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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.

The transformations described here will allow you to simulate color blindness and apply conversions of the kind shown in the images below to your own work.

The conversion from RGB values to their color blindness equivalents for protanopia, deuteranopia and tritanopia consists of the following steps

- convert from sRGB to linear RGB
- convert from linear RGB to XYZ
- convert from XYZ to LMS
- apply color blindness transformation in LMS space
- convert from LMS to XYZ (inverse of #3)
- convert from XYZ to linear RGB (inverse of #2)
- convert from linear RGB to sRGB (clip and inverse of #1)

The greyscale conversion for achromatopsia does not require the XYZ and LMS steps. We can go straight to `Y`.

- convert from sRGB to linear RGB
- `Y = 0.02126r + 0.7152g + 0.0722b`
- convert from linear RGB to sRGB (inverse of #1)

The details of each of the steps are shown below. If you just want the `\mathbf{T}` matrices, scroll down to the bottom or download the code.

Depending on the implementation, this process may have an additional step that reduces the color domain (e.g. step 2 in Viénot, Brettel & Mollon, 1999). This makes sure that none of the transformed colors are outside of the sRGB domain. Here, I instead of doing this I just clip the transformed colors at the end. This simplifies the process and (my sense is that) the difference is negligible.

Different simulators will yield slightly different results, too.

The RGB input color is `\{R,G,B\} = \{ V \in [0,255] \}` and assumed to be sRGB. This is first linearized with `\gamma = 2.4` to obtain `\{r,g,b\} = \{ v \in [0,1] \}`. $$v = \begin{cases} \dfrac{V/255}{12.92} & \text{if $V/255 \le 0.04045$} \\[2ex] \left({\dfrac{V/255+0.055}{1.055}}\right)^\gamma & \text{otherwise} \end{cases} $$

Next, convert the linearized `(r,g,b)` to XYZ by multiplying by `\mathbf{M}_\textrm{XYZ}`. $$ \left[ \begin{matrix} X \\Y \\Z \end{matrix} \right] = \underbrace{ \left[ \begin{matrix} 0.4124564 & 0.3575761 & 0.1804375 \\0.2126729 & 0.7151522 & 0.0721750 \\0.0193339 & 0.1191920 & 0.9503041 \end{matrix} \right] }_{\mathbf{M}_\textrm{XYZ}} \left[ \begin{matrix} r \\g \\b \end{matrix} \right] $$

Multiply by `\mathbf{M}_\textrm{LMS,D65}` to convert XYZ to LMS. This matrix is normalized to the D65 illuminant, which will ensure that greys will be preserved.

The LMS (long, medium, short) is a color space which represents the response of the three types of cones of the human eye, named for their sensitivity peaks at long (560 nm, red) medium (530 nm, green), and short (420 nm, blue) wavelengths. $$ \left[ \begin{matrix} L \\ M \\ S \end{matrix} \right] = \underbrace{ \left[ \begin{matrix} 0.4002 & 0.7076 & -0.0808 \\ -0.2263 & 1.1653 & 0.0457 \\ 0 & 0 & 0.9182 \end{matrix} \right] }_{\mathbf{M}_\textrm{LMS,D65}} \left[ \begin{matrix} X \\ Y \\ Z \end{matrix} \right] $$

If for some reason you don't want to normalize to D65, you would use `\mathbf{M}_\textrm{LMS}` below but whites will now appear pinkish. $$ \left[ \begin{matrix} L \\ M \\ S \end{matrix} \right] = \underbrace{ \left[ \begin{matrix} 0.0.38971 & 0.68898 & -0.07868 \\ -0.22981 & 1.18340 & 0.04641 \\ 0 & 0 & 1 \end{matrix} \right] }_{\mathbf{M}_\textrm{LMS}} \left[ \begin{matrix} X \\ Y \\ Z \end{matrix} \right] $$

There are other XYZ to LMS matrices and the R code lists them all.

Now that we have the RGB color represented in LMS space, we can correct for color receptor dysfunction in this space, since color blindness affects one of the L, M or S receptors.

