, the images of the contour plot models were also converted into greyscale using Adobe Photoshop CC (2020) (Image -> Adjustments -> Black & White and using the default setting for greyscale mode). (f ) A model of the manual claw of the therizinosaurian dinosaur Nothronychus graffami [52] to illustrate the effect of colour maps against a different background colour. The boundary conditions for these models have no direct effect on the colour map interpretation. Therefore, please refer to the original publications for further details on the boundary conditions of the respective models.
To quantify the discriminative power of the individual colour maps (i.e. relating individual colour values to their respective FEA stress magnitudes), the correlation between the colour maps and stress results of the FEA models was calculated. For consistency across the models, 20 points (i.e. elements)  royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 8: 211357 along with a line across the FEA model were selected that covered the morphology evenly. Stress magnitudes (von Mises and compressive/tensile stresses) were recorded for all sampled points. This approach follows the practice to sample a subset of elements of an FEA model [11,12,50] to quantify its biomechanical properties.
In the next step, the colour values for the sampled points for each tested colour map were recorded as RGB values. Although defining colour as RGB values is a common practice for many (web-based and digital) applications, they were specifically designed for use on monitors and do not reflect human colour perception as the RGB colour space is not uniform [53]. A solution to this problem is using the CIELAB (also known as CIE L Ã a Ã b Ã ) colour space [54] which has been designed to be perceptually uniform. Here, the distance between two points defining individual colours is proportional to the perceptual difference between them [55,56]. Therefore, the collected RGB values were converted into CIELAB colour values. The collection of the RGB colours from images of the FEA models and subsequent conversion to CIELAB colour space was done via the convertColor function in R [57] (see electronic supplementary material). For a colour map to represent the underlying data correctly, it must reflect changes in magnitude between two sampled points accordingly. To test this correlation, the absolute difference ΔS in stress magnitude was calculated for subsequent points sampled for each model (equation (2.1)).
DS ¼ stress magnitude of point 1-stress magnitude of point 2: ð2:1Þ Similarly, the difference in colour value ΔE was determined for each colour map [53] by calculating the Euclidean distances between two subsequent points (equation (2.2)) In a final step, ΔE and ΔS were subjected to an ordinary least square regression and the R 2 -value was obtained as a measure for the discriminative power of the individual colour maps (see electronic supplementary material).
It should be noted that Abaqus applies a shading algorithm when displaying FEA contour plots in that it simulates an artificial light source positioned to the top left of the three-dimensional space. To avoid the effects of artificial shadows on the colour representation, all measurements were performed with the model exposed to the maximum light intensity (usually with models in left lateral or dorsal view).

Results
Overall, 10 different colour maps were tested for their accuracy to represent the underlying stress magnitudes of finite-element models in the form of contour plots. In addition to the default Rainbow map, nine further colour maps were tested and the R 2 -value was used as a measure for the correlation between stress magnitudes and colour maps.
Across the different colour maps, models, stress types and visual appearances, the R 2 values range from nearly no (R 2 = 0.008) to strong correlations (R 2 = 0.967) (table 1). No single colour map was found to show consistently the strongest correlation for the different test settings, with rather more nuanced variations in representative performance for the different colour maps. It is noteworthy that the Rainbow colour map performed worse than most of the other colour maps.
For results in the form of ratio data, such as von Mises stress (figure 2; electronic supplementary material, figure S1), the sequential colour map Inferno produced the highest correlation (R 2 = 0.802). By contrast, the commonly used Rainbow map showed only a weak correlation (R 2 = 0.571), and only the colour maps Roma (R 2 = 0.563) and YlGnBu (R 2 = 0.547) had a weaker performance. The rainbow variant Turbo performed only moderately better than the classic Rainbow (R 2 = 0.664).
For interval data, such as compressive (i.e. negative) and tensile (i.e. positive) stresses plotted together (figure 3; electronic supplementary material, figure S2), the sequential colour maps YlGnBu (R 2 = 0.967) and Parula (R 2 = 0.959), as well as the diverging colour map Polar (R 2 = 0.934) showed the highest correlation between stress magnitudes and colour representation. The Rainbow colour map produced a strong, although not the highest, correlation (R 2 = 0.887), whereas Turbo performed worst in this scenario but with still a strong correlation (R 2 = 0.852).
