An experimental method for the assessment of color simulation tools

Abstract

Abstract

Abstract:

Abstract  The Simulcheck method for evaluating the accuracy of color simulation tools in relation to dichromats is described and used to test three color simulation tools: Variantor, Coblis, and Vischeck. A total of 10 dichromats (five protanopes, five deuteranopes) and 10 normal trichromats participated in the current study. Simulcheck includes two psychophysical tasks: the Pseudoachromatic Stimuli Identification task and the Minimum Achromatic Contrast task. The Pseudoachromatic Stimuli Identification task allows determination of the two chromatic angles (h_uv values) that generate a minimum response in the yellow–blue opponent mechanism and, consequently, pseudoachromatic stimuli (greens or reds). The Minimum Achromatic Contrast task requires the selection of the gray background that produces minimum contrast (near zero change in the achromatic mechanism) for each pseudoachromatic stimulus selected in the previous task (L_R values). Results showed important differences in the colorimetric transformations performed by the three evaluated simulation tools and their accuracy levels. Vischeck simulation accurately implemented the algorithm of Brettel, Viénot, and Mollon (1997). Only Vischeck appeared accurate (similarity in h_uv and L_R values between real and simulated dichromats) and, consequently, could render reliable color selections. It is concluded that Simulcheck is a consistent method because it provided an equivalent pattern of results for h_uv and L_R values irrespective of the stimulus set used to evaluate a simulation tool. Simulcheck was also considered valid because real dichromats provided expected h_uv and L_R values when performing the two psychophysical tasks included in this method.

Introduction

Most human beings are trichromats. They have three different retinal cones and need a minimum of three primaries to match any color. Some people (slightly above 2% of males; Sharpe, Stockman, Jägle, & Nathans, 1999) are, as are most mammals (Hemmi, James, & Taylor, 2002; Vorobyev, 2004), dichromats. They have only two cone types and require a minimum of two primary colors for matching any color. The clinical denominations of dichromatism (protanopia, deuteranopia, and tritanopia) specify the absent cone type (proto, deutera, and trita mean first, second, and third, respectively, in Greek). In the standard scientific nomenclature (Carroll, Neitz, & Neitz, 2002; Hunt & Pointer, 2011, chapter 1; Lewis & Zhaoping, 2006) such cones are frequently identified as “L,” “M,” and “S” because they are relatively more sensitive to long, medium, and short wavelengths, respectively. 

Chromatic stimuli frequently improve visual memory performance for both normals and dichromats (Gegenfurtner, Wichmann, & Sharpe, 1998; Spence, Wong, Rusan, & Rastegar, 2006; Velisavljević & Elder, 2008). Color variations are frequently used to facilitate some visual tasks (Breslow, Trafton, & Ratwani, 2009; Yamani & McCarley, 2010). They include the localization (Remington, Johnston, Ruthruff, Gold, & Romera, 2000) and identification (Spence, Kutlesa, & Rose, 1999) of some kind of information. Hue changes (color qualitative changes) are especially accurate for supporting identification tasks on complex visualizations (categorization, absolute numeric value judgments, etc.), whereas ordered lightness or saturation scales are superior for relative comparison tasks (differentiation between greater or lesser amounts of something; Breslow et al., 2009). All the previously mentioned color uses (localization, identification, categorization, etc.) require seeing different colors in response to the stimuli used for representing different things. 

Dichromats have more metamers than do normal trichromats (Lillo & Moreira, 2004); that is, they see the same color when responding to stimuli only differentiable thanks to the activity in the absent cone (L for protanopia, M for deuteranopia). According to clinical nomenclature (Birch, 2001; Fletcher & Voke, 1985), pseudoisochromatic stimuli are perceived as the same color by some people (i.e., protanopes or deuteranopes) but as different colors by normal people. Such stimuli are frequently used in classical color clinical diagnosis tests (Birch, 2001) and in the tests designed to be applied using new gadgets, such as different screen types (Pardo, Pérez, & Suero, 2001, 2004) and handheld game console displays (Suero, Pardo, & Pérez, 2010). Of course, new gadgets also can be used for performing everyday tasks requiring color differentiation or categorization, and the presence of pseudoisochromatic stimuli is a possible source of problems (Lillo & Moreira, 2013, chapter 5). For example, if red backgrounds are used to indicate some kind of emergency, their localization would be more difficult for people confusing such reds with other colors used. 

According to the universal design framework (Vanderheiden, 2006), things and environments must be designed to allow easy use by normal and disabled people. In relation to color (Lillo & Moreira, 2013, chapter 5) the main goal of universal design is to promote similar, accurate uses by people with and without normal color perception. For example, when designing a political map its colors must be differentiable both for normals and dichromats. If we use an accurate color simulation tool, it will be easy to reach this goal. 

Color simulation tools transform colors (original colors) to make them similar to the ones that people who are color blind presumably see (transformed colors). Most of the times, color transformation is done using some kind of software (Capilla, Díez-Ajenjo, Luque, & Malo, 2004; Luque, Fez, & Acevedo, 2014; Viénot, Brettel, & Mollon, 1999). Alternatively, optic filtering can be used (http://www.variantor.co.uk/page.php?sub=2&id=2). Colors differing in the original image but similar in the transformed one are considered pseudoisochromatic. Some simulation tools incorporate a “daltonization function” (Nakauchi & Onouchi, 2008) to avoid pseudoisochromatic colors selectively changing some colors in the original image. Along the same lines, the CIE (International Commission on Illumination) technical committee TC1-89 (http://www.cie.co.at/index.php/Technical+Committees) has been created to “study, evaluate, and recommend image enhancing techniques for color defective observers.” 

Color simulation tools assume (implicit or explicitly) a “reduction view” theoretical framework (Broackes, 2010b; Byrne & Hilbert, 2010; Logvinenko, 2014) based on the following two premises: (1) Dichromats see only a subset of the colors experienced by normal trichromats and (2) there are pseudometameric (pseudoisochromatic) stimuli groups—that is, groups of stimuli that produce different colors (C₁₁, C₁₂, …, C_1n) for people with normal color vision but only one color (C′₁) for dichromats (i.e., Vischeck is based on the algorithm of Brettel, Viénot, & Mollon, 1997).¹ From the two previous premises, differences between color simulation tools can appear about (1) the stimuli included in each pseudoisochromatic group and (2) the color perceived when seeing any stimulus included in a specific pseudoisochromatic group. As will be shown, this last aspect is not relevant when making universal design. 