I show the calculations here in a lot of detail—they're not as complicated as things look. Once you understand what's happening for one of the color blindness types, the other two are analogously treated.

Each of the color blindness types will have its own correction matrix `\mathbf{S}`. $$ \left[ \begin{matrix} L' \\ M' \\ S' \end{matrix} \right] = \mathbf{S} \left[ \begin{matrix} L \\ M \\ S \\ \end{matrix} \right] $$

This matrix is the identity matrix with the row for the malfunctioning receptor (e.g. S for protanopia) replaced by two free parameters `a` and `b`. $$ \begin{align} \mathbf{S}_\textrm{protanopia} & = \left[ \begin{matrix} 0 & a & b \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \right] \\ \mathbf{S}_\textrm{deuteranopia} & = \left[ \begin{matrix} 1 & 0 & 0 \\ a & 0 & b \\ 0 & 0 & 1 \end{matrix} \right] \\ \mathbf{S}_\textrm{tritanopia} & = \left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & 0 \end{matrix} \right] \end{align} $$

The reason why these matrices have this format is so that we can satisfy two conditions. First, this matrix should not affect how white appears—if one of the rows was just zero then white would be altered. Second, we expect that one of the primaries won't be affected, depending on the color blindness type. For protanopia and deuteranopia this is blue and for tritanopia this is red.

If we set `\mathbf{M} = \mathbf{M}_\textrm{LMS,D65} \mathbf{M}_\textrm{XYZ}` then these conditions (using the blue case for protanopia) can be expressed as $$ \mathbf{S}_\textrm{protanopia} \mathbf{M} \left[ \begin{matrix} 1 \\ 1 \\ 1 \end{matrix} \right] = \mathbf{M} \left[ \begin{matrix} 1 \\ 1 \\ 1 \end{matrix} \right] = \left[ \begin{matrix} L_0 \\ M_0 \\ S_0 \end{matrix} \right] $$ $$ \mathbf{S}_\textrm{protanopia} \mathbf{M} \left[ \begin{matrix} 0 \\ 0 \\ 1 \end{matrix} \right] = \mathbf{M} \left[ \begin{matrix} 0 \\ 0 \\ 1 \end{matrix} \right] = \left[ \begin{matrix} L \\ M \\ S \end{matrix} \right] $$

where `(L_0,M_0,S_0)` are the LMS coordinates of white and `(L_b,M_b,S_b)` of the primary that is not affected (e.g. blue). Using the form for `\mathbf{S}_\textrm{protanopia}`, these lead to the following equations $$ \begin{align} a M_b + b S_b & = L_b \\a M_0 + b S_0 & = L_0 \end{align} $$

which can be written as $$ \left[ \begin{matrix} a \\b \end{matrix} \right]_\textrm{protanopia} = { \left[ \begin{matrix} M_b & S_b \\M_0 & S_0 \end{matrix} \right] }^{-1} \left[ \begin{matrix} L_b \\L_0 \end{matrix} \right] $$

For deuteranopia and tritanopia the calculation of `a` and `b` is analogous, except that because now `a` and `b` change position in `\mathbf{S}`, the equations are slightly different. $$ \left[ \begin{matrix} a \\b \end{matrix} \right]_\textrm{deuteranopia} = { \left[ \begin{matrix} L_b & S_b \\L_0 & S_0 \end{matrix} \right] }^{-1} \left[ \begin{matrix} M_b \\M_0 \end{matrix} \right] $$

and for tritanopia (here `(L_r,M_r,S_r)` refers to the coordinates of red). $$ \left[ \begin{matrix} a \\b \end{matrix} \right]_\textrm{tritanopia} = { \left[ \begin{matrix} L_r & M_r \\L_0 & M_0 \end{matrix} \right] }^{-1} \left[ \begin{matrix} S_r \\S_0 \end{matrix} \right] $$