To test for the discriminative performance of the different colour maps when perceived with a CVD, contour plots were converted to deuteranopia-and protanopia-type appearances (figures 4 and 5; royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 8: 211357     royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 8: 211357 In a final analysis, contour plots were converted to greyscale and the discriminative performance of the colour maps was tested ( figure 6; electronic supplementary material, figure S5). In this scenario, all colour maps produced only a moderate to no correlation (0.578 < R 2 < 0.008). royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 8: 211357 The best performance was found for Inferno (R 2 = 0.578); Rainbow, Roma and Turbo showed the least correlation (R 2 < 0.098).

Discussion
The Rainbow colour map has been a ubiquitous tool in data visualization for decades [25,37]. Engineering techniques such as FEA, which has increasingly been used in palaeontological studies over the last 20 years, are no exception. Results from FEAs are routinely visualized in the form of contour plots using the Rainbow colour map. Contour plots typically display von Mises stress, a common measure to evaluate the stability of a model under loading conditions. However, as shown by the results from this study, the Rainbow colour map correlates only poorly with the underlying von Mises stress data (table 1 and figure 2), and its discriminative power is equally poor when perceived with different types of CVDs (figures 4 and 5). This should not come as a surprise as the Rainbow colour map has been considered problematic and misleading in other disciplines [18,19,22,[23][24][25].
Other colour maps tested here performed considerably better. However, no one colour map was found to be optimally suited for all types of stress and visual perception. For interval-type stresses, such as compressive and tensile stresses plotted on the same model, the Rainbow colour map showed a high correlation (table 1, figure 3) similar to or even better than the diverging colour maps in this study. Interestingly, diverging colour maps did not necessarily perform better for interval data, whereas sequential colour maps were not always found to show the best correlation for ratio data (i.e. von Mises stress). Inferno, Batlow and Parula generally showed the highest discriminative power, but not consistently so (table 1). It is noteworthy that differences in the performance were recorded when colour maps were tested in CVD settings. The same colour maps (Inferno, Batlow and Parula, and to a lesser degree Polar and Viridis) represented the underlying stress values reasonably well despite the reduced colour information. However, this means that a single colour map cannot be used as a silver bullet to perform equally well under all conditions. Similar to the alternative text describing figures, the second set of contour plots with a different colour map could be provided in the electronic supplementary material accessible to those with CVD. More generally, the use of specific colour maps may have to be decided on a case to case basis using custom-made or existing colour maps (see, for example, [38] for available colour maps). For interval-type data, other considerations than the discriminative power (expressed as the R 2 value here) may need to be considered. For such data, the central zero value can be an important identifier of stress-free regions in the model, which can be recognized more easily when diverging colour maps are used.
In this context, it should be noted that the correlation analysis used here to discriminate stress/colour changes is not perfect. Human colour perception is not uniform, often subjective and dependent on other factors such as age and individual variation and as such does not correspond to Euclidean distances in colour space [58]. The CIELAB colour space is an attempt to replicate human colour differentiation. As the correlation analysis only considers absolute changes along with a trajectory, the analysis may not record the exact correlation when non-monotonic changes on the stress scale are associated with changes in different directions in the CIELAB space. However, this is less likely to be a problem for the perceptually based colour maps.
The choice of an appropriate colour map may further depend on the nature of the results of an FEA. Models spanning a wide range of stress magnitudes, but with an uneven distribution of values will be biased towards certain regions of the colour map. This situation could result in a lower resolution of stress (and thereby colour) values towards the lower end of the colour map to encompass the full range of stress magnitudes present. Sequential colour maps will be a better option in such cases as their colour gradient is expressed along with the whole range of the colour map in comparison to divergent colour maps.