When red–green dichromats (protanopes or deuteranopes) do a Rayleigh match (Birch, 2001, p. 53), the colors they perceive when seeing any mix of the green (545 nm) and the red (670 nm) lights can be matched by the yellow (589 nm, orangish-yellow) light adjusted to a specific intensity. This means that, for these dichromats, some pseudoisochromatic stimuli include lights that belong to very different color categories (red, orange, green, or yellow) for normal people. Consequently, such lights must not be used to symbolize different things (for example, different power levels in a mobile phone) regardless of the specific color seen by the dichromats. In conclusion, the key aspect when evaluating a color simulation tool is to know if it creates pseudoisochromatic stimuli groups equal to the ones existing in the simulated dichromat type. The notorious differences existing between the colors provided by different simulation tools (i.e., in lightness level) for the same type of dichromat make obvious that some tools do not work accurately. 

Our main research goal was the design and evaluation of a method, Simulcheck, that aims to be useful for evaluating color simulation tools. Data collected in this research allow comparisons between real dichromats, simulation tools (Vischeck, Variantor, and Coblis), and a dichromat's color appearance algorithm (Brettel et al., 1997; see ). On one hand, as is usually the case, there was no information about the algorithm that supposedly was used by two simulation tools (Variantor and Coblis). Therefore, we took colorimetric measurements on the transformation performed by each tool. On the other hand, such information was available for Vischeck (based on Brettel et al., 1997; see ), making it possible to test the algorithm's implementation accuracy. 

The real dichromats' behavior in the two psychophysical tasks performed (see next paragraph) are analyzed. Later, the real dichromats' data are compared with the behaviors predicted by the algorithm and with those of simulated dichromats in order to check the algorithm's validity and the tool's accuracy in simulating color vision. Finally, we compare the behaviors predicted by the algorithm with those provided by simulated dichromats. 

Simulcheck includes two psychophysical tasks named Pseudoachromatic Stimuli Identification and Minimum Achromatic Contrast. Each task is related to a well-known fact of dichromatic color vision and with the performance of an opponent mechanism (the yellow–blue mechanism or the achromatic mechanism). 

Pseudoachromatic Stimuli Identification task. Color derives from the activity of red–green, yellow–blue, and achromatic mechanisms. It is commonly assumed that protanopes and deuteranopes have no functionality in the red–green mechanism and, consequently, see no differences between the grays and the stimuli producing a neutral response in their yellow–blue mechanism. Several works (Birch, 2001; Fletcher & Voke, 1985; Smith & Pokorny, 2003) have shown that each dichromat type has two dominant wavelengths (λ_d) and, consequently, two chromatic angles (h_uv) that identify pseudoachromatic angles. One (Smith & Pokorny, 2003) is located near λ_d = 494 nm (h_uv = 183.95°) for protanopes and λ_d = 499 nm (h_uv = 167.39°) for deuteranopes.² We name stimuli with such λ_d-h_uv values “pseudoachromatic greens” because they are categorized as greens (more specifically, bluish greens) by normal people. The other dominant wavelength associated with pseudoachromatic color (Smith & Pokorny, 2003) is located near λ_d = −494 nm (h_uv = 3.95°) for protanopes and λ_d = −499 nm (h_uv = 347.39°) for deuteranopes.³ We name stimuli with such λ_d-h_uv values “pseudoachromatic reds” because they are categorized as reds (or reddish purples) by normal people. 

The chromatic angles (h_uv) corresponding to the pseudoachromatic greens and reds are the first type of variable provided by Simulcheck to evaluate simulation tools' accuracy. They allow specifying the stimuli that produce a minimum response in the yellow–blue mechanism of real and simulated dichromats. 

Minimum Achromatic Contrast. Only the achromatic mechanism responds to the highest visible spatial frequencies (De Valois & De Valois, 1988, chapter 7; Kaiser & Boynton, 1996, chapter 9). It makes it possible to see small details and provides maximum visual acuity. The achromatic mechanism performance depends essentially on the response of L and M cones (Hunt & Pointer, 2011, chapter 1). For people with trichromatic color vision, its spectral sensitivity is very similar to the spectral luminous efficiency function for photopic vision (Hunt & Pointer, 2011, pp. 21–22), V(λ), and, consequently, to the weighting performed by the standard photometric measurement apparatuses (Kaiser & Boynton, 1996, appendixes 1 and 2). The achromatic mechanism spectral sensitivity is anomalous for people lacking L (protanopes) or M (deuteranopes) cones (Birch, 2001, figure 4.2). Such anomaly is bigger for the protanopes because L cones are more frequent and contribute more to the achromatic mechanism response (Carroll et al., 2002; Hunt & Pointer, 2011, chapter 1). 

Previously, we (Lillo, Collado, Martín, & García, 1999; Lillo & Moreira, 2005) developed the AMLA (AM = achromatic measurement, LA = luminosity adjustment) method to compute the transformed luminance (L_T) variable in people who are color blind.⁴ AMLA includes tasks similar to some others classically used for defining V(λ). The Minimum Achromatic Contrast task is the second Simulcheck task. It serves to determine the luminance that an achromatic background must have to make most difficult the reading of a colored text. It is assumed that the L_T value for such color is equal to the standard luminance (L) of the selected background. In the Simulcheck method, the Minimum Achromatic Contrast task is used to measure the L_T value for the pseudoachromatic greens and reds previously selected in the Pseudoachromatic Stimuli Identification task. 

Comparing L_T with L provides relative luminance (L_R = L_T/L).⁵ L_R is the second variable used by Simulcheck to evaluate simulation tools' accuracy—specifically, how much the responses of the achromatic mechanism differ between dichromats and normals. L_R values provide a very intuitive specification of such differences. Values over one correspond to stimuli that are perceived to be lighter by people who are color blind (real or simulated) than by normal people. The reverse is true when L_R is less than one. 

We also evaluated Simulcheck's consistency using two different stimulus sets in the Pseudoachromatic Stimuli Identification task. Although such sets had important colorimetric differences, we expected this fact did not substantially influence the pattern of results obtained for the h_uv and L_R values provided by Simulcheck. 

Simulcheck was applied to real dichromats and to simulated dichromats using three color simulation tools. One of the tools was the Variantor goggles. Variantor goggles perform an optical transformation that works for any kind of visual stimulation and hence it is not restricted to screen colors. Variantor's main limitation is that it does not provide different transformations for different types of dichromatism. The other two tools were free access tools: Vischeck (http://www.vischeck.com/vischeck/) and Coblis (http://www.color-blindness.com/coblis-color-blindness-simulator/). Both Vischeck and Coblis use software for transforming screen colors into other colors that are supposedly similar to the ones seen by protanopes or deuteranopes. For Vischeck the transformation is based in the algorithm developed by Brettel et al. (1997; see ). There is no comparable information for Coblis and Variantor. We provide information about chromatic coordinates in order to specify the results produced by these software simulation tools. We used two stimulus sets (Maximum Chroma and Constant Chroma) for evaluating Simulcheck's consistency. 