Using the following LMS coordinates (calculated by linearizing the corresponding RGB values and then muptiplying by `\mathbf{M}`) $$ \begin{align} \left[ \begin{matrix} L_0 \\ M_0 \\ S_0 \end{matrix} \right] & = \left[ \begin{matrix} 1 \\ 0.999683 \\ 0.9997637 \end{matrix} \right] \\ \left[ \begin{matrix} L_b \\ M_b \\ S_b \end{matrix} \right] & = \left[ \begin{matrix} 0.04649755 \\ 0.08670142 \\ 0.87256922 \end{matrix} \right] \\ \left[ \begin{matrix} L_r \\ M_r \\ S_r \end{matrix} \right] & = \left[ \begin{matrix} 0.31399022 \\ 0.15537241 \\ 0.01775239 \end{matrix} \right] \end{align} $$

Using protanopia as the example $$ \begin{align} \left[ \begin{matrix} a \\b \end{matrix} \right]_\textrm{protanopia} & = { \left[ \begin{matrix} 0.08670142 & 0.87256922 \\0.999683 & 0.9997637 \end{matrix} \right] }^{-1} \left[ \begin{matrix} 0.04649755 \\1 \end{matrix} \right] \\ & = \left[ \begin{matrix} -1.27219 & 1.1103359 \\1.27245 & -0.1103267 \end{matrix} \right] \left[ \begin{matrix} 0.04649755 \\1 \end{matrix} \right] \\ & = \left[ \begin{matrix} 1.05118294 \\-0.05116099 \end{matrix} \right] \end{align} $$

The `(a,b)` calculations for deuteranopia and tritanopia are analogous and once they're done we can write the correction matrices as $$ \left[ \begin{matrix} L' \\ M' \\ S' \end{matrix} \right] = \underbrace{ \left[ \begin{matrix} 0 & 1.05118294 & -0.05116099 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \right] }_{\mathbf{S}_\textrm{protanopia}} \left[ \begin{matrix} L' \\ M' \\ S' \end{matrix} \right] $$ $$ \left[ \begin{matrix} L' \\ M' \\ S' \end{matrix} \right] = \underbrace{ \left[ \begin{matrix} 1 & 0 & 0 \\ 0.9513092 & 0 & 0.04866992 \\ 0 & 0 & 1 \end{matrix} \right] }_{\mathbf{S}_\textrm{deuteranopia}} \left[ \begin{matrix} L' \\ M' \\ S' \end{matrix} \right] $$ $$ \left[ \begin{matrix} L' \\ M' \\ S' \end{matrix} \right] = \underbrace{ \left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -0.86744736 & 1.86727089 & 0 \end{matrix} \right] }_{\mathbf{S}_\textrm{tritanopia}} \left[ \begin{matrix} L' \\ M' \\ S' \end{matrix} \right] $$

These `(a,b)` values are for the D65-normalized XYZ-to-LMS matrix and will change if you use a different matrix. The R code calculates `(a,b)` for whatever matrix you provide.

Once the color blindness correction has been applied in LMS space, we convert back to XYZ using the inverse `\mathbf{M}_\text{LMS,D65}^{-1}`. $$ \left[ \begin{matrix} X' \\ Y' \\ Z' \end{matrix} \right] = \underbrace{ \left[ \begin{matrix} 1.8600666 & -1.1294801 & 0.2198983 \\ 0.3612229 & 0.6388043 & 0 \\ 0 & 0 & 1.089087 \end{matrix} \right] }_{\mathbf{M}_\textrm{LMS,D65}^{-1}} \left[ \begin{matrix} L' \\ M' \\ S' \end{matrix} \right] $$

If you used `\mathbf{M}_\textrm{LMS}` and didn't normalize to D65, then you'd use its inverse `\mathbf{M}_\text{LMS}^{-1}` instead. $$ \left[ \begin{matrix} X' \\ Y' \\ Z' \end{matrix} \right] = \underbrace{ \left[ \begin{matrix} 1.9101968 & -1.1121239 & 0.2019080 \\ 0.3709501 & 0.6290543 & 0 \\ 0 & 0 & 1 \end{matrix} \right] }_{\mathbf{M}_\textrm{LMS}^{-1}} \left[ \begin{matrix} L' \\ M' \\ S' \end{matrix} \right] $$