It is important to note that CVD is only one form of visual impairment and of course further improvements for accessibility should be aimed for when considering the presentation of results from FEA (and other analyses more broadly). The Web Content Accessibility Guidelines (WCAG) [59] provide further recommendations to improve accessibility, including appropriate contrast ratios between colours to allow their distinction. For example, Cork, Polar and Roma have high contrast ratios, whereas Inferno and Viridis have poorer contrast ratios. These ratios are, of course, lower if subsequent colours along with the gradient were to be tested, not just extreme and mid-point values. However, this goes to show that not only the uniform sequence of colours but also the contrast between them plays a role in making contour plots accessible.
royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 8: 211357 This situation is further complicated in that FEA contour plots cannot be regarded in isolation but need to be considered in the context of background and environmental settings. In the simplest of cases, this could mean that the choice of background colour can influence the readability of the contour plots [25,60,61] (figure 7). Especially colour maps with a large amount of dark components can become invisible against a black background. Colourmaps with strong contrast and luminosity (e.g. Parula, Polar) can work well in such a case. For the presentation on a white background, colour maps with a decreasing chroma (=colour intensity), such as Batlow, Inferno and, in particular, YlGnBu are more appropriate to convey the results [25,61,62].
Within a digital, three-dimensional environment pseudo-colouring creates a further difficulty as the choice of colours interact with the shading and perception of spatial cues [24]. Properties such as the number, direction and intensity of light sources, specularity (i.e. reflectiveness of a surface) and other settings can have an impact on the appearance of colour maps as well. Further, different devices and display technologies display colours variably and for consistent perception colour calibration would have to be performed first.
Most FEA software allows turning off shading effects. However, this could possibly result in a reduced perception of the model morphology, especially for flattened surfaces with low topography (e.g. figure 5). Although not tested here, it should further be taken into consideration that different FEA software packages may use slightly different variations/colour definitions of the Rainbow colour map, further exacerbating comparisons between outputs from different software. royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 8: 211357 The eye-catching quality of the Rainbow colour map with its high luminance and contrast is likely the reason for its continued prevalence despite its problems with data distortion. Different reasons have been discussed in the past [24,25] for why the Rainbow colour map is still the visualization tool of choice for many studies and applications. For finite-element models, this has likely historic reasons and it is the default colour map in most software. Furthermore, the colour distribution of the Rainbow map has a very strong signalling function and communicative power: cold colours (i.e. blue) are associated with no or low stresses, whereas warm colours (i.e. yellow, red) indicate high-stress magnitudes. For von Mises stress, high magnitudes indicate possible material failure and an association with a colour such as red which is commonly used to convey danger is intuitive [63]. However, this concept can also be conveyed with other colour maps such as Inferno.
A recent study has used a variety of colour maps to display the results from FEAs [64]. Similarly, for palaeontological studies using other engineering tools, different colour maps have started to appear in publications. CFD, an engineering technique to simulate fluid flow within or around objects, uses a similar approach to FEA to represent data with pseudo-colour plots [7]. Although the Rainbow colour map is routinely used to visualize CFD results, different colour maps have been used recently in some studies [65].

Conclusion
Results from this study demonstrate that the Rainbow colour map is not well suited to represent the underlying stress distribution of FEA models. Although most of the other colour maps tested here showed a higher discriminative power, no single colour map was found to perform consistently well throughout all scenarios and for all stress types. It is therefore recommended that different colour maps without data distortion are used to present results. This could mean using different colour maps for ratio (e.g. von Mises stress) and interval data (e.g. compressive and tensile stresses). Alternatively, the second set of contour plots with a different colour map could be provided in the electronic supplementary material to increase accessibility.
The perception of colour is highly dependent on multiple factors, including display devices, colour standards for display and printing, and differences in the human visual apparatus. A variety of different colour maps displayed on different models of palaeontological specimens have been presented here. It is hoped that the reader will use these examples alongside the quantitative evaluation as guidance for their applications and studies. However, the tested colour maps in this study are far from exhaustive and a variety of tools exist to access pre-designed colourmaps (see [25], Box 2).