Predicted effectiveness of EnChroma multi-notch filters for enhancing color perception in anomalous trichromats

EnChroma filters are aids designed to improve color vision for anomalous trichromats. Their use is controversial because the results of lab-based assessments of their effectiveness have so far largely failed to agree with positive anecdotal reports. However, the effectiveness of EnChroma filters will vary depending on the conditions of viewing, including whether the stimuli are broadband reflective surfaces or colors presented on RGB displays, whether illumination spectra are broadband or narrowband, the transmission spectra of particular filters, and the cone spectral sensitivity functions of the observer. We created a model of anomalous trichromatic color vision to predict the effects of EnChroma filters on the color signals impaired in anomalous trichromacy. Using the model we varied illumination, filter type and observer cone sensitivity functions, and tested the effect of presenting colors as broadband reflective surfaces or on RGB displays. We also used hyperspectral images to assess the impact of the filters on anomalous trichromats ’ color vision for natural scenes. Model results predicted that the filters should be broadly effective at enhancing anomalous trichromats ’ equivalent to L/(L + M) chromatic contrasts under a range of viewing conditions, but are substantially more effective for deuteranomals than for protanomals. The filters are predicted to be more effective for broadband reflective surfaces presented under broadband illuminants than for surfaces presented under narrowband illuminants or for colors presented on RGB displays. Since the potential impacts of contrast adaptation and perceptual learning are not considered in the model, it needs to be empirically validated. Results of empirical tests of the effects of EnChroma filters on deuteranomalous color vision in comparison with model predictions are presented in an accompanying paper (Somers et al., in prep.).


Introduction
EnChroma glasses (EnChroma Ltd., Berkeley, California, USA; Schmeder & McPherson, 2019) are aids designed to enhance the color vision of people with anomalous trichromacy, a form of congenital color vision deficiency (CVD).Color vision is impaired in anomalous trichromacy because the sensitivity functions of the two cone types sensitive in the medium and long wavelength part of the spectrum overlap to a greater degree than in normal trichromacy.Spectral notches in the transmission spectra of EnChroma filters are designed to increase the difference between signals from the cone types sensitive in the medium and long wavelength part of the spectrum by filtering out light in the spectral region where the cone sensitivity functions are most similar.However, thus far, there has been limited empirical support for their effectiveness, and several studies have concluded that they are largely ineffective ( Álvaro et al., 2022;Gómez-Robledo et al., 2018;Patterson et al., 2022;Pattie et al., 2022).Since their effectiveness is likely to depend on a range of factors, including illumination spectra, reflectance spectra, observer-specific cone sensitivity functions and the precise filter, we conducted a modelling investigation to predict the effectiveness of the filters under a variety of conditions.The paper is accompanied by an empirical paper (Somers et al., in prep.), which tests the predictions of the model in psychophysical experiments with anomalous trichromats.
Normal human color vision is based on comparisons between the signals of three classes of retinal cone sensitive to short (S), medium (M) and long (L) wavelengths of light.Most congenital CVDs arise from polymorphisms in the genes that encode the L and M cone opsins, which are located in a gene array on the X-chromosome.Anomalous trichromacy is a mild form of CVD, where either the L or the M cone class is replaced by an anomalous cone class containing a photopigment with a spectral sensitivity that is much more similar to that of the other X-linked cone class than in normal trichromacy.There are two types of anomalous trichromacy: In the more common deuteranomaly (affecting about 5% of men) the normal M cone class is replaced by a variant L′ cone class, and in the rarer protanomaly (affecting about 1% of men) the normal L cone class is replaced by a variant M′ cone class (note that these labels have sometimes been reversed, e.g., DeMarco et al., 1992).
Postreceptorally, color information is thought to be processed predominantly via two 'cardinal' retinogeniculate color mechanisms.One color subsystem comparing the activities of the L and M cones signals colors along a roughly red-teal color axis.The other subsystem compares the activities of the S cones with those of the other two cone types to signal colors that lie along a violet-chartreuse axis.The strengths of redteal color signals are reduced in anomalous trichromacy compared to those of normal trichromats because of the greater spectral overlap of the sensitivities of the two inputting cone classes.Anomalous trichromats' color vision along this axis is therefore typically impaired.
Filters have long been used to try to enhance the color vision of people with CVD.In an early report, Seebeck (1837, cited in Lanthony, 2013) found that people with CVD were able to distinguish red and green surfaces when looking at them sequentially though red and green filters, by noticing how their lightness changed between the filters.Similarly, filters of different colors can be placed in front of each of the two eyes, creating a binocular luminance conflict perceived as binocular lustre (Formankiewicz & Mollon, 2009;Wendt & Faul, 2022).Several commercial products employ this strategy, including Chromagen (Birkenhead, UK; Harris, 1999), Colormax (Birmingham, Alabama, USA) and X-Chrom (Ipswich, Massachusets, USA; Zeltzer, 1991).Interocularly discrepant filters are not thought to enhance color appearance but may allow CVD observers to perceive the figures in pseudoisochromatic plate diagnostic tests for CVD by providing a difference in binocular lustre between figure and ground (Heath, 1974;Formankiewicz & Mollon, 2009;Sheedy & Stocker, 1984).
Colorlite filters (Colorlite USA LLC, Boca Raton, Fl, USA; Ábrahám et al., 1998) employ particular combinations of multiple filter layers to selectively attenuate certain regions of the spectrum with the aim of enhancing color vision for the individual observer.Functionally, the resulting filter often contains a spectral notch, but transmission spectra vary because they are 'tailored' to the individual.The first filter-based aids to explicitly employ a spectral notch were EnChroma filters and Oxy-Iso filters (Vino Optics, US Virgin Islands; Barber & Changizi, 2012).Oxy-Iso filters were later developed for medical contexts as aids for enhancing perception of blood oxygenation in normal trichromats, though have more recently launched as VINO filters for CVD, while EnChroma filters have been developed to improve color vision in CVD.More recent notch filters have been developed using dyes (Badawy et al., 2018) and using gold nanostructures (Karepov & Ellenbogen, 2020).In contrast to interocularly discrepant filters which are expected to have roughly the same effect for any class of CVD observer, notch filters can only be effective for anomalous trichromats and not for dichromats who lack a cone class.The mode of operation of notch filters is to increase the effective difference between the activities of different cone classes when exposed to colored light, so dichromats who have only one cone class sensitive in the medium and long wavelength region of the spectrum cannot benefit.
There are several existing models to predict the impacts of filterbased aids on color vision in anomalous trichromats.Moreland et al. (2010) calculated chromaticity coordinates in versions of the MacLeod-Boynton (1979) chromaticity diagram constructed for protanomalous and deuteranomalous observers using the DeMarco et al. (1992) anomalous trichromatic cone fundamentals, corrected to be approximately perceptually uniform by considering the spectra that form a set of grid intersections in a revised version of the uniform CIE u′v′ color space.They modelled the effects of ChromaGen, ColorMax and Coloryte (Colorlite) filters on the color 'gamuts' of a set of Munsell surfaces, the D15 test surfaces and red and green traffic lights.For each filter they computed an 'expansion factor': the ratio of filtered versus unfiltered standard deviations along the cardinal axes of the observer-specific MacLeod-Boynton (1979) chromaticity diagrams corrected for perceptual non-uniformity.They found that most filters were predicted to contract color gamuts, particularly along the anomalous analogues of the L/(L + M) axis in the MacLeod-Boynton chromaticity (1979) diagram.They found that only a few aids produced a positive expansion factor for both MacLeod-Boynton axes, which fell far short of the factor of ~ 2 they considered a feasibly useful enhancement.Later, Moreland et al. (2022) revised their model to test a more representative set of 80 colored surfaces equally spaced in saturation and hue in their modified CIE u′v′ space.They used the model to predict the effects of EnChroma filters, Oxy-Iso filters, Carelust filters (Carelust Ltd., London, UK) and Color-Correct (ColorMax) filters for deuteranomalous and protanomalous observers.They found predicted gamut expansion along the anomalous analogues of the MacLeod-Boynton (1979) chromaticity diagram's L/(L + M) axis for Oxy-Iso and EnChroma filters but not for Carelust or ColorCorrect filters.Of the EnChroma filters they tested, one type (Cx3 SP) created the largest expansions (factors of 1.8 for protanomals and 1.7 for deuteranomals), but at the expense of gamut contraction along the S/(L + M) axis.
Gómez-Robledo et al. ( 2018) created a model based on a model of anomalous trichromacy by Lucassen & Alferdinck, 2006 to predict the effectiveness of EnChroma filters for anomalous trichromats.The model combines proportions of normal trichromatic L and M cone activities to estimate the activities of cone types with intermediate spectral sensitivities.Mild deuteranomaly, for example, is simulated by combining 30% of the activity of the normal trichromatic L cone with 70% of the activity of the normal trichromatic M cone.This method goes some way towards representing the reduction in cone-opponent signals in anomalous trichromacy, but does not fully capture the interaction between the anomalous cone sensitivity functions and the spectral notch of the EnChroma filter.They used the model to assess the impact of the filters on simulated observer representations of median lightness, median chroma and mean hue angle for stimulus datasets.Their model predicted that EnChroma filters increase the chroma of uniform stimuli for all the observer types tested, but for hyperspectral images of natural scenes it predicted that EnChroma filters tend to decrease chroma.They concluded that overall EnChroma filters should cause only small changes in perception of certain colors.Martínez-Domingo et al. (2020) extended this work to estimate changes in the number of discernible colors in CIE Lab space conferred by filters for standard uniform colored surfaces and hyperspectral images of natural scenes.Their model predicted that EnChroma and Oxy-Iso filters should decrease the number of discernible colors, probably at least in part due to the significant reduction in luminance they confer.
Recently, Pattie et al. (2022) created a model based on the DeMarco et al. (1992) cone sensitivity functions.They applied their model to a subset of matte Munsell surfaces and predicted that EnChroma filters should confer a small gamut expansion for anomalous trichromats.They also constructed an ideal observer model of performance on the Farnsworth-Munsell 100 hue test (FM100), which predicted that EnChroma filters would cause an overall increase in Total Error Score, and improve color discrimination for some regions of the hue circle but worsen it for others.This was in agreement with their behavioural results: they found that total error score on the FM100 increased for a large sample of anomalous trichromats when using EnChroma filters.Nascimento and Foster (2022) used an information theoretic approach to estimate the improvement in the number of discriminable colors conferred by EnChroma and Oxy-Iso filters for hyperspectral images of natural scenes.Using their model of anomalous trichromatic cone sensitivity functions that included receptor noise they calculated the mutual information between the spectra in the hyperspectrally imaged scenes and observer-specific cone responses with and without the addition of notch filters.From this they estimated that the filters are able to confer a modest (9-15%) increase in the number of discriminable surfaces for anomalous trichromats, less than the 27-42% increase predicted for normal trichromats using the same filters.
Though most existing models of the effectiveness of EnChroma filters for improving anomalous trichromatic color vision have concluded positively, they have not systematically assessed the influence of different viewing conditions.The effect of EnChroma filters will depend on the interaction between the spectral notch(es) in the filter transmission spectra, observer cone sensitivity functions, and light spectra.The latter is influenced by reflectance spectra and illumination, and varies between broadband reflective surfaces in the real world and colors rendered on RGB displays.We sought to systematically test the influence of observer cone sensitivity functions, illumination, filter transmission spectra for different EnChroma models and RGB displays on the effectiveness of EnChroma filters for enhancing anomalous trichromatic color vision.

