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Designing for Color Blindess
Color choices and transformations for deuteranopia and other afflictions
Here, I help you understand color blindness and describe a process by which you can make good color choices when designing for accessibility. You can also delve into the mathematics behind the color blindness simulations.
Different color blindness simulations don't all agree on the luminance of the simulated color. See methods for details.
color blindness R code
R code for converting an RGB color for color blindness. For details see the math tab and the resources section for background reading.
--- title: 'RGB color correction for color blindess: protanopia, deuteranopia, tritanopia' author: 'Martin Krzywinski' web: http://mkweb.bcgsc.ca/colorblind --- ```{r} gamma = 2.4 ############################################### # Linear RGB to XYZ # https://en.wikipedia.org/wiki/SRGB XYZ = matrix(c(0.4124564, 0.3575761, 0.1804375, 0.2126729, 0.7151522, 0.0721750, 0.0193339, 0.1191920, 0.9503041), byrow=TRUE,nrow=3) SA = matrix(c(0.2126,0.7152,0.0722, 0.2126,0.7152,0.0722, 0.2126,0.7152,0.0722),byrow=TRUE,nrow=3) ############################################### # XYZ to LMS, normalized to D65 # https://en.wikipedia.org/wiki/LMS_color_space # Hunt, Normalized to D65 LMSD65 = matrix(c( 0.4002, 0.7076, -0.0808, -0.2263, 1.1653, 0.0457, 0 , 0 , 0.9182), byrow=TRUE,nrow=3) # Hunt, equal-energy illuminants LMSEQ = matrix(c( 0.38971, 0.68898,-0.07868, -0.22981, 1.18340, 0.04641, 0 , 0 , 1 ), byrow=TRUE,nrow=3) # CIECAM97 SMSCAM97 = matrix(c( 0.8951, 0.2664, -0.1614, -0.7502, 1.7135, 0.0367, 0.0389, -0.0685, 1.0296), byrow=TRUE,nrow=3) # CIECAM02 LMSCAM02 = matrix(c( 0.7328, 0.4296, -0.1624, -0.7036, 1.6975, 0.0061, 0.0030, 0.0136, 0.9834), byrow=TRUE,nrow=3) ############################################### # Determine the color blindness correction in LMS space # under the condition that the correction does not # alter the appearance of white as well as # blue (for protanopia/deuteranopia) or red (for tritanopia). # For achromatopsia, greyscale conversion is applied # to the linear RGB values. getcorrection = function(LMS,type="p",g=gamma) { red = matrix(c(255,0,0),nrow=3) blue = matrix(c(0,0,255),nrow=3) white = matrix(c(255,255,255),nrow=3) LMSr = LMS %*% XYZ %*% apply(red,1:2,linearize,g) LMSb = LMS %*% XYZ %*% apply(blue,1:2,linearize,g) LMSw = LMS %*% XYZ %*% apply(white,1:2,linearize,g) if(type == "p") { x = matrix(c(LMSb[2,1],LMSb[3,1], LMSw[2,1],LMSw[3,1]),byrow=T,nrow=2) y = matrix(c(LMSb[1,1],LMSw[1,1]),nrow=2) ab = solve(x) %*% y C = matrix(c(0,ab[1,1],ab[2,1],0,1,0,0,0,1),byrow=T,nrow=3) } else if (type == "d") { x = matrix(c(LMSb[1,1],LMSb[3,1], LMSw[1,1],LMSw[3,1]),byrow=T,nrow=2) y = matrix(c(LMSb[2,1],LMSw[2,1]),nrow=2) ab = solve(x) %*% y C = matrix(c(1,0,0,ab[1,1],0,ab[2,1],0,0,1),byrow=T,nrow=3) } else if (type == "t") { x = matrix(c(LMSr[1,1],LMSr[2,1], LMSw[1,1],LMSw[2,1]),byrow=T,nrow=2) y = matrix(c(LMSr[3,1],LMSw[3,1]),nrow=2) ab = solve(x) %*% y C = matrix(c(1,0,0,0,1,0,ab[1,1],ab[2,1],0),byrow=T,nrow=3) } else if (type == "a" | type == "g") { C = matrix(c(0.2126,0.7152,0.0722, 0.2126,0.7152,0.0722, 0.2126,0.7152,0.0722),byrow=TRUE,nrow=3) } return(C) } # rgb is a column vector convertcolor = function(rgb,LMS=LMSD65,type="d",g=gamma) { C = getcorrection(LMS,type) if(type == "a" | type == "g") { T = SA } else { M = LMS %*% XYZ Minv = solve(M) T = Minv %*% C %*% M } print(T) rgb_converted = T %*% apply(rgb,1:2,linearize,g) return(apply(rgb_converted,1:2,delinearize,g)) } # This function implements the method by Vienot, Brettel, Mollon 1999. # The approach is the same, just the values are different. # http://vision.psychol.cam.ac.uk/jdmollon/papers/colourmaps.pdf convertcolor2 = function(rgb,type="d",g=2.2) { xyz = matrix(c(40.9568, 35.5041, 17.9167, 21.3389, 70.6743, 7.98680, 1.86297, 11.4620, 91.2367),byrow=T,nrow=3) lms = matrix(c(0.15514, 0.54312, -0.03286, -0.15514, 0.45684,0.03286, 0,0,0.01608),byrow=T,nrow=3) rgb = (rgb/255)**g if(type=="p") { S = matrix(c(0,2.02344,-2.52581,0,1,0,0,0,1),byrow=T,nrow=3) rgb = 0.992052*rgb+0.003974 } else if(type=="d") { S = matrix(c(1,0,0,0.494207,0,1.24827,0,0,1),byrow=T,nrow=3) rgb = 0.957237*rgb+0.0213814 } else { stop("Only type p,d defined for this function.") } M = lms %*% xyz T = solve(M) %*% S %*% M print(T) rgb = T %*% rgb rgb = 255*rgb**(1/g) return(rgb) } ############################################### # RGB to Lab rgb2lab = function(rgb,g=gamma) { rgb = apply(rgb,1:2,linearize,g) xyz = XYZ %*% rgb delta = 6/29 xyz = xyz / (c(95.0489,100,108.8840)/100) f = function(t) { if(t > delta**3) { return(t**(1/3)) } else { return (t/(3*delta**2) + 4/29) } } L = 116*f(xyz[2]) - 16 a = 500*(f(xyz[1]) - f(xyz[2])) b = 200*(f(xyz[2]) - f(xyz[3])) return(matrix(c(L,a,b),nrow=3)) } # CIE76 (https://en.wikipedia.org/wiki/Color_difference) deltaE = function(rgb1,rgb2) { lab1 = rgb2lab(rgb1) lab2 = rgb2lab(rgb2) return(sqrt(sum((lab1-lab2)**2))) } clip = function(v) { return(max(min(v,1),0)) } ############################################### # RGB to/from linear RGB #https://en.wikipedia.org/wiki/SRGB linearize = function(v,g=gamma) { if(v <= 0.04045) { return(v/255/12.92) } else { return(((v/255 + 0.055)/1.055)**g) } } delinearize = function(v,g=gamma) { if(v <= 0.003130805) { return(255*12.92*clip(v)) } else { return(255*clip(1.055*(clip(v)**(1/g))-0.055)) } } pretty = function(x) { noquote(formatC(x,digits=10,format="f",width=9)) } # a dark red rgb1 = matrix(c(0,209,253),nrow=3) # dark green rgb2 = matrix(c(60,135,0),nrow=3) # simulate deuteranopia convertcolor(rgb1,type="d") convertcolor(rgb2,type="d") # get color distance before and after simulation deltaE(rgb1,rgb2) deltaE(convertcolor(rgb1,type="d"),convertcolor(rgb2,type="d")) # transformation matrices for each color blindness type M = LMSD65 %*% XYZ pretty(solve(M) %*% getcorrection(LMSD65,"p") %*% M) pretty(solve(M) %*% getcorrection(LMSD65,"d") %*% M) pretty(solve(M) %*% getcorrection(LMSD65,"t") %*% M) pretty(SA) # method by Vienot, Brettel, Mollon, 1999 convertcolor2(rgb1,type="d",g=2.2) convertcolor2(rgb2,type="d",g=2.2) ``` ```{r} library(tidyverse) library(colorscience) printf <- function(...) invisible(print(sprintf(...))) # let's get grey #rgbg = matrix(c(120,120,120),nrow=3) #XYZgrey = XYZ %*% rgbg #xyYgrey = XYZgrey/sum(XYZgrey) # define an (x,y,Y) point #x = xyYgrey[1,1] #y = xyYgrey[2,1] #Y = xyYgrey[3,1] # convert to XYZ #X = x/y*Y #Y = Y #Z = (1-x-y)*Y/y # find combination of eigenvectors that solves this #Q = solve(ev,c(X,Y,Z)) # now we can add any amount of ev3 to this and still have the same answer palette = c(S="#333333", p="#a0d848", d="#f15a24", t="#29abe2", V="#8cc63f", C="#ed1e79") aspect = 1 p = ggplot() p = p + geom_path(data=rbind(cccie31,head(cccie31,1)),aes(x,y)) # copunctal