Methods

Participants

A total of 10 normal trichromats (four men, six women; age range 22–30 years) and 10 dichromats (five protanopes, five deuteranopes; nine men, one woman; age range 21–34 years) took part in the experiment. Their color vision was tested with a set of psychophysical tests (Fletcher, 1980 [City University Color Vision Test]; Ishihara, 1996; Lanthony, 1985). In order to assign them to groups, participants with color vision deficiencies also performed a Rayleigh match in a Nagel anomaloscope (Tomey AF-1, Tomey, Nagoya, Japan). All participants collaborated voluntarily in the research and could stop their participation at any moment. 

Materials and stimuli

Stimuli were presented using a Sony Trinitron Multiscan 17 SEII screen (Sony, Tokyo, Japan) with a gamma correction equal to 2.15. Primaries luminance (Y) and chromatic coordinates (u', v') were as follows: red (Y = 14.6; u' = 0.404, v' = 0.530), green (Y = 31.2; u' = 0.119, v' = 0.561), and blue (Y = 3.5; u' = 0.170, v' = 0.155). Equivalent values for the screen white were Y = 49.3; u' = 0.204, v' = 0.467. During part of the experiment normal trichromats used Variantor glasses (http://www.variantor.co.uk/page.php?sub=2&id=2; Cambridge Research Systems, Kent, UK). The measurements of primaries through the goggles provided the following results: red (Y = 0.95; u' = 0.268, v' = 0.524), green (Y = 6.65; u' = 0.205, v' = 0.546), and blue (Y = 0.35; u' = 0.185, v' = 0.161). Equivalent values for the screen white were Y = 7.95; u' = 0.211, v' = 0.447. All the measurements were performed using a Minolta CL-200 luxocolorimeter (Minolta, Tokyo, Japan) with the required accessories. The room where the screen was located had an illuminance level near 3 lux in the area near the screen. 

Figure 1 gives the colorimetric characteristics of the stimuli included in the two stimulus sets of the Pseudoachromatic Stimuli Identification task. The 40 radii departing from the coordinates of white (u' = 0.204, v' = 0.467) correspond to chromatic angles (h_uv) changing in 9° steps (3°, 12°, 21°, … , 354°). Coordinates of stimulus sets were defined by the intersection of the radii and the triangle defined by the three screen primaries (Maximum Chroma set) or the radii and the circle (Constant Chroma set). The circle radius was defined by the less-saturated angle available in the monitor color gamut (h_uv = 183°, saturation, s_uv = 0.884). The stimuli included in this last set had similar C*_uv (chroma) and L* (lightness) values (C*_uv = 66.5, maximum value possible in all the chromatic angles; L* = 75). For the Maximum Chroma set, C*_uv and L* changed for the different chromatic angles (C*_uv was the maximum possible in each angle, and L* was the value that made possible such maximum C*_uv). 

Figure 1

 

Representation of the chromatic coordinates for the stimuli included in the two stimulus sets used in the Pseudoachromatic Stimuli Identification task. Maximum Chroma set: intersections between radii (40 different chromatic angles) and triangle perimeter; Constant Chroma set: intersections between radii and circle. The center of the circle corresponds to the achromatic point. The triangle vertices to the three screen primaries. Two confusion lines cross the achromatic point. Black line is for protanopes; gray line is for deuteranopes. Dotted segments are for pseudoachromatic greens; dashed segments are for pseudoachromatic reds.

Procedure

Participants performed 20 trials (10 for each stimulus set; five for pseudoachromatic green and five for red). Each trial included a Pseudoachromatic Stimuli Identification task (task 1) and, after it, a Minimum Achromatic Contrast task (task 2). Real dichromats performed the Pseudoachromatic Stimuli Identification task responding to the original stimuli included in both sets (Maximum Chroma and Constant Chroma; Figure 1), but normal trichromats responded, in different moments, to the stimuli transformed by each simulation tool (Variantor, Coblis, and Vischeck). One half of the normal observers performed all the trials with the optical tool (Variantor) before beginning with the software tools (Coblis and Vischeck; their trials were randomized); the other half did so in the opposite order. Figure 2 provides a schematic version of the spatial configuration used in the Pseudoachromatic Stimuli Identification task. A total of 20 stimuli (half set, a semiset) were simultaneously presented. They belonged to one of the two stimulus sets (Maximum Chroma or Constant Chroma). Chromatic angle (h_uv) value was increased from left to right by 9° steps (h₁, h₂, h₃, … , h₂₀). Each stimulus was a 1.3-cm-wide square that projected 1.5° by 1.5° when viewed from 50 cm. In half of the trials the leftmost stimulus h_uv was 84° ± 18° (pseudoachromatic green selection). In the other half it was 264° ± 18° (pseudoachromatic red selection). The ±18° random variation was used to avoid the presentation of the stimuli in exactly the same position across trials. The order of the pseudoachromatics (green and red) and the sets (Maximum Chroma and Constant Chroma) was randomized. In each of the 20 trials of the Pseudoachromatic Stimuli Identification task, participants were required to select the stimulus most similar to gray. If participants indicated that a hint of hue was still present, they were required to select the less-saturated stimulus. 

Figure 2

 

Schematic version of the spatial configuration used in the Pseudoachromatic Stimuli Identification task. Twenty stimuli (a semiset) were simultaneously presented. Their chromatic angles (h_uv) changed from left to right.

Colors selected in the Pseudoachromatic Stimuli Identification task were used in the Minimum Achromatic Contrast task. They appeared in text (the word “ENSAYO” written two times) presented on 20 achromatic backgrounds (Figure 3). Each background measured 5.6 cm × 1 cm (6.4° × 1.2°). The full horizontal length for the semiset was 27.2 cm. The vertical length was 10 cm. The lightest background (L* = 100) was located at the upper left and the darkest background (L* = 2.04) was located in the lower right. L* changed in five L* steps between the more similar backgrounds. Both tasks' stimuli configurations were presented on a gray background (L* = 90). In the Minimum Achromatic Contrast task, participants were required to select the background that made it most difficult to read the text. 

Figure 3

 

Minimum Achromatic Contrast task stimuli configuration. Selected colors in the Pseudoachromatic Stimuli Identification task appeared in text in the Minimum Achromatic Contrast task. Achromatic backgrounds changed from white (up and left) to black (down and rightmost) by steps of five L* units.

Results

Colorimetric transformation and the implementation accuracy of the algorithm

In Figures 4 through 6, the blue and black symbols correspond to the Maximum Chroma and Constant Chroma stimulus sets, respectively. The solid lines correspond to the triangle and the circle used to define the Maximum Chroma and Constant Chroma sets without transformation. Figure 4 is for protanope simulations and Figure 5 is for deuteranope simulations. Figure 6 provides equivalent information for Variantor (no different transformation for protanopes and deuteranopes). 