The model
We created observer-specific variants of the MacLeod-Boynton (1979) chromaticity diagram to model postreceptoral color signals (e.g., Thomas et al., 2011) in response to any combination of illumination, surface, filter, and observer.Fig. 1a depicts the model.The radiance spectrum from a given illuminant (1) is attenuated by the reflectance spectrum of a given surface (2), and an EnChroma filter (3).Anomalous trichromatic cone sensitivity functions (4) are then multiplied by the incident spectrum to estimate each cone class's response to the light at each wavelength (5).The activity of each cone class is summed over wavelengths to give the total cone response to the light (6).We then estimate postreceptoral retinogeniculate cone-opponent signals as observer-specific analogues of MacLeod-Boynton (1979) chromaticity coordinates S/(L + M) and L/(L + M) (7).Equations 1-3 describe the calculation of cone activities.Here, the three cone sensitivity functions available to the particular modelled observer are l(λ), m(λ) and s(λ).R(λ) is the reflectance spectrum, I(λ) is the illumination spectrum, and τ(λ) is the transmission spectrum of the EnChroma filter.L, M and S are the tristimulus cone responses to the combination of illumination, reflectance, and filter.For deuteranomals we subsequently label the three cone activities L, L′ and S, and for protanomals, M′, M and S (note that anomalous trichromatic cone types have sometimes been labelled oppositely, e.g., DeMarco et al., 1992). (1)

Spectral datasets
We used three spectral datasets: (i) Uniform broadband surfaces; (ii) uniform colored patches rendered on RGB displays; and (iii) hyperspectral images.Dataset (i) was the full set of glossy Munsell surfaces of Value 5 (N = 259) obtained from an online database of 1600 glossy Munsell surfaces (Haanpalo n.d.).Dataset (ii) was created by converting the spectra in Dataset (i) to RGB values (metameric for anomalous trichromats) for three types of RGB display (see Section 2.5 for details).Dataset (iii) was a selection of 75 hyperspectral images (see Section 2.6 for details).1) is multiplied by the surface reflectance spectrum (2), and by the EnChroma transmission spectrum (3) to model the light reaching the eye.This product is then multiplied by the three cone sensitivity functions (4) to model the cone responses at each wavelength (5).The result is summed over all wavelengths (6) to model the cone activities elicited by the surface under the illuminant viewed through the EnChroma filter (7).Finally, the chromaticity of the surface is plotted in an observer-specific version of the MacLeod-Boynton (1979) chromaticity diagram (8), and compared to the chromaticity of the same surface without the EnChroma filter.(b) Deuteranomalous cone sensitivity functions from DeMarco et al. (1992), and the low transmission Explorer CX14 EnChroma transmission spectrum.