Finally, one last matrix multiplication from XYZ back to linear RGB using `\mathbf{M}_\textrm{XYZ}^{-1}`. $$ \left[ \begin{matrix} r' \\ g' \\ b' \end{matrix} \right] = \underbrace{ \left[ \begin{matrix} 3.24045484 & -1.5371389 & -0.49853155 \\ -0.96926639 & 1.8760109 & 0.04155608 \\ 0.05564342 & -0.2040259 & 1.05722516 \end{matrix} \right] }_{\mathbf{M}_\textrm{XYZ}^{-1}} \left[ \begin{matrix} X' \\ Y' \\ Z' \end{matrix} \right] $$

The first step is now inverted to obtain the final sRGB values as perceived by someone with color blindness. $$V = \begin{cases} 255(12.92v) & \text{if $v \le 0.0031308$} \\[1ex] 255(1.055 v^{1/\gamma} - 0.055) & \text{otherwise} \end{cases} $$

Make sure to clip `v` to `[0,1]` before applying the final transformation back to sRGB.

The matrix multiplication steps can be written compactly as $$ \left[ \begin{matrix} r' \\ g' \\ b' \end{matrix} \right] = \mathbf{M}_\textrm{XYZ}^{-1} \mathbf{M}_\textrm{LMS,D65}^{-1} \mathbf{S} \mathbf{M}_\textrm{LMS,D65} \mathbf{M}_\textrm{XYZ} \left[ \begin{matrix} r \\ g \\ b \end{matrix} \right] = \mathbf{T} \left[ \begin{matrix} r \\ g \\ b \end{matrix} \right] $$

where `\mathbf{T}` is the product of all the matrices for a given color blindness correction `\mathbf{S}`, which are $$ \begin{align} \mathbf{T}_\textrm{protanopia} & = \left[ \begin{matrix} 0.170556992 & 0.829443014 & 0 \\ 0.170556991 & 0.829443008 & 0 \\ -0.004517144 & 0.004517144 & 1 \end{matrix} \right] \\ \mathbf{T}_\textrm{deuteranopia} & = \left[ \begin{matrix} 0.33066007 & 0.66933993 & 0 \\ 0.33066007 & 0.66933993 & 0 \\ -0.02785538 & 0.02785538 & 1 \end{matrix} \right] \\ \mathbf{T}_\textrm{tritanopia} & = \left[ \begin{matrix} 1 & 0.1273989 & -0.1273989 \\ 0 & 0.8739093 & 0.1260907 \\ 0 & 0.8739093 & 0.1260907 \end{matrix} \right] \\ \mathbf{T}_\textrm{achromatopsia} & = \left[ \begin{matrix} 0.2126 & 0.7152 & 0.0722 \\ 0.2126 & 0.7152 & 0.0722 \\ 0.2126 & 0.7152 & 0.0722 \end{matrix} \right] \end{align} $$

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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.

The uncertainty may be structural (which model?), parametric (what is `R_0`?), and/or operational (how well do masks work?).

This month, we continue our exploration of epidemiological models and look at how uncertainty affects forecasts of disease dynamics and optimization of intervention strategies.

We show how the impact of the uncertainty on any choice in strategy can be expressed using the Expected Value of Perfect Information (EVPI), which is the potential improvement in outcomes that could be obtained if the uncertainty is resolved before making a decision on the intervention strategy. In other words, by how much could we potentially increase effectiveness of our choice (e.g. lowering total disease burden) if we knew which model best reflects reality?

This column has an interactive supplemental component (download code) that allows you to explore the impact of uncertainty in `R_0` and immunity duration on timing and size of epidemic waves and the total burden of the outbreak and calculate EVPI for various outbreak models and scenarios.

Bjørnstad, O.N., Shea, K., Krzywinski, M. & Altman, N. (2020) Points of significance: Uncertainty and the management of epidemics. *Nature Methods* **17**.

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.