Figure 4

 

Chromatic coordinates provided by the protanope transformations for Vischeck (A) and Coblis (B). Solid lines include the chromatic coordinates of both original stimulus sets (triangle = Maximum Chroma; circle = Constant Chroma). Small symbols (triangles = Maximum Chroma; circles = Constant Chroma) show the chromatic coordinates after being transformed.

Figure 5

 

Chromatic coordinates provided by the deuteranope transformations for Vischeck (A) and Coblis (B). Solid lines include the chromatic coordinates of both original stimulus sets (triangle = Maximum Chroma; circle = Constant Chroma). Small symbols (triangles = Maximum Chroma; circles = Constant Chroma) show the chromatic coordinates after being transformed.

Figure 6

 

Chromatic coordinates provided by the Variantor transformation. Solid lines include the chromatic coordinates of both original stimulus sets (triangle = Maximum Chroma; circle = Constant Chroma). Small symbols (triangles = Maximum Chroma; circles = Constant Chroma) show the chromatic coordinates after Variantor transformation.

In order to test the accuracy of Vischeck in the implementation of Brettel et al.'s (1997) algorithm, a series of linear regression analyses of the chromatic coordinates of the transformed stimuli were performed. Table 1 provides the main results related to these analyses. The three leftmost columns specify the simulation tool, the stimulus set, and the location of transformed stimuli (over [Y, yellow] or below [B, blue] the achromatic point). The three next columns indicate the R² values (proportion of variance explained by the model), F-test statistic values, and degrees of freedom for each linear regression analysis. The rightmost columns indicate the dominant wavelengths matched by the best-fitting lines and the corresponding h_uv values. 

Table 1

 

Linear regression analyses values (R², F, and df), dominant wavelengths (λ_d), and chromatic angles (h_uv) matched by each simulation tool. Dichromat type: P = protanope, D = deuteranope. Stimulus set: MC = Maximum Chroma, CC = Constant Chroma; Y = yellow, B = blue. **p < 0.01; ***p < 0.001.

Simulation tool	Stimulus set		R2	F	df	λd	huv (°)
Vischeck P	CC	Y	0.993***	3313.47	1, 22	574.53	70.81
		B	0.980***	693.607	1, 14	475.54	251.98
	MC	Y	0.948***	379.042	1, 21	574.57	70.65
		B	0.716***	37.871	1, 15	472.56	257.04
Vischeck D	CC	Y	0.993***	2980.21	1, 22	574.61	70.51
		B	0.982***	766.97	1, 14	476.02	251.03
	MC	Y	0.973***	746.151	1, 21	574.58	70.60
		B	0.865***	96.336	1, 15	473.61	255.42
Coblis P	CC	Y	0.852***	120.51	1, 21	571.26	83.80
		B	0.928***	194.325	1, 15	466.78	263.21
	MC	Y	0.726***	47.757	1, 18	571.21	83.98
		B	0.968***	541.563	1, 18	466.12	264.54
Coblis D	CC	Y	0.127	3.046	1, 21	574.97	69.09
		B	0.117	1.995	1, 15	469.50	261.00
	MC	Y	0.095	1.999	1, 19	577.11	60.78
		B	0.393**	10.985	1, 17	469.16	261.38

The Vischeck simulations for the Constant Chroma set were very close to straight lines (protanopia minimum R² = 0.980; deuteranopia minimum R² = 0.982). Best-fitting lines matched dominant wavelengths λ_d ≈ 575 and 476 nm (yellow and blue, respectively; the same for protanopia and deuteranopia), which are almost identical to the ones that Brettel et al.'s (1997) algorithm assumes to be equally perceived by dichromats and normal trichromats (λ_d = 575 and 475 nm, respectively; h_uv = 71.86° and 252.19°, respectively). The Vischeck lines corresponding to the Maximum Chroma set matched the following dominant wavelengths: protanope λ_d ≈ 575 and 473 nm; deuteranope λ_d ≈ 575 and 474 nm. The yellow dominant wavelength was the same irrespective of the stimulus set, but this was not true for the blue dominant wavelength. As can be observed in Figures 4A and 5A, monitor color gamut did not make it possible for the Maximum Chroma set to align along a unique straight line in both Vischeck simulations. Instead, two segments can be appreciated. The first segment radiates from the achromatic point to the triangle and matched dominant wavelength λ_d ≈ 475 nm (protanope R² = 0.988, F(1, 9) = 771.164, p < 0.001; deuteranope R² = 0.994, F(1, 13) = 2065.141, p < 0.001). The second segment overlaps the line of the triangle defined by the green and blue primaries. 

Only for comparison purposes, the same analyses were performed using the chromatic coordinates of the stimuli transformed by Coblis (we omit Variantor because its stimuli are close to an ellipse or a triangle but not to a straight line; Figure 6). Coblis' simulation for protanopes was closer to a straight line (Figure 4B; minimum R² = 0.726) than its simulation for deuteranopes (Figure 5B; deuteranopia maximum R² = 0.393). Coblis' protanopia simulation lines matched the same dominant wavelengths irrespective of the stimulus set (Maximum Chroma set λ_d ≈ 571 and 466 nm; Constant Chroma set λ_d ≈ 571 and 467 nm). Coblis' deuteranopia simulation was not close to a straight line, but we show its results for contrast purposes. Such simulation matched the following dominant wavelength: Maximum Chroma set λ_d ≈ 575 and 470 nm; Constant Chroma set λ_d ≈ 577 and 469 nm. 

Real dichromats' behavior

Results provided by real dichromats are now commented on. Next, we compared these results with (1) the predictions made by Brettel et al.'s (1997) algorithm and (2) the simulated dichromats. 

Figures 7 (Pseudoachromatic Stimuli Identification) and 8 (Minimum Achromatic Contrast) show mean h_uv and L_R values both for real protanopes (left) and real deuteranopes (right). Each group of four bars shows the results for the two stimulus sets (Maximum Chroma and Constant Chroma) and the two pseudoachromatic selections (green and red). Bar colors and saturation are consistent with the type of pseudoachromatic selected (green and red) and the set (Maximum Chroma and Constant Chroma). 

Figure 7

 

Mean h_uv selected by real protanopes (left) and deuteranopes (right). Each group of four bars represents the results for the two stimulus sets (MC = Maximum Chroma; CC = Constant Chroma) for the two pseudoachromatic selections (green and red). Its colors are consistent with the set and type of pseudoachromatic selected. Error bars show standard error of the mean (SEM).

Figure 8

 

Mean L_R selected by real protanopes (left) and deuteranopes (right). See Figure 7 caption for a detailed description.

As can be observed in Figures 7 and 8, even though h_uv and L_R values showed some differences depending on stimulus set, such differences were of reduced magnitude (h_uv) and did not appear on a qualitative level (h_uv and L_R).⁶ Specifically, protanope h_uv values were near 180° for the green pseudoachromatic and near 360° for the red pseudoachromatic (Figure 7, left), and deuteranope h_uv values always represented a clockwise change (fewer degrees) in relation to protanopes (Figure 7, left bars higher). Protanope L_R values were always lower for pseudoachromatic reds but the opposite was true for the pseudoachromatic greens (Figure 8, left); deuteranopes showed the reverse pattern (Figure 8, right). 