Illuminants
The 'default' LED illuminant used for the results of all sections other than Section 3.3 was a S63 5DL 30 W bulb (Maplins, Rotheram, UK), with a correlated color temperature (CCT) of 5100 K and a color rendering index (CRI) of 77.9.Our reason for choosing this LED as the default is that we used it as an illuminant for two out of the three experiments we report in the accompanying empirical paper (Somers et al., in prep.).In Section 3.3 the model was applied to four further illuminants.A halogen illuminant (CCT 2900, CRI 98.9) was a Diall R7S 499 W bulb (Kingfisher, London), blue daylight (CCT11500 K, CRI 93.9) and yellow daylight (CCT 5400 K, CRI 97.7) were taken from the high and low temperature extremes of the Granada Daylights dataset (Hernández-Andrés et al., 2001), and a fluorescent illuminant (CCT 2800 K; CRI 84.0) was a Philips TL-D bulb obtained from an online dataset (Aphalo, 2015), converted from photons to energy.

Filters
The 'default' filter used in the model was the Explorer CX14 filter which we label 'low transmission', intended for outdoor use (Fig. 1b).This filter is used for results presented in all sections other than Section 3.4 where the Mento CX25 'high transmission' and Lumi CX14 'triple notch' filters are additionally investigated.16 Filter transmission spectra were measured using a SpectraScan PR655 spectroradiometer (PhotoResearch, Chatsworth, CA, USA) in reference to a polytetrafluouethylene white plaque (Sphere Optics, Uhldingen, Germany), or a PR650 SpectraScan spectroradiometer (PhotoResearch, Chatsworth, CA, USA) in reference to a barium sulphate white plaque.All 16 filter transmission spectra are shown in Section 3.4 (Fig. 5).The three filters selected for further analysis were all measured using the PR655 SpectraScan.

RGB displays
To compare the effect of EnChroma filters on colors rendered on displays with their effect on the colors of real-world reflective surfaces, we used Dataset (ii), modelled RGB display-rendered versions of the same Munsell surfaces as in Dataset (i).We included 3 display types in the model: a GDM-FW900 (Sony, Tokyo, Japan) cathode ray tube (CRT), a DreamColor z27 (Hewlett Packard, Palo Alto, California USA) "inplane switching" liquid-crystal display (IPS LCD), and a 5th Generation iPad (Apple, Cupertino, California USA) light emitting diode backlit liquid-crystal display (LED LCD).The spectral power distributions of the RGB primaries were measured using the PR655 SpectraScan spectroradiometer, and interpolated to have a resolution of 1 nm in the range 380 nm to 780 nm.The spectra for RGBs metameric with the 259 Munsell surfaces of Value 5 under LED illumination were calculated for each display type for anomalous trichromatic observers with the DeMarco et al. (1992) cone sensitivity functions.

Hyperspectral images
We used 75 hyperspectral images of natural scenes collected from 5 publicly available datasets: Párraga et al., (1998;29 scenes), Ruderman et al., (1998;12 scenes), Nascimento et al., (2002;8 scenes), Foster et al., (2006;5 scenes) and Arad and Ben-Shahar (2016;ICVL database: 21 scenes).For each hyperspectral scene, the spectral radiance was used, or reconstructed using the reflectance and illuminant spectra recorded at the time.The Nascimento et al. (2002) dataset contains spectral reflectances only, so an estimate of radiance was constructed using equal energy white as an illuminant.To reduce chromatic noise, the lowest luminance 0.5% of pixels were removed from each hyperspectral image.Three images were omitted from the Foster et al. (2006) dataset due to high levels of chromatic noise in pixels with low luminance which was not sufficiently addressed by removing the 0.5% of pixels with lowest luminance from each image.All hyperspectral data had a spectral resolution of 10 nm from 400 nm to 700 nm.Data not originally in 10 nm resolution was interpolated.

Observers
The 'default' observers are a 'standard' deuteranomalous observer with S, L and L′ cones, where the S (peak: 440 nm) and L (peak: 566 nm) cone sensitivity functions are from Smith and Pokorny (1975), and the L′ (peak: 560 nm) sensitivity function is from DeMarco et al., (1992; labelled M′ in their tabulation), and a 'standard' protanomalous observer with S, M′ (peak: 553 nm) and M cones, where the S and M (peak: 543 nm) cone sensitivity functions are from Smith and Pokorny (1975), and the M′ sensitivity function is from DeMarco et al., (1992;labelled L′ in their tabulation).
We chose the DeMarco et al. cone fundamentals for consistency with previous work (e.g., Moreland et al., 2022), but in Section 3.5 we also model the effects of the EnChroma filters for anomalous observers with cone fundamentals based on the Stockman and Sharpe (2000) nomogram.In that section we consider individual differences by modelling 5 deuteranomalous observers and 5 protanomalous observers, where the different observers in each group have different 'severities' of anomalous trichromacy based on the spectral difference in the peak sensitivities of the two medium or long wavelength cone types (Δλ max ).Within each group, the 5 observers had Δλ max of 1 nm, 3 nm, 6 nm, 9 nm and 12 nm.The simulated cone sensitivities as a function of log wavelength Log 10 (S(x)) were calculated according to Equation ( 4 log 10 (S(x) ) = a + bx 2 + cx 4 + dx 6 + ex 8 + fx 10 + gx 12 + hx 14  (4) Pre-receptoral filtering was modelled using the macular pigment transmittance function used by Stockman at al. (1999), and a lens transmittance function calculated for a 20-year-old observer using equations provided by Pokorny et al. (1987).Optical densities were estimated as 0.38 0.38, and 0.3 for L, M and S cones, respectively (Stockman & Sharpe, 2000).Scaling factors were applied before the prereceptoral filters to replicate the relative sensitivities of the Smith and Pokorny (1975) cone sensitivity functions.In Section 3.5 we also present results from a normal trichromatic observer with the Smith and Pokorny (1975) cone fundamentals.

