Steps on the Way. Getting There.

A formula for determining whether text of a given colour or shade is readable over a background of a different colour or shade.

I am not a mathematician, I am essentially an artist, and also a computer programer, hence much of what I have discovered about this has been through trial and error. Here are the results of my current researches.

Non-linear

First of all, a computer monitor (and any type of display screen) does not get brighter linearly, the distance between two brightnesses of a given hue will be relatively greater, the brighter the colours are. For a knowledgeable explanation of this, see Frequently Asked Questions about Gamma by Charles Poynton.

This means that any formula that gives constants as factors to multiply out by, where you multiply the red, green and blue elements of the RGB triad each by a number to get a brightness value, e.g. formulas such as those in the W3C Guidelines, won’t work when you are wanting to compare two brightnesses. They can’t, and they don’t.

Photoshop Seems to Get it Right at First Sight

Photoshop gets it right. Or does it? Those greyscales you see in the examples below look right. So what formula does Photoshop use?

Here is a block produced in Photoshop showing red, green and blue at values 255, 192, 128 and 64:

Now we convert that to greyscale and see what grey values Photoshop gives us.

That looks pretty good perceptually. It goes like this:

Colour value | Red | Green | Blue |

255 | 130 | 220 | 70 |

192 | 96 | 165 | 50 |

128 | 62 | 110 | 30 |

64 | 28 | 54 | 10 |

I don’t know for sure what formula Photoshop uses as I’m not privy to Photoshop’s internal code, but I do know of a formula that gives exactly the same greyscale results as Photoshop, and I know of another that is simpler and more rough-and-ready and which gives results that are pretty close to Photoshop’s. (Tested against Photoshop 5.5 thru CC, the greyscale conversion seems to have stayed consistent across the versions).

The formula for exactly equivalent results to Photoshop uses a gamma calculation. I got this code from StackOverflow, expressed there in C and I give it below in Javascript. As I explain below, these functions use classic sRGB gamma formulas.

As with all the various formulas for obtaining a greyscale, the like-Photoshop formula uses percentage values to multiply the red (R), green (G) and blue (B) numeric levels by. (see Puzzling Greys). The like-Photoshop formula uses R=0.2235, G=0.7154, B=0.0611 to get a result equivalent to what you get with Photoshop.

I say in the title to this section that Photoshop seems to get it right at first sight. It does so far as the examples on the left show, but as I explain below Photoshop gives a linear result for greyscale, which is not perfect for comparing the brightness levels of two shades of grey. Read on . . .

So, to the formulas:

The like-Photoshop Formula

First some functions in Javascript:

/* Inverse of sRGB "gamma" function. ********************** */

function inv_gam_sRGB( ic) {

var c = ic/255.0;

return c <= 0.04045 ? c/12.92 : Math.pow(((c+0.055)/(1.055)),2.4);

}

/* sRGB "gamma" function ******************************** */

function gam_sRGB(vi) {

var v=vi<=0.0031308 ? vi * 12.92 : 1.055*Math.pow(vi,1.0/2.4)-0.055

return (v*255);

}

/* GRAY VALUE ("brightness") *************************** */

function gamma_grey(r, g, b, rY, gY, bY) {

return gam_sRGB( rY*inv_gam_sRGB(r) + gY*inv_gam_sRGB(g) + bY*inv_gam_sRGB(b) );

}

You get a brightness (i.e. greyscale) value by calling:

brightness_value = gamma_grey(R, G, B, 0.2235, 0.7154, 0.0611);

Where R, G and B are the colour values of red, green and blue, of course.

You’ll find that gives you a brightness value exactly equivalent to Photoshop’s (well I do).

What the Formula is Doing

The formulas use a classic linear to sRGB and sRGB to linear conversion. Mathematically that can be seen on the page A close look at the sRGB formula and in structural code on Optimizing conversion between sRGB and linear.

The two Javascript functions shown above,

*gam_sRGB*and*inv_gam_sRGB*are the inverse of one another, a colour value passed through*gam_sRGB*and its result through*inv_gam_sRGB*, or vice-versa, will give you back the colour value. To be more precise, if you use these functions standalone, you have to divide the colour value by 255 before passing it as a parameter to*gam_sRGB*– note that*inv_gam_sRGB*does that division as its first operation, whereas*gam_sRGB*doesn’t.
The

*gamma_grey*function passes the colour value, R, G or B, though the inverse gamma sRGB*inv_gam_sRGB*and then multiplies the result by the R, G, or B constant factor. It then adds the R, G and B results from*inv_gam_sRGB*together and passes the result though the gamma sRGB*gam_sRGB*function to get a greyscale.*inv_gam_sRGB*will always give a result from 0 to 1. This fraction is then multiplied by the factor for each of the R, G and B values.

Although these are gamma functions, the resulting greyscale will in effect be linear, that is because the colour values go through both

*gam_sRGB*and*inv_gam_sRGB*, which has a cancelling-out outcome, see my page Colour Brightness Experiment which shows that, without further adjustment, a brightness range for a given colour triad is linear.
Values 0-255 calculated as sRGB give a curve, which is said to be very useful for adjusting colours on photographs, but it is a curve that it inverse to what we need when comparing two brightnesses. The sRGB curve with an exponent of 2.4 looks like this:

With this curve the brightnesses get less the higher the colour value, not more, so we need the inverse of this:

That is the chart for 0-255 through the

*inv_gam_sRGB*formula.
Similar but Rougher

You can get almost identical results, slightly further out in the low values values, by forgetting all about complex gamma calculations and using something much simpler.