Several statistical analyses were performed to study the above patterns. There were no significant differences (Wilcoxon signed-rank test, null hypothesis: h_uv Maximum Chroma = h_uv Constant Chroma, p > 0.05) in h_uv values of real dichromats depending on the stimulus set (exception made for red in deuteranopes, p = 0.043). On the other hand, there were significant differences (Mann-Whitney U test, null hypothesis: h_uv or L_R real protanope = h_uv or L_R real deuteranope, p < 0.05) in seven of the eight possible comparisons (two types of pseudoachromatics [red and green] × two stimulus sets × two variables [h_uv and L_R]). The results can be summarized as follows. All deuteranope h_uv values were different in the clockwise direction in relation to protanope h_uv values (p < 0.05). Only for the green Maximum Chroma condition the difference was not significant (z = −1.362, p = 0.111, N = 10). Protanope L_R values were significantly greater than the deuteranope L_R values for the green pseudoachromatics (p < 0.05), but the opposite was true for the red pseudoachromatics (p < 0.05). 

Brettel et al.'s (1997) algorithm versus real dichromats: Algorithm validity

From the rationale of the algorithm of Brettel et al. (1997), we derived the following predictions related to the performance of real and simulated dichromats in the experimental tasks. 

Pseudoachromatic Stimuli Identification task. The h_uv predicted values were obtained from the pseudoachromatic confusion line (Smith & Pokorny, 2003) using the white of the monitor as the reference white (the predicted h_uv was the one corresponding to the nearest stimulus). Strictly speaking, the algorithm assumes that the equienergetic stimulus produces the same perception in dichromats as in normal trichromats (see ). Nevertheless, we slightly modified the predictions because our reference white did not match the equienergetic stimulus. 

Minimum Achromatic Contrast. Because protanopes and deuteranopes lack L and M cones, respectively, and because it is commonly assumed that S cones barely contribute to luminance (Hunt & Pointer, 2011, chapter 1), the background selected in this task should be the one that produces the response that is most similar to the one produced by the colored text in M (protanopes) or L (deuteranopes) cones. We specify this prediction using the L_R variable (L_R = L_T/L) because, as explained before, it gives very intuitive values. Two L_R values were computed for each stimulus, taking into account protanopes' lack of L cones and deuteranopes' lack of M cones. The procedure was as follows. First, we estimated L and M cone responses for each stimulus using Stockman and Sharpe (2000; see also Stockman, MacLeod, & Johnson, 1993) L₂(λ) and M₂(λ) cone spectral sensitivities (these are the fundamentals used by the algorithm of Brettel et al., 1997) and stimulus spectra. The relative contribution of L and M cones to luminosity was derived from V₂*(λ) = 1.5L₂(λ) + M₂(λ) (Stockman & Sharpe, 2000, equation 7). These estimations were used to compute a specific transformed luminance for protanopes and deuteranopes on the assumption that the luminance of the reference white is exactly the same for normal trichromats and dichromats, which makes it possible to compute a scaling factor for protanopes and another for deuteranopes. Finally, we computed the corresponding relative luminance values (L_R) for both types of dichromats (for an analogous procedure applied to surface stimuli, see Moreira, Lillo, Álvaro, & Davies, 2014). 

We computed confidence intervals (CI) for the mean of the selected stimuli in task 1 (h_uv) and task 2 (L_R) in order to check if the predicted values fell within that range. Table 2 provides the main results related to these analyses. The leftmost column specifies the dependent variable (h_uv or L_R) and the type of pseudoachromatic selection (green or red). The next two columns show data corresponding to the Maximum Chroma set. The rightmost two columns show data corresponding to the Constant Chroma set. For each set, two types of information are provided: CIs (outside brackets) for the two types of real dichromats considered (protanope, left; deuteranope, right) and predictions derived from the rationale of Brettel et al.'s (1997) algorithm (between brackets). 

Table 2

 

The h_uv and L_R values of pseudoachromatic stimuli: 95% CIs for the mean of selected stimuli by real dichromats (outside brackets) and predicted values (between brackets) for each set (Maximum Chroma and Constant Chroma). **p < 0.01.

Dependent variable	Maximum Chroma		Constant Chroma
	Protanope	Deuteranope	Protanope	Deuteranope
huv green	179.41°–188.62° (182.86°)	165.41°–188.03° (169.31°)	181.41°–186.35° (183°)	167.17°–182.05° (169.5°)
huv red	349.62°–364.67° (2.86°)	334.63°–347.72° (344.89°)	349.15°–366.18° (3°)	332.9°–351.3° (349.5°)
LR green	1.11–1.27 (1.13)	0.83–1.02 (0.93)	1.11–1.28 (1.14)	0.85–1.04 (0.92)
LR red	0.42–0.73 (0.73)	0.90–1.29 (1.13)	0.75–0.88 (0.94)**	0.97–1.10 (1.03)

Even though h_uv and L_R predicted values were different depending on the stimulus set, such differences did not appear on a qualitative level. Specifically, (1) h_uv predicted values for deuteranopes always represented a clockwise change in relation to protanopes, and (2) L_R predicted values were always lower for protanopes for pseudoachromatic reds whereas the opposite was true for pseudoachromatic greens.⁶ 

No comparison (out of 16) of real dichromats' empirical values and predicted values was significant (one-sample t-test, null hypothesis: the predicted value fell within the empirical 95% CI) except for L_R red Constant Chroma in protanopes, 95% CI [0.75, 0.88], predicted L_R = 0.94, p < 0.01. 

Real versus simulated dichromats: Tool accuracy

Figures 9 and 10 show mean differences between simulated and real dichromats Display Formula . In order to facilitate comparisons, leftmost bars are for Vischeck, middle bars are for Variantor, and rightmost bars are for Coblis. Each group of four bars shows the results for a tool with the two stimulus sets (Maximum Chroma and Constant Chroma) and the two pseudoachromatic selections (green and red). Bar colors are consistent with the type of pseudoachromatic selected. In Figures 9 and 10, the upper parts (Figures 9A and 10A) are for protanopes and the lower parts (Figures 9B and 10B) are for deuteranopes. In Figures 9 and 10, the asterisks indicate the results of a series of Mann-Whitney U tests (null hypothesis: h_uv or L_R real dichromat = h_uv or L_R simulated dichromat;*p < 0.05, **p < 0.01). 