Model output
The model calculates the difference caused by EnChroma filters in chromaticity coordinates in an analogue of the MacLeod-Boynton (1979) chromaticity diagram for the particular modelled observer.This is a useful diagram for presenting results, since the impact of the filters on the color subsystem that is impaired in anomalous trichromacy is isolated to the horizontal axis (L/(L + M)).In these variants of the MacLeod-Boynton (1979) chromaticity diagram chromaticities are translated so that the they are centered at (0,0) as the achromatic point.This translation is not intended to be a model of adaptation, but is simply intended to define a convenient metric in which the effects of the EnChroma filter can be expressed.For the Munsell datasets (spectral datasets i and ii) the achromatic point was defined as the lightest neutral surface, Neutral 9.5.For the hyperspectral dataset (spectral dataset iii), the achromatic point for each scene was defined as that scene's mean chromaticity.
We refer to changes in chromaticities in the MacLeod-Boynton (1979) chromaticity diagram away from the achromatic point as changes in 'saturation'.To quantify the effect of EnChroma filters in changing saturation we collapse across increments and decrements (relative to the white point) in the MacLeod-Boynton (1979) chromaticity diagram by extracting the absolute changes in chromaticity along each cardinal axis.For example, for deuteranomals, we define Δ(L/(L + L′)) as |L/(L + L′) w | E -|L/(L + L′) w |, where Δ(L/(L + L′)) is the change in L/(L + L′) saturation conferred by the EnChroma filter, |L/(L + L′) w | E is the L/(L + M) saturation of the target (relative to the white point) with the EnChroma filter and |L/(L + L′) w | is the L/(L + M) saturation of the target (relative to the white point) without the EnChroma filter.Using this metric, any increase in the saturation of the target with the EnChroma filter is positive, while any decrease is negative.Thus, any systematic increase in Δ(L/(L + L′)) represents 'gamut expansion' and suggests an increase in chromatic diversity, while any systematic decrease in Δ(L/(L + L′)) represents 'gamut contraction' and a decrease in chromatic diversity.
To express the color changes conferred by EnChroma filters using a meaningful scale, we express, for example, for deuteranomals, Δ(L/(L + L′)) in relation to their usual chromatic gamut.Specifically, we present the percentage changes in Δ(L/(L + L′)) in relation to the 99.6% of |L/(L + L′) w | values (i.e., values between the 0.2th and 99.8th percentiles) without the EnChroma filter across all 75 hyperspectrally imaged scenes included in spectral dataset iii, according to Equation ( 5).We chose the 99.6% of saturations rather than 100% because the most extremely saturated pixels in hyperspectral images contain a high proportion of outliers which may be attributable to chromatic noise.Thus, the effect sizes we report are scaled by the denominator in Equation ( 5) and would change with any change in the denominator (e.g., to include 75% of saturations in hyperspectrally imaged scenes rather than 99.6%).

Broadband uniform surfaces
Using our model of anomalous trichromatic color vision we predicted the effect of a low transmission CX14 EnChroma filter on the chromatic signals available to anomalous trichromats from spectral dataset (i), uniform broadband surfaces.Fig. 2 shows the changes in predicted color signals conferred by the EnChroma filter on the 259 Munsell surfaces.Fig. 2a shows, for deuteranomals, a predicted systematic shift in chromaticities by the EnChroma filter vertically and slightly to the right in the MacLeod-Boynton (1979) chromaticity diagram, suggesting that the lenses have a purple tint.Fig. 2b shows the predicted effect of the EnChroma filter for protanomals, where there is a systematic shift in chromaticities with the EnChroma filter upwards and to the left, suggesting a bluish tint for protanomals.
Fig. 2c and 2d shows the same results, but with both sets of chromaticities (with and without the EnChroma filter) translated so that the achromatic point is at (0,0).For both deuteranomals and protanomals, the EnChroma filter confers changes in chromaticities away from the centre of the chromaticity diagram.This represents increases in cone opponent signals relative to the white point.The increase in cone opponent signals is evident in all color directions away from the white point, shifting both S/(L + L′) and L/(L + L′) in both incremental and decremental directions.Once exception to this generalized 'expansion' of the Munsell color gamut is for protanomalous M′/(M′+M) decrements, where chromaticities with the EnChroma filter (orange points) are closer to the achromatic point than without the EnChroma filter (black points).
When the impact of the filter on L/(L + L′) or M′/(M′+M) signals is expressed as a proportion of the 'chromatic gamut' of natural scenes, the mean changes in saturation along the horizontal axis in the MacLeod-Boynton (1979) chromaticity diagram across all the surfaces in Dataset (i) are 9.81% for deuteranomals and 3.93% for protanomals.This indicates an overall increase in the saturation of the Munsell surfaces along anomalous trichromats' 'impaired' color dimension.The maximum predicted increases over the 259 Munsell surfaces were 35.38% for deuteranomals and 33.08% for protanomals.
Across the 259 Munsell surfaces, there were significant correlations between unfiltered L/(L + L′) and M′/(M′+M) saturations and the predicted changes in saturation along the same axis conferred by the EnChroma filter (ρ = 0.92, p < 0.0001 for deuteranomals; ρ = 0.43, p < 0.0001 for protanomals).This suggests that the sizes of the saturation enhancements conferred by the EnChroma filter increase with the unfiltered saturations.