point for(type in c("p","d")) { M = solve(LMSD65) %*% getcorrection(LMSD65,type) %*% LMSD65 ev = eigen(M)$vectors xyp = (ev[,3]/sum(ev[,3]))[1:2] #for(angle in seq(0,360,by=1)) { for(b in seq(-0.1,1.1,by=0.05)) { # pick any xy point m = (xyp[2]-b)/xyp[1] #m = tan(angle*pi/180) #b = xyp[2] - m*xyp[1] if(b < 0 | b > 1.2) { #next } # the line of xy points between these two points is the confusion line p = p + geom_abline(slope=m,intercept=b,color=palette[type]) } } #p = p + scale_x_continuous(lim=c(0,2.5)) #p = p + scale_y_continuous(lim=c(-1.5,1)) p = p + scale_x_continuous(lim=c(0,0.8)) p = p + scale_y_continuous(lim=c(0,0.8)) p dev.copy2pdf(file = "fig1.pdf",useDingbats=FALSE,width=4,height=4) ```
# a dark red rgb1 = matrix(c(225,0,30),nrow=3) # dark green rgb2 = matrix(c(60,135,0),nrow=3) # simulate deuteranopia convertcolor(rgb1,type="d") [,1] [1,] 136.7002 [2,] 136.7002 [3,] 0.0000 convertcolor(rgb2,type="d") [,1] [1,] 116.76071 [2,] 116.76071 [3,] 16.73263 # get color distance before and after simulation deltaE(rgb1,rgb2) [1] 116.9496 deltaE(convertcolor(rgb1,type="d"),convertcolor(rgb2,type="d")) [1] 12.72204 # transformation matrices for each color blindness type M = LMSD65 %*% XYZ pretty(solve(M) %*% getcorrection(LMSD65,"p") %*% M) [,1] [,2] [,3] [1,] 0.1705569911 0.8294430089 0.0000000000 [2,] 0.1705569911 0.8294430089 -0.0000000000 [3,] -0.0045171442 0.0045171442 1.0000000000 pretty(solve(M) %*% getcorrection(LMSD65,"d") %*% M) [,1] [,2] [,3] [1,] 0.3306600735 0.6693399265 -0.0000000000 [2,] 0.3306600735 0.6693399265 0.0000000000 [3,] -0.0278553826 0.0278553826 1.0000000000 pretty(solve(M) %*% getcorrection(LMSD65,"t") %*% M) [,1] [,2] [,3] [1,] 1.0000000000 0.1273988634 -0.1273988634 [2,] -0.0000000000 0.8739092990 0.1260907010 [3,] 0.0000000000 0.8739092990 0.1260907010 pretty(SA) [,1] [,2] [,3] [1,] 0.2126000000 0.7152000000 0.0722000000 [2,] 0.2126000000 0.7152000000 0.0722000000 [3,] 0.2126000000 0.7152000000 0.0722000000 # method by Vienot, Brettel, Mollon, 1999 convertcolor2(rgb1,type="d",g=2.2) [,1] [,2] [,3] [1,] 0.29275003 0.70724967 -2.978356e-08 [2,] 0.29275015 0.70724997 1.232823e-08 [3,] -0.02233659 0.02233658 1.000000e+00 [,1] [1,] 131.81223 [2,] 131.81226 [3,] 36.37274 convertcolor2(rgb2,type="d",g=2.2) [,1] [,2] [,3] [1,] 0.29275003 0.70724967 -2.978356e-08 [2,] 0.29275015 0.70724997 1.232823e-08 [3,] -0.02233659 0.02233658 1.000000e+00 [,1] [1,] 122.71798 [2,] 122.71801 [3,] 48.34316
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When modelling epidemics, some uncertainties matter more than others.
Public health policy is always hampered by uncertainty. During a novel outbreak, nearly everything will be uncertain: the mode of transmission, the duration and population variability of latency, infection and protective immunity and, critically, whether the outbreak will fade out or turn into a major epidemic.
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Bjørnstad, O.N., Shea, K., Krzywinski, M. & Altman, N. (2020) Points of significance: Uncertainty and the management of epidemics. Nature Methods 17.
Background reading
Bjørnstad, O.N., Shea, K., Krzywinski, M. & Altman, N. (2020) Points of significance: Modeling infectious epidemics. Nature Methods 17:455–456.
Bjørnstad, O.N., Shea, K., Krzywinski, M. & Altman, N. (2020) Points of significance: The SEIRS model for infectious disease dynamics. Nature Methods 17:557–558.