Figure 9

 

Mean differences in h_uv values between simulated and real dichromats (tool accuracy evaluation) for protanopes (A) and deuteranopes (B). Each group of four bars represents the results of the simulated dichromats for the two stimulus sets (MC = Maximum Chroma; CC = Constant Chroma) for the two pseudoachromatic selections (green and red). Its colors are consistent with the set and type of pseudoachromatic selected. Error bars show standard error of the mean (SEM). *p < 0.05; **p < 0.01.

Figure 10

 

Mean differences in L_R values between simulated and real dichromats (tool accuracy evaluation) for protanopes (A) and deuteranopes (B). See Figure 9 caption for a detailed description.

As for real dichromats, although h_uv and L_R values showed some differences in each simulation tool depending on stimulus set, such differences did not appear on a qualitative level (see ), so it is possible to summarize our results as follows. First, there were not big differences between Vischeck-simulated dichromats and real dichromats. Mann-Whitney U tests showed only one significant difference (Figure 9B) for h_uv values (deuteranope simulation red Constant Chroma condition). For L_R there were significant differences for both protanope Constant Chroma conditions (Figure 10A), while for deuteranope also appeared significant differences for two of the four possible comparisons (both greens; Figure 10B). Second, conversely, all the differences between Coblis-simulated dichromats and real dichromats were very big and significant (**p < 0.01; Figures 9 and 10). Finally, Variantor produced results halfway between both previous tools. Comparison with real protanopes showed small differences and frequently were not significant (Figures 9A and 10A), whereas for deuteranopes all differences were significant (Figures 9B and 10B) and bigger. 

The high similarity for each tool between the two pseudoachromatic greens and reds (Maximum Chroma and Constant Chroma ones) shows that the change of stimulus set produced only minor variations in the results of h_uv (Figure 9). Such impression agrees with the results provided by a series of Wilcoxon analyses where no significant differences (null hypothesis: h_uv Maximum Chroma = h_uv Constant Chroma; p < 0.05) were found (with the exceptions of Variantor green and Coblis protanopia, both pseudoachromatics). 

Brettel et al.'s (1997) algorithm versus simulated dichromats' behavior

As previously done for real dichromats (Table 2), we computed CIs for the mean of the selected stimuli by Vischeck-simulated dichromats (normal trichromats using Vischeck) in task 1 (h_uv) and task 2 (L_R) to check if the predicted values derived from Brettel et al.'s (1997) algorithm fell within that range (one-sample t-test, null hypothesis: h_uv or L_R predicted = h_uv or L_R real or simulated dichromat;*p < 0.05, **p < 0.01, ***p < 0.01). This kind of analysis can be especially useful because it allows potential users of Simulcheck to test if the pattern of results is compatible with a supposed underlying algorithm, even when no real dichromats are available. Only for comparison purposes, the same analyses were performed for the stimulus selections of normal trichromats using Coblis and Variantor. Figures 11 (Pseudoachromatic Stimuli Identification) and 12 (Minimum Achromatic Contrast) show the differences between the mean of empirical data and predicted values Display Formula . In order to facilitate comparisons, the first bars are for real dichromats, the second bars are for Vischeck, the third bars are for Variantor, and the fourth bars are for Coblis. Each group of four bars shows the results for a real or simulated dichromat with the two stimulus sets (Maximum Chroma and Constant Chroma) and the two pseudoachromatic selections (green and red). Bar colors are consistent with the type of pseudoachromatic selected. In Figures 11 and 12, the upper parts (Figures 11A and 12A) are for protanopes and the lower parts (Figures 11B and 12B) are for deuteranopes. Error bars represent 95% CIs for the empirical values of real and simulated dichromats, so significant differences arise when zero is not included inside error bars. 

Figure 11

 

Differences in h_uv between the mean of dichromats' data (real or simulated) and predicted values for protanopes (A) and deuteranopes (B). Each group of four bars represents the results of the real or simulated dichromats for the two stimulus sets (MC = Maximum Chroma; CC = Constant Chroma) for the two pseudoachromatic selections (green and red). Its colors are consistent with the set and type of pseudoachromatic selected. Error bars show 95% CIs of real or simulated values. *p < 0.05; ***p < 0.001.

Figure 12

 

Differences in L_R between the mean of dichromats' data (real or simulated) and predicted values for protanopes (A) and deuteranopes (B). See Figure 11 caption for a detailed description. *p < 0.05; **p < 0.01; ***p < 0.001.

Although several comparisons (11 of 16) between Vischeck-simulated dichromats and predictions derived by Brettel et al. (1997) were significant, the magnitude of such differences was minor (Figures 11 and 12). Conversely, the predictions of the algorithm were always out of the CIs for Coblis-simulated dichromats. All the differences were very big and strongly significant (***p < 0.001; Figures 11 and 12). On the other hand, Variantor produced results halfway between both previous tools. Variantor simulation CIs better fit Brettel et al.'s (1997) predictions for protanopia (four of eight differences were small and significant; Figures 11A and 12A), whereas all the differences were big and significant for deuteranopia (Figures 11B and 12B). 

Discussion

New tools aimed at making colors similar to the ones seen by people with color vision deficiencies are becoming increasingly common. A recent paper (Luque et al., 2014) shows that similar transformations are possible even with video streams. Several websites (i.e., http://play.google.com/store/apps) provide free applications ready for working in mobile phones, tablets, and other electronic gadgets. Of course, this availability carries with it the prospect of detecting and eliminating pseudoisochromatic stimuli with the aim of promoting the use of accurate colors for people with and without normal color vision (“universal design”; Vanderheiden, 2006). On the other hand, as the application of the Simulcheck method in this paper has shown, there are good reasons to be cautious about the accuracy of some simulation tools. 

Simulcheck is a versatile method that allows the evaluation of simulation tools in different ways. Each one requires comparing two of the following data: (1) predictions derived from the algorithm theoretically used by the tool, (2) the behaviors really provided by responding to the stimuli transformed by the tool (simulated dichromats), and (3) the behaviors of real dichromats. Because in our research the color transformation algorithm was known only for Vischeck, only for this tool was fair the comparison of the algorithm-based predictions and the results provided by real dichromats (algorithm validity; see Table 2 and bars labeled “real” in Figures 11 and 12), or the ones produced when using the tool (bars labeled “Vischeck” in Figures 11 and 12). Obviously, for the three evaluated tools (Vischeck, Variantor, and Coblis) the comparison between real and simulated dichromats (tool accuracy evaluation; Figures 9 and 10) was appropriate. 

Simulcheck is a consistent method because it leads to similar conclusions even when using colorimetrically different sets (Maximum Chroma or Constant Chroma sets). For example, when considering the results summarized in Figures 9 and 10 (tool accuracy evaluation), both the Maximum Chroma and Constant Chroma bars lead to conclude the following: (1) Coblis simulations were inaccurate for both dichromats types, (2) Variantor-simulated dichromats were very similar to real protanopes (small and frequently no significant differences) but not to real deuteranopes, and (3) only Vischeck simulations were accurate for both protanopes and deuteranopes. Nevertheless, we recommend the use of the Constant Chroma set for potential users of Simulcheck because it is not so dependent on the color gamut of the monitor used to implement the method. 