Screen-rendered stimuli
We modelled the impact of the EnChroma filter on color signals for stimuli rendered on RGB displays.In RGB images, colors are typically rendered to be roughly metameric (for normal trichromats) with the colors of the real-world surfaces represented in the images, but are spectrally very different from real-world surfaces.Since the spectra of the RGB primaries underlying display RGB values are relatively narrowband compared to the typically broadband spectra reflected from real-world objects, the effects of notch filters on light from RGB displays is likely to be different from the effects on light from the real-world surfaces that the colors in the rendered scenes are metameric with (Nascimento & Foster, 2022).If the notches of the EnChroma filters coincide with non-emissive regions of the display spectrum, we might expect the filters to have minimal impact on displayed colors.
The predicted effect of EnChroma filters on deuteranomalous cone opponent signals for screen-rendered colors are shown in Fig. 3, for a subset of 40 of the 259 Munsell surfaces with Chroma 8. Results are shown for three different RGB displays, for which the spectra for white (maximum RGB) are shown in Fig. 3a1-3a3.For deuteranomals, across all 259 modelled stimuli the largest predicted saturation enhancements by the EnChroma filter for colors rendered on the CRT monitor were about half the size of those predicted for broadband surfaces (5.37% against 9.81%, Fig. 3b1, 3c and 3d).Saturation enhancements were predicted to be smaller still for the IPS display (Fig. 3b2, 3c and 3d), and somewhat larger for the LED backlit display (Fig. 3b3, 3c and 3d), but still smaller than for the broadband surfaces.The predicted changes in saturation for deuteranomals conferred by the EnChroma filter for stimuli displayed on the IPS LCD monitor were almost entirely confined to the L/(L'+L) axis (Fig. 3b2 and 3c).
The predicted changes in saturation along the impaired color axes were compared for stimulus types (the three displays and physical surfaces) in repeated measures ANOVAs using all 259 Munsell surfaces of Value 5, separately for deuteranomals and for protanomals.There were significant main effects of stimulus type for both deuteranomals (F (1.24,320.44)= 250.17,p < 0.001) and protanomals (F(1.29,323.36)= 33.30,p < 0.001).Post-hoc Tukey tests (Table 1) revealed that for deuteranomals the EnChroma filter is predicted to provide significantly smaller enhancements in L/(L + L′) saturation for colors rendered on any of the RGB displays than for the physical surfaces.For deuteranomals the predicted effect of the EnChroma filter also differed significantly between the three displays: the IPS display was predicted to mediate the smallest effects (mean 2.25%), followed by the CRT (mean 5.37%), and the LED display (mean 7.39%) was predicted to mediate the largest effects.For protanomals surfaces (mean 3.93%) were predicted to mediate significantly larger enhancements in M′/(M′+M) saturation than the IPS display (mean 1.44%) but not than the other two displays.Both the CRT (mean 3.37%) and the LED display (mean 3.34%) were also predicted to mediate significantly larger enhancements than the IPS display.

Illumination
We modelled the impact of the EnChroma filter on color signals in response to Munsell surfaces under 5 different illuminants (see Fig. 4b1 and b2 for normalized radiance spectra).The predicted effects of EnChroma filters on deuteranomalous cone opponent signals for a subset of 40 of the 259 Munsell surfaces with Chroma 8 under the different illuminants are shown in Fig. 4. For all 5 illuminants there are predicted increases in average L/(L + L′) saturation.To test the impact of illuminant on the predicted effectiveness of the EnChroma filter, we conducted one-way repeated measures ANOVAs using all 259 Munsell surfaces of Value 5, separately for deuteranomals and protanomals.The ANOVAs revealed a significant effect of illuminant for both deuteranomals (F (1.88,483.91)= 202.80,p < 0.001) and protanomals (F(2.37,610.20)= 248.63,p < 0.001).Table 2 shows the results of post-hoc tests: For protanomals the halogen illuminant was predicted to mediate the greatest enhancements in M′/(M′+M) saturation by the filter (mean 9.16%), significantly greater than those mediated by LED (mean 3.93%), yellow daylight (mean 4.26%), blue daylight (mean 1.58%) or fluorescent illuminants (mean 4.07%).For deuteranomals, robust enhancements in L/(L + L′) saturation by the filter were predicted for all illuminants, though the fluorescent was predicted to mediate significantly weaker effects (mean 4.54%) than LED (mean 9.81%), halogen (mean 12.77%), yellow daylight (mean 11.59%) or blue daylight (mean 10.12%).

Filter
Fig. 5a2 shows transmission spectra for the 16 filters we measured.These can be sorted into three categories.'Low transmission' filters (plotted in blue) have two spectral notches at 470-500 nm and at 570-600 nm.'High transmission' filters (plotted in orange) cause lower overall light attenuation, and also have two spectral notches.Within the 'low transmission' group there may be two families.One has a similar spectral profile to that of the 'high transmission' group but with higher overall attenuation.The other has a short wavelength notch that is spectrally narrower and centered at longer wavelengths than for the 'low transmission' group, and transmits relatively more light at long wavelengths.We measured one example of a spectrally distinct "triple notch" filter (plotted in chartreuse) which has a third spectral notch at 630-660 nm.One filter of each type was selected for further analysis (Fig. 5a1).
The impact of filter type on the predicted changes in deuteranomalous L/(L + L′) for 40 Munsell surfaces of Value 5 and Chroma 8 is shown in Fig. 5b1-5c.Despite substantial differences in the spectral transmission profiles of the three EnChroma filters (Fig. 5a1), their predicted impacts on L/(L + L′) saturations are similar.The most notable difference between the predicted effects of the 3 filter types is in the direction of the systematic shift of the chromaticities.The high transmission filter is predicted to confer a shift mainly in the direction of L/(L + L′) increments (Fig. 5b2), while the 'low transmission' (Fig. 5b1) and 'triple notch' (Fig. 5b3) filters are predicted to cause shifts in the direction of S/ (L + L′) increments.Once plotted relative to a white-point (Fig. 5c), the rough equivalence between filter types in predicted L/(L + L′) saturation enhancements is evident.
The distributions of predicted changes in L/(L + L′) saturation for the full set of 259 Munsell surfaces of Value 5 are shown in Fig. 5d.To test the effect of filter type on its predicted effectiveness, the three EnChroma filters were compared using one-way repeated measures ANOVAs separately for deuteranomals and protanomals.There were significant effects of filter for both deuteranomals (F(1.60,412.46)= 104.23,p < 0.001) and protanomals (F(1.42,366.31)= 419.34,p = < 0.001).Posthoc Tukey tests revealed that for both groups all the filters were predicted to mediate significantly different effects from one another (Table 3).For deuteranomals, the low transmission filter was predicted to mediate significantly larger enhancements in L/(L + L′) saturation (mean 9.81%) than the triple notch filter (mean 7.95%) or the high transmission filter (mean 9.00%).For protanomals, the high transmission filter was predicted to mediate significantly larger enhancements in M′/(M′+M) saturation (mean 6.32%) than the low transmission filter (mean 3.98%) and the triple notch filter (mean 3.49%).

