Designing for Color blindness
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.
The opposite of color blindness is seeing all the colors and I can help you find 1,000 (or more) maximally distinct colors.
You can also delve into the mathematics behind the color blindness simulations and learn about copunctal points (the invisible color!) and lines of confusion.
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) ```
# 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
Happy 2025 π Day—
TTCAGT: a sequence of digits
Celebrate π Day (March 14th) and sequence digits like its 1999. Let's call some peaks.
Crafting 10 Years of Statistics Explanations: Points of Significance
I don’t have good luck in the match points. —Rafael Nadal, Spanish tennis player
Points of Significance is an ongoing series of short articles about statistics in Nature Methods that started in 2013. Its aim is to provide clear explanations of essential concepts in statistics for a nonspecialist audience. The articles favor heuristic explanations and make extensive use of simulated examples and graphical explanations, while maintaining mathematical rigor.
Topics range from basic, but often misunderstood, such as uncertainty and P-values, to relatively advanced, but often neglected, such as the error-in-variables problem and the curse of dimensionality. More recent articles have focused on timely topics such as modeling of epidemics, machine learning, and neural networks.
In this article, we discuss the evolution of topics and details behind some of the story arcs, our approach to crafting statistical explanations and narratives, and our use of figures and numerical simulations as props for building understanding.
Altman, N. & Krzywinski, M. (2025) Crafting 10 Years of Statistics Explanations: Points of Significance. Annual Review of Statistics and Its Application 12:69–87.
Propensity score matching
I don’t have good luck in the match points. —Rafael Nadal, Spanish tennis player
In many experimental designs, we need to keep in mind the possibility of confounding variables, which may give rise to bias in the estimate of the treatment effect.
If the control and experimental groups aren't matched (or, roughly, similar enough), this bias can arise.
Sometimes this can be dealt with by randomizing, which on average can balance this effect out. When randomization is not possible, propensity score matching is an excellent strategy to match control and experimental groups.
Kurz, C.F., Krzywinski, M. & Altman, N. (2024) Points of significance: Propensity score matching. Nat. Methods 21:1770–1772.
Understanding p-values and significance
P-values combined with estimates of effect size are used to assess the importance of experimental results. However, their interpretation can be invalidated by selection bias when testing multiple hypotheses, fitting multiple models or even informally selecting results that seem interesting after observing the data.
We offer an introduction to principled uses of p-values (targeted at the non-specialist) and identify questionable practices to be avoided.
Altman, N. & Krzywinski, M. (2024) Understanding p-values and significance. Laboratory Animals 58:443–446.
Depicting variability and uncertainty using intervals and error bars
Variability is inherent in most biological systems due to differences among members of the population. Two types of variation are commonly observed in studies: differences among samples and the “error” in estimating a population parameter (e.g. mean) from a sample. While these concepts are fundamentally very different, the associated variation is often expressed using similar notation—an interval that represents a range of values with a lower and upper bound.
In this article we discuss how common intervals are used (and misused).
Altman, N. & Krzywinski, M. (2024) Depicting variability and uncertainty using intervals and error bars. Laboratory Animals 58:453–456.