The high validity of the Vischeck algorithm was indicated by the high agreement found between the predictions derived from Brettel et al. (1997)⁷ and the empirical h_uv and L_R values provided by real dichromats (see Table 2 and the bars labeled “real” in Figures 11 and 12). The only exception was for L_R (Table 2; empirical 95% CI [0.72, 0.88]; predicted, 0.94) in the red pseudoachromatic Constant Chroma condition. Even in this case the value type (<1) was accurately predicted. 

Apart from Simulcheck's results, there was high colorimetric similarity between the performed color transformation (Figures 4A and 5A; Table 1) and the predictions derived from Brettel et al. (1997). As expected, the transformed color coordinates were near to λ_d = 575 and 475 nm, indicating high implementation accuracy. Returning to Simulcheck, qualitative analysis (see ) based on h_uv and L_R values can be used not only to synthesize the results provided by the evaluation of a color transformation tool but also to define a series of requisites that must be fulfilled by any algorithm or simulation tool that can be considered accurate: (1) It must provide different color transformations for each dichromat type (something not done by Variantor; see Figure 6), and (2) the specific h_uv and L_R values for the pseudoachromatic reds and greens must show specific relations. The h_uv values are more in the clockwise direction for deuteranopes (see Figure 7 and Table 2, upper part), and L_R values for the pseudoachromatic reds are smaller when simulating protanopes (the opposite for the greens; reverse pattern when simulating deuteranopes; see Figure 8 and Table 2, lower part). 

Assessing a simulation tool using the h_uv and L_R variables has two important advantages. First, it explicitly differentiates between two aspects of color vision: the activity in the yellow–blue mechanism (a qualitative aspect, measured through h_uv) and the activity in the achromatic mechanism (a quantitative aspect, measured through L_R). It must be remembered that each of these aspects is usually associated with a different color use (Breslow et al., 2009). Second, this makes it easier to understand the consequences of a simulation error. Let's use an example to illustrate this. The rightmost bars in Figure 9 indicate that, as we commented before, there were important differences between the values of the pseudoachromatic angles of Coblis simulations and real dichromats. Real protanopes and deuteranopes both selected h_uv values near 180° (Figure 7), a bluish green emerald for normal observers. On the contrary, Coblis values were near 140° (−40° in Figure 9), a chromatic angle that for normal observers is related with pure (no bluish) greens. Consequently, if Coblis is used to decide, for example, which colors must not be side by side on a political map (to help image perceptual segmentation, see Francis, Bias, & Shive, 2010), important errors can appear. The probability of errors associated with Coblis use is increased when considering its inaccuracy in measuring L_R (Figure 10). More specifically, although we found that relative to normals real protanopes see green pseudoachromatics as lighter (L_R values near 1.17) but the red ones as darker (L_R under 0.84), Coblis simulations provided exactly the opposite pattern. This is a very significant error considering that the reduced sensitivity of the protanopes to long wavelengths (see, e.g., Birch, 2001) has led some authors (see McIntyre, 2002, for a recent example) to refer to “red blindness” for naming protanopes. 

There is great conceptual similarity between Simulcheck and the test designed to assess color vision in macaque monkeys (Koida et al., 2013). Both methods use two psychophysical tasks. The first task (Pseudoachromatic Stimuli Identification task in our nomenclature) allows the determination of pseudoachromatic angles (h_uv). Both for monkeys and humans, protanope h_uv values were near 0° (pseudoachromatic reds) and 180° (pseudoachromatic greens). The second task (Minimum Achromatic Contrast task in Simulcheck) had a similar goal in both studies: the evaluation of the relative response of the achromatic mechanism. Again, similar results appeared in this case. Protanopes (the only type of dichromat among Koida's monkeys) showed reduced sensitivity to long-wavelength stimuli (the only stimuli type used in the second task with the monkeys). 

Simulcheck is a method that can be described as versatile (can be used for evaluating tool accuracy and algorithm validity), consistent (provides qualitatively similar results using different stimulus sets), selective (evaluates, through h_uv and L_R, two different color visions mechanisms), and useful (makes it easy to understand the inaccuracies of simulation tools). In our opinion, its main limitation is that it assumes permanent functional dichromatism in people diagnosed as protanope or deuteranope. This is inaccurate because it leads to an underestimation of the color vision capacities of these groups of people. 

In a paper originally published in German titled “Dichromatische Fovea, Trichromatische Peripherie” (“Dichromatic Fovea, Trichromatic Periphery”; for a short description of the Nagel works, see Broackes [2010a] or Smith & Pokorny [1977]), Wilibald Nagel (1905) showed that many diagnosed dichromats, like himself, behaved as expected for dichromats when responding to small stimuli (e.g., the ones used in the standard Nagel anomaloscope) but as expected for anomalous trichromats when responding to bigger stimuli. Although this discovery was ignored for a long time, several works (Boynton & Scheibner, 1967; Lillo, Moreira, Álvaro, & Davies, 2014; Moreira et al., 2014; Nagy, 1980; Scheibner & Boynton, 1968; Smith & Pokorny, 1977) have confirmed and extended Nagel's discovery: Big stimuli (>3°) improve the capacity of clinical dichromats to discriminate stimuli (this changes from dichromacy to anomalous trichromacy). This improvement can be related to some kind of residual activity in the red–green mechanism. 

As stated in the Introduction, valid and accurate simulation tools make isochromatic those stimuli that differ only in terms of the response in the cone type that a dichromat lacks. Such transformation is fully valid when, as happens in conventional underground maps, the size of visual elements is small. For example, the tool makes it possible to know which lines are seen by the dichromat as having similar color. However, this color transformation can be less accurate when, as happens for some political maps, there are relatively big visual elements (i.e., color background). In this case, some clinically diagnosed dichromats will become anomalous trichromats and the residual activity in their red–green mechanism will allow the differentiation between some stimuli that are pseudoisochromatic for a true dichromat. 

Because of the influence of red–green residual activity, it is possible that the best simulation tool leads us to the erroneous conclusion that big stimuli are pseudoisochromatic. Adopting such a conservative criterion is not a serious problem when performing universal design because this error only leads to the avoidance of the simultaneous use of stimuli that are very similar for some people (clinical diagnosed dichromats). Consequently, such a conservative criterion avoids difficult differentiations and makes performance in some color-based tasks easier. 