Observer
We predicted the effects of EnChroma filters for deuteranomals and protanomals with different cone sensitivity functions, as well as for normal trichromats.Fig. 6 shows results for deuteranomals and protanomals with 5 'severities', modelled by varying the spectral separation (Δλ max ) between the peak sensitivities of the L and L′ cone sensitivity functions and the M′ and M cone sensitivity functions between 1 nm and 12 nm (Fig. 6a1 and 6a2).
The raw sizes of changes in L/(L + L′) and M′/(M′+M) saturation by the EnChroma filter decrease with severity, but the unfiltered saturations of course decrease correspondingly (Fig. 6b1 and 6b2).For chromaticities expressed as a percentage of each observer's usual color gamut, the predicted sizes of saturation enhancements along the L/(L + L′) and M′/(M′+M) axes are relatively uniform across severities of anomalous trichromacy, yet Fig. 6c shows that there are small systematic differences.To test the impact of variation in Δλ max on the predicted changes in saturation conferred by the EnChroma filter, a two-way   repeated measures ANOVA were conducted with 5 levels for severity of anomalous trichromacy, and 2 levels for group (deuteranomals and protanomals).There was a significant main effect of group (F(1,258) = 808.18,p < 0.001), a significant main effect of severity (F(1.44,371.56)= 101.80,p < 0.001), and a significant interaction between group and severity (F(1.43,368.91) = 522.61,p < 0.001).Post-hoc tests showed that deuteranomals are predicted to gain significantly greater enhancements in L/(L + L′) saturation compared to protanomals in M′/ (M′+M) saturation (t = 28.43,p < 0.001).For deuteranomals, predicted enhancements in L/(L + L′) saturation increase significantly with severity (mean predicted enhancements 7.9% for a Δλ max of 12 nm and 9.0% with a Δλ max of 1 nm), while for protanomals predicted enhancements in M′/(M′+M) saturation decrease with severity (mean predicted enhancement 4.8% for a Δλ max of 12 nm and 2.4% with a Δλ max of 1 nm).EnChroma filters are not predicted to confer any enhancement of chromatic signals for dichromatic observers along the affected color axis, as the notch filter works by increasing the functional difference between the two X-linked cone spectral sensitivities.In dichromacy there is only one X-linked cone class, so there are no L/(L + M) signals to be enhanced.The filters are therefore expected to have no ability to ameliorate the color vision deficiency but are predicted to have some  effect on the S/L or S/M dimensions.We compared predicted enhancements in L/(L + M) saturation for normal trichromats with predicted enhancements in L/(L + L′) saturation for deuteranomals and predicted enhancements in M′/(M′+M) saturation for protanomals, for observers defined using the Smith and Pokorny (1975) and DeMarco et al. (1992) cone fundamentals.There was a significant effect of group (F(1.16,298.41) = 690.51,p < 0.001).Predicted saturation enhancements were largest for deuteranomals (mean 9.81%), intermediate for normal trichromats (mean 6.56%) and smallest for protanomals (mean 3.93%).

Natural scenes
To predict effects of the filters on perception of real-world scenes, changes in chromaticity were calculated for hyperspectral images of natural scenes, which contain spectral power distributions for each pixel.The predicted changes in L/(L + L′) and M′/(M′+M) conferred by the EnChroma filter for the 75 hyperspectral scenes are represented in Fig. 7a1 and 7a2.Predicted changes in saturation are mainly positivemedian changes are greater than zero for all 75 hyperspectrally imaged scenes for deuteranomals and for most for protanomals.As for the Munsell surfaces, the EnChroma filter was predicted to confer a reduction in saturation for some pixels, but an increase for the majority: 89.3% and 60.4% of pixels were predicted to receive an enhancement in L/(L + L′) and M′/(M′+M), respectively.Fig. 7c1-7d4 represents predicted changes in L/(L + L′) saturation by the EnChroma filter for 4 example images.The greatest predicted enhancements in L/(L + L′) saturation were for the scene containing flora (Fig. 7c1 and 7d1), where there were moderate predicted enhancements for vegetation of about 10%, and large predicted enhancements for red flowers of 20-30%.For man-made objects there were large predicted enhancements in L/(L + L′) saturation for saturated reddish and greenish surface colors (Fig. 7c2-7c4).The predicted reductions in L/(L + L′) saturation were mostly restricted to areas of low saturation, such as the white sections of the curb stone in the scene shown in Fig. 7c4.
The predicted impact of EnChroma filters on luminance is shown in Fig. 7e1-7f4.There is a relatively uniform reduction in luminance to a mean of 27.1% of the original value for deuteranomals and 26.3% of the original value for protanomals.Red surfaces tend to be have a somewhat lower luminance reduction to about 33% of their original value for detueranomals (e.g., Fig. 7e1) and to about 30% of their original value for protanomals.