Simulcheck's first task (Pseudoachromatic Stimuli Identification) allows us to know which chromatic angles are associated with minor responses in the dichromats' yellow–blue mechanism. This choice makes the task relatively easy because h_uv variation in the proximities of the pseudoachromatic stimuli produces qualitative changes (hue varies from blue to yellow or vice versa) that are easy to appreciate. These changes would not appear if the same kind of choice task were used to identify the h_uv that produces maximum responses in the yellow–blue mechanism. This kind of information (maximum chroma identification) would complement the one actually provided by Simulcheck (good simulation tools must be accurate in both aspects) but would require the exclusive use of the Constant Chroma set (see Figure 1) to avoid unwanted variations in the chromatism perceived by dichromats. We are now considering how to change Simulcheck's first task to get information about h_uv values associated with the yellow–blue mechanism's minimum and maximum responses. 

Even when considering the effects of the residual activity of the red–green mechanism or possible improvements related to the availability of information about yellow–blue maximum responses, it can be concluded that Simulcheck is a useful method for selecting among an increasing number of simulation tools the ones that make it possible to perform color universal designs. Simulcheck requires the measurement of only two key colorimetric variables—h_uv and L_R—for two pseudoachromatic grays and can be described as versatile, consistent, selective, and useful. Its application in the present study indicated that, among the evaluated tools, only Vischeck provided an accurate simulation of the colors for both protanopes and deuteranopes. 

Supplementary Materials

docx S1 

Acknowledgments

This research was supported by grant PSI2008-04166 from Ministerio de Ciencia e Innovación, grant PSI2012-37778 from Ministerio de Economía y Competitividad, and by research fellowship BE48/09 from Universidad Complutense de Madrid. The authors would like to thank M. C. Durán and D. Majarín for help with data processing, J. A. Aznar for thoughtful and insightful comments on earlier versions of this article, and two anonymous reviewers for their helpful comments and suggestions. 

Commercial relationships: none. 

Corresponding author: Julio Lillo. 

Email: [e-mail]. 

Address: Departamento de Psicología Diferencial y del Trabajo, Universidad Complutense de Madrid, Madrid, Spain. 

Footnotes

¹  Brettel et al. (1997) used Q for an original stimulus and Q′ for the one that normals and dichromats saw with the same color (Q′ = transformed stimulus). They used the LMS colorimetry (based on the response of L, M, S cones) to decide which Q′ (Q′ = L_Q′, M_Q′, S_Q′) correspond to every specific Q (Q = L_Q, M_Q, S_Q). The specification is based on the following assumptions: (1) Dichromats and trichromats have the same perceptions when seeing equienergetic stimuli, and (2) for dichromats, LMS space is reduced to two half-planes (those corresponding to 575 or 475 nm). In the LMS space, each confusion line is parallel to an axis: the L (M_Q′ = M_Q and S_Q′ = S_Q) for the protanopes and the M (L_Q′ = L_Q and S_Q′ = S_Q) for the deuteranopes. Q′ coordinates are determined by the intersection between (A) the confusion line and (B) the half-plane defined by the origin (O), the equienergetic stimulus E, and the corresponding monochromatic stimuli (575 or 475 nm).

Footnotes

²  Two radii of Figure 1 specify the chromatic angle (h_uv) of a stimulus. One (not shown) corresponds to the horizontal line extended from the achromatic point to the right of the diagram. The other is the radius including the stimulus. All the stimuli sharing a dominant wavelength (i.e., λ_d = 494 nm) also share a chromatic angle (i.e., h_uv = 183.95°).

Footnotes

³  The intersection of dashed and dotted lines in Figure 1 defines achromatic point coordinates (the ones corresponding to the screen white). The black line originates in the protanope convergence point (not shown; u' = 0.656, v' = 0.5051) and finishes in 494 nm. Stimuli represented by a point in the radius, extended between the achromatic point and the 494-nm position (black dotted segment), have such dominant wavelength (λ_d = 494 nm). Stimuli represented by the complementary radius (black dashed segment) have a negative dominant wavelength (λ_d = −494 nm). Negative values are used for radii ending in the purple line (the one extended between 400 and 700 nm in the diagram) because there are no monochromatic stimuli included in the purple line.

Footnotes

⁴  Transformed luminance (L_T) was named “effective luminance (Y_E)” in previous AMLA (AM = achromatic measurement, LA = luminosity adjustment) descriptions. The main variable provided by the AMLA method was named “relative effective luminance (Y_ER)” and derives from comparing target stimulus (Y_E) and reference white (Y_EN) effective luminances (Y_ER = Y_E/Y_EN). This variable (Y_ER) is very similar to the luminance factor (β) in standard photometry (i.e., Hunt & Pointer, 2011, appendix 1). AMLA assumes that standard and effective luminances have the same level (Y_E = Y) for achromatic stimuli but not (Y_E ≠ Y) for chromatic ones. Consequently, their Y_E level is determined using the following procedure: (1) Some psychophysical task is used to know which achromatic stimulus has the same Y_E as the target chromatic stimulus (LA = luminosity adjustment). (2) The achromatic stimuli standard luminance (Y) is measured (AM = achromatic measurement). It is assumed that the result is the target chromatic stimulus effective luminance (Y_E).

Footnotes

⁵  Despite the similarity in their denominations, Y_ER (relative effective luminance, included in previous AMLA descriptions; see Footnote 4) and L_R (relative luminance, included in Simulcheck) refers to two different kind of comparisons where stimulus effective luminance (Y_E) is used. Y_ER results from comparing Y_E with the assumed maximum (luminance of the reference white). L_R results from comparing Y_E with normal observers' effective luminance. To avoid conceptual confusion between both variables, in this paper (1) naming of variables related to luminance includes “L” (as is usual in photometry; i.e., Hunt & Pointer, 2011, appendix 1) and not Y (as is usual in colorimetry; i.e., Hunt & Pointer, 2011, chapter 2) and (2) instead of “effective luminance (Y_E),” the expression “transformed luminance (L_T)” is used.

Footnotes

⁶  Figure 1 helps to differentiate quantitative and qualitative predictions. As can be seen, for example, the radius ending near 700 nm is intersected in different localizations by the circle (Constant Chroma set) and the triangle (Maximum Chroma set). Because the radius is the same (one corresponding to reddish colors), the same qualitative prediction was done for both sets. The L_R value was under one for protanopes because for this type of colors the effective luminance must be under the standard. On the other hand, because the intersections differ in their distance to the achromatic point, there were different quantitative predictions for both sets. The predicted L_R was smaller for the Maximum Chroma set.

Footnotes

⁷  The algorithm of Brettel et al. (1997) uses LMS colorimetry to specify the colors that supposedly elicit the same perceptions in normal trichromats and dichromats (see Footnote 1). We used LMS colorimetry to predict L_R values and a standard chromaticity diagram to predict h_uv values. This is a traditional and usual way to represent the confusion lines for dichromats (Hunt & Pointer, 2011; Kaiser & Boynton, 1996; Smith & Pokorny, 2003; Wyszecki & Stiles, 1982), so we have used it for simplicity and to directly link the predictions to the expected behavior of our subjects.