Discussion
Our results predict that EnChroma filters can increase the saturation of colors for anomalous trichromats.Once chromaticities with and without the filters are expressed relative to their respective white points, the model predicted that for the most common form of color vision deficiency, deuteranomaly, EnChroma filters shift chromaticities of broadband surfaces away from the white point, along all color axes, in both incremental and decremental directions (e.g., Fig. 2).For deuteranomals, the mean increase in saturation along the impaired L/(L + L′) color axis is predicted to be 9.85% for Munsell surfaces, expressed using our metric of a percentage of 99.6% of the usual deuteranomalous L/(L + L′) color gamut.However, though the model predicts EnChroma filters to be effective to some degree under most viewing conditions, it predicts that the degree of effectiveness depends on the interaction between the filter transmission spectrum, the particular light spectra, and the spectral sensitivity functions of the particular observer's long and medium wavelength sensitive cones.
Our model predicted that EnChroma filters are less effective for colors rendered on displays than for broadband surfaces with the same chromaticities (Fig. 3).This is because the notch in the EnChroma transmission spectrum at medium to long wavelengths typically coincides with a gap in the emission spectra of display primaries (Fig. 3a-3c), meaning that the EnChroma filter has a limited effect on spectra emitted from displays.Similarly, the effectiveness of EnChroma filters was predicted to depend on illumination (Fig. 4).Halogen illuminants, which are broadband, were predicted to mediate strong enhancements in L/(L + L′) saturation for deuteranomals and in M′/(M′+M) saturation for protanomals (Fig. 4b2).However, fluorescent illuminants were predicted to mediate weaker enhancements by the EnChroma filter for both groups of observers because the medium to long wavelength spectral notch in the EnChroma transmission spectrum aligns with a gap in the spectrum of the fluorescent illuminant (Fig. 4b2).
Different classes of EnChroma filter have distinctive transmission spectra (Fig. 5a1-a2).It is therefore surprising that our model predicted that they should mediate similar effects on anomalous color vision.However, for both protanomals and deuternomals the triple notch filter was predicted to mediate somewhat smaller effects than the other two classes of filter we tested.For protanomals the high transmission filter was predicted to be most effective, and for deuteranomals the low transmission filter.In agreement with earlier work (Nascimento & Foster, 2022;Pattie et al., 2022), the filters were predicted to be substantially more effective for deuteranomals than protanomals, with intermediate results predicted for normal trichromats.For deuteranomals the filters are expected to enhance the saturations both of L/(L + L′) increments and of L/(L + L′) decrements (e.g., Fig. 6b1).However, for protanomals the filters are expected to enhance saturations of M′/ (M′+M) increments, but may reduce saturations of M′/(M′+M) decrements (e.g., Fig. 6b2).Surprisingly, effectiveness was predicted to decrease with Δλ max for deuternomals but increase with Δλ max for protanomals.
Our model predicts that EnChroma filters should also be broadly effective for enhancing anomalous trichromatic color vision for natural scenes.For deuteranomals, 89.3% of pixels in hyperspectral images were predicted to receive an enhancement in L/(L + L′) saturation, but the filters are predicted to be less effective for protanomals for natural scenesonly 60.4% of pixels in hyperspectral images were predicted to receive an enhancement in M′/(M′+M) saturation.
We chose a particular metric in which to express predicted effect sizes for enhancements in saturation by EnChroma filters, which was as a percentage of 99.6% of the observer's usual gamut.This decision is quite arbitrary but was motivated by the desire to have a somewhat meaningful metric by comparing the changes caused by the filter with the typical range of colors observers encounter in natural scenes (captured in the hyperspectral image sets).We excluded the 0.4% most saturated pixels from the hyperspectral images in this calculation as they may contain chromatic noise as well as highly saturated surface colors.However, had we chosen to compare saturation changes by the EnChroma filter with a smaller percentage of the observer's usual color gamut (e.g., 75%), the effect sizes we reported would have been correspondingly larger.Another reasonable metric would have been to express the change in saturation of each surface by the EnChroma filter as a percentage of the unfiltered saturation of the particular surface.Under this metric, for Munsell surfaces under LED illumination EnChroma filters cause a more uniform enhancement of about 45% in deuteranomalous L/(L + L′) saturation.For protanomals, because the filters tend to enhance the saturation of M′/(M′+M) increments but decrease the saturation of M′/(M′+M) decrements, the effect sizes expressed as a percentage of the unfiltered saturation for each surface are bimodally distributed around 0.
Of course, any gains in L/(L + L′) or M′/(M′+M) saturations are accompanied by losses in L + L′ or M′+M (luminance).The low transmission EnChroma filter we have used as default reduces luminance to about 27% of its original value for deuteranomals and 26% of its original value for protanomals.Could this reduction in luminance impact observers' abilities to discriminate colors or perceive suprathreshold saturation contrasts, and offset gains provided by the EnChroma filter in chromatic contrast?A classic study by Brown (1951) found that sensitivity to color differences remains constant until luminance falls below about 3.5 cdm − 2 , and results of later studies have been in broad agreement with this finding (Jennings & Barbur, 2010;Yebra et al., 2001).However, older observers can experience a greater loss in chromatic contrast sensitivity at low light levels than younger observers (Barbur & Konstantakopoulou, 2012;Knoblauch et al., 1987), and thus may not experience the enhancements in color contrast predicted by the model at lower light levels.Aside from its potential effects on chromatic contrast sensitivity, light level is known to impact color appearance.Hunt (1952Hunt ( , 1953) ) reported that perceived saturation increases substantially with light level between 8 and about 1000 cdm − 2 , while Valberg (1975) found modest increases in perceived saturation with luminance for luminances greater than 25 cdm − 2 , and much larger increases with increasing luminance below about 13 cdm − 2 , still in the photopic range.Thus, while saturation discrimination might not change much with a luminance attenuation of the size provided by the low transmission EnChroma filter (so long as light levels remain in the photopic range), perceived saturations may reduce with reduced light level, potentially offsetting gains in perceived saturation by the filters.Considering the impacts of low light on color appearance and on color discrimination for older observers, the high transmission filter (with lower overall light attenuation) may lead to better overall results for users, despite performing worse in our model than the low transmission filter (Fig. 5).
Our model simulates only retinal color signals, and does not accurately model color appearance.Changes in perceived saturation may be different from changes in signals carried by the modelled cone-opponent mechanisms, as contrast adaptation, perceptual learning as well as any nonlinearities in cortical color representations may have an impact.In particular, anomalous trichromats are thought to show 'postreceptoral compensation' of their impaired retinal L/(L + M) color signals (Regan & Mollon, 1997;Boehm et al., 2014), where representations of anomalous trichromats' equivalents of L/(L + M) color differences are thought to be expanded in the cortex so that the range of contrasts may appear roughly the same for anomalous trichromats as for normal trichromats (Boehm et al., 2014;Tregillus et al., 2021).Although postreceptoral compensation may be incomplete (Emery et al., 2023;Robinson et al., 2023), and there may be individual differences in its extent (Boehm et al., 2021), it may limit the extent to which EnChroma glasses can have an effect on color appearance.The model requires validation from behavioral experiments to determine whether the predicted changes in saturation as represented in the cone-opponent mechanisms lead to equivalent changes in perceived saturation.
In conclusion, our model, in agreement with several existing models (Moreland et al., 2022;Nascimento & Foster, 2022;Pattie et al., 2022), predicts that EnChroma filters should be broadly effective for enhancing anomalous trichromatic color vision.Also in agreement with the predictions of existing models (Nascimento & Foster, 2022;Pattie et al., 2022), we predicted that the filters should be substantially more effective for deuteranomals than protanomals.When the spectral notches in the EnChroma filter transmission spectra which underlie their effect on color signals align with gaps in incident light spectra, the filters will have little impact.The predicted effectiveness of EnChroma filters is therefore greater for broadband reflective surfaces under broadband illuminants than for narrowband illuminants or colors rendered on RGB displays, where spectral gaps in light spectra align with the EnChroma filter spectral notches.The EnChroma filters are predicted to be effective for natural scenes, where reflectance spectra and illumination spectra are broadband.Since our model is based on retinal color signals and does not account for contrast adaptation or perceptual learning, the predicted effects of the filters need to be tested on the color vision of anomalous trichromatic observers.Empirical results presented in a sister paper show that the impact of EnChroma filters on deuteranomalous color vision matches model predictions for a color matching task, but that their impact on color appearance maybe smaller than predicted (Somers et al., in prep.).

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 1 .
Fig. 1.(a) Schematic of the steps of the model.The illuminant spectrum (1) is multiplied by the surface reflectance spectrum (2), and by the EnChroma transmission spectrum (3) to model the light reaching the eye.This product is then multiplied by the three cone sensitivity functions (4) to model the cone responses at each wavelength (5).The result is summed over all wavelengths (6) to model the cone activities elicited by the surface under the illuminant viewed through the EnChroma filter (7).Finally, the chromaticity of the surface is plotted in an observer-specific version of the MacLeod-Boynton (1979) chromaticity diagram (8), and compared to the chromaticity of the same surface without the EnChroma filter.(b) Deuteranomalous cone sensitivity functions fromDeMarco et al. (1992), and the low transmission Explorer CX14 EnChroma transmission spectrum.

Fig. 3 .
Fig. 3. Predicted effects of an EnChroma filter on screen-rendered colors for deuteranomals.(a1) Spectrum of white (max R, G and B) for the CRT monitor.Also shown is the transmission spectrum for the Explorer CX14 EnChroma filter, and the DeMarco et al. (1992) deuteranomalous cone fundamentals.(a2) As for (a1) but for the IPS backlit LCD display.(a3).As for (a1) but for the LED backlit LCD display.(b1) Chromaticities of 40 Munsell surfaces of Value 5 and Chroma 8 (black points), rendered on the CRT display to be metameric for anomalous trichromats with the physical surfaces.The chromaticities of the same colors rendered on the CRT monitor and viewed through the EnChroma filter are indicated by the grey circles, and the chromaticities of the physical surfaces viewed through the EnChroma filter are also shown for comparison (blue circles).(b2) As for (b1) but for stimuli rendered on the IPS backlit LCD display.(b3) As for (b1) but for stimuli rendered on the LED backlit LCD display.(c) Changes in chromaticities by the EnChroma filter for all display types and for physical surfaces plotted relative to the achromatic point.(d) Violin plots showing the mean predicted changes in L/(L + L′) saturation (black bars), the median predicted changes (red bars), the standard deviations (dotted vertical lines) and the distributions of predicted changes in L/(L + L′) for the full set of 259 Munsell colors of Value 5 rendered on the three displays and for physical surfaces.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) p-values for post-hoc tests comparing predicted enhancements of L/(L + L′) and M′/(M′+M) saturation by an EnChroma filter for Munsell surfaces under different illuminants, for deuteranomals (DA) and protanomals (PA).* indicates p < 0.05, ** indicates p < 0.01.

Fig. 4 .
Fig. 4. Predicted effects of an EnChroma filter for deuteranomals on Munsell surfaces under 5 different illuminants.(a1)-(a5) chromaticities of 40 Munsell surfaces of Value 5 and Chroma 8 without the EnChroma filter (black points), and chromaticities of the same surfaces with the EnChroma filter (colored points).The illuminant is indicated on each panel.(b1) and (b2) normalised radiance spectra for each illuminant (colored to match panels a1-a5), and the transmission spectrum of the EnChroma filter (dotted grey line).(c) Changes in chromaticity by the EnChroma filter for the Munsell surfaces under the 5 illuminants plotted relative to the achromatic point.(h) Violin plots showing the mean predicted changes in L/(L + L′) saturation for the full set of 259 Munsell surfaces of Value 5 (black bars), the median predicted changes (red bars), the standard deviations (dotted vertical lines) and the distributions of predicted changes in L/(L + L′) for the 5 illuminants.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 5 .
Fig. 5. Predicted effect of filter type on deuteranomalous L/(L + L′) saturation.(a1) Transmission spectra for a representative 3 EnChroma filters selected for further analysis.(a2) Transmission spectra for 16 EnChroma filters sorted into 3 types.(b1)-(b3).Predicted changes in chromaticity by the EnChroma filters for 40 Munsell surfaces of Value 5 and Chroma 8. Black points show chromaticities without the EnChroma filter and colored points with the EnChroma filter.Results for different filters are plotted in the 3 panels, as labeled.(c) Predicted changes in chromaticity by the three EnChroma filters for the Munsell surfaces plotted relative to the achromatic point.Black points show chromaticities without the EnChroma filter and colored points with the EnChroma filter.(d) Violin plots showing the predicted changes in L/(L + L′) saturation for the full set of 259 Munsell surfaces of Value 5 by the 3 filters, with mean predicted changes (black bars), median predicted changes (red bars), standard deviations (dotted vertical lines), and distributions of predicted changes.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 6 .
Fig. 6.Effects of observer type on predicted enhancements in L/(L + L′) and M′/(M′+M) saturation by the EnChroma filter.(a1 and a2) Cone fundamentals used in the model.(a1) Nomogram-based L′ cone sensitivity functions for deuteranomals (orange solid lines), the DeMarco et al. (1992) L′ cone sensitivity function (black dotted line), L (red dashed line) and S (blue dashed line) cone sensitivity functions.The missing M cone sensitivity function is shown for comparison (green dot dashed line).(a2) Nomogram-based M′ cone sensitivity functions for protanomals (green solid lines), the DeMarco et al. (1992) M′ cone sensitivity function (black dotted line), M (green dot dashed line) and S (blue dashed line) cone sensitivity functions.The missing L cone sensitivity function is shown for comparison (red dashed line).(b1 and b2) Predicted changes in chromaticities by the EnChroma filter for Munsell surfaces of Value 5 and Chroma 8 plotted relative to the achromatic point.Black points show chromaticities without the EnChroma filter and colored points with the EnChroma filter: (b1) for deuteranomals; (b2) for protanomals.(d) Violin plots showing the predicted changes in observer-specific equivalent of L/(L + M) saturation for the full set of 259 Munsell surfaces of Value 5.The plots show mean predicted changes (black bars), median predicted changes (red bars), standard deviations (dotted vertical lines), and distributions of predicted changes in L/(L + M) for all modelled observers, also including normal trichromats.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 7 .
Fig. 7. Predicted effects of the EnChroma filter on colors in hyperspectrally imaged natural scenes.(a1)-(a2) summary results for the predicted effects of the EnChroma filter on individual hyperspectrally imaged scenes for deuteranomalous |L/(L + L′)| (a1) and protanomalous |M′/(M′+M)| (a2).In each case, predicted changes in chromaticity are expressed as a percentage of the 'usual gamut' (the range of values of 99.6% pixels in hyperspectrally imaged natural scenes for the axis in question).Median changes are indicated by the white points, and the percentiles indicated by the legend.Bars represent the ranges of changes between the 0.2nd and 99.8th percentiles, colored according to the hyperspectral dataset (see legend).(b1)-(b4) four example hyperspectrally imaged scenes from the ICVL dataset.(c1)-(c4) heat maps showing the predicted changes in |L/(L + L′)| as a percentage of usual gamut for each pixel in the four example hyperspectral images.(d1-d4) violin plots showing the distributions of predicted changes in |L/(L + L′)| plotted in the heat maps in (d1)-(d4).(e1)-(e4) heat maps showing changes in luminance for deuteranomals (L + L′), where the luminance of the filtered image is expressed as a percentage of the luminance of the unfiltered image.(f1)-(f4) violin plots showing the distributions of predicted changes (L + L′) plotted in the head maps in (e1)-(e4).

Table 1 P
-values for post-hoc Tukey tests comparing predicted enhancements of saturation by the EnChroma filter for different stimulus types, for deuteranomals (DA) and protanomals (PA).* indicates p < 0.05, ** indicates p < 0.01.