My Rough-and-Ready Formula

brightness_value = (0.22475*R^2.235 + 0.7154*G^2.235 + 0.05575*B^2.235)^(1/2.235)

Thanks to Roland Miyamoto for pointing out to me the simplicity of this formula.

My Rough-and-Ready Formula in Javascript:

var inv_p=1/2.235;

var brightness_value=Math.pow(0.22475*Math.pow(R,2.235) + 0.7154*Math.pow(G,2.235) + 0.05575*Math.pow(B,2.235), inv_p);

BUT . . .

The gamma calculation that gives a result identical to what you get with Photoshop still gives a brightness R where R = G = B. Although the formula uses exponents, the brightness results come out linear as I explain above. To get a perceptual relative brightness we need a curved graph along the lines of the inverse gamma sRGB curve as shown above, see my page Colour Brightness Experiment where the formulas can be seen in practice and in graphic form and can be experimented with as desired.

The

*brightness weighted*option on my page Colour Brightness Experiment takes the greyscale values as calculated, from whatever method, and runs them through the*inv_gam_sRGB*function to get a curved graph on output for the relative brightness of grey.Now as I said above, I am not a mathematician, so there may be something I am missing here, but from where I am standing at the moment it seems that unless you get a curved graph of the inverse gamma type for relative levels of brightness, you will never be able to judge mathematically, whether a brightness of level A is going to be readable over a brightness of level B, for the reasons I touched on at the beginning of this long page.

## 11 comments:

This is great. We're trying to help our customers create email newsletters that are accessible for the visually impaired. I'm going to see if we can make use of your javascript to give a warning to designers and contributors when their contrast is insufficient.

Thanks for your work!

Hey there. Thank you for sharing your work.

I've made some experiments and I think the gamma-decompress, weighted sum then gamma-recompress method (method 6), shown here (http://www.roguish.com/blog/?p=775), seems to have the same result, or at least have no Just-Noticeable-Difference.

I think what Photoshop does is apply the summed weights when they're gamma-decompressed (it makes the sum in linear RGB), then recompresses them into sRGB.

Kryzon thank you everso much for pointing me towards the StackOverflow thread. I have amended this page to reflect the links between my formula and the inverse-gamma adjusted values. I have also done a page that shows the greyscale converted value for any colour by each of nine different greyscale conversion formulas: http://www.casamatita.co.uk/nineshades

In your formulae for b1 and brightness_value, first dividing and then multiplying by 841.685 cancels out against one another. Therefore, you would arrive at the same result for brightness_value if you omitted both. Then you have the simpler formula

brightness_value = (0.22475*R^2.2155 + 0.7195*G^2.2155 + 0.05575*B^2.2155)^(1/2.2155).

Thank you extremely much Roland Miyamoto, that's excellent, I can now move forward with my experiments so much more easily.

I shall be updating this page soon to incorporate then simplified formula. (I did say I wasn't a mathematician!)

Hi Dave. I am indeed a mathematician. If you are willing to adjust a misconception in your above text, saying that "...any type of display screen does ... get brighter exponentially" is mathematically incorrect. Exponential growth would mean that the variables, here R, G and B, occurred in the *exponent* of your brightness formula, which they don't. One *could* say that the growth is "non-linear" or "super-linear" or "polynomial" or "a power growth with non-integer exponent", but it is certainly not exponential.

It is a common misconception - caused by narrow maths teaching at school (I know this because I do teach maths at school) - that many people nowadays think, growth should be either linear or exponential, and that there is nothing in between and maybe even no other type of growth at all. The truth is that we distinguish infinitely many types of growth, infinitely many types slower than linear (e.g. logarithmic), infinitely many types between linear and exponential (e.g. quadratic, cubic, quartic, etc) and even infinitely many types faster than exponential (e.g. doubly exponential, etc.). Brightness depending on R, G, B is, as your research has disclosed, a growth type somewhere between quadric and cubic.

Many thanks again, Roland, for helping me out so much with this. I'm going to do a major revamp on this page just as soon as I've got my head round some experiments that I am doing. I'm finding that the constants in that formula: the R, G ad B factors and the exponent of 2.2155 (if I got my terminologies right there) don't have to be so fixed. Further fiddlings with the figures and then a writeup, just a few other coding jobs to get out of the way first. And thanks again for your inspiration.

Updated this page on Friday 12 February 2016 to reflect all the excellent and useful comments and to give a formulas that really, really work.

Thank you, Dave, for your good work!

Hi David, Thanks for doing all this work I was looking for how Photoshop does it conversion and you saved me a lot of time.

I have used you formula for conversions but have a question. In the section "The like-Photoshop Formula" an example is given under the code "brightness_value = gamma_grey(R, G, B, 0.2235, 0.7154, 0.0611)" later in the post, in the section "My Rough-and-Ready Formula in Javascript:" a simpler version of the code is given. However it appears that the values for rY, gY, bY have change from 0.2235, 0.7154, 0.0611 to 0.22475, 0.7154, 0.05575 and the two formulas (obviously) give slightly different results. My question, is this a typo or have the values changed due to your work? I only ask as you note an update.

Yes the factors and exponent are slightly different for my rough-n-ready formula than the gamma calculation factors and exponent. I think that the explanation may be that, though the two formulas work in different ways, they happen to come close on results, with those numbers. If you look at my page http://landofinterruptions.co.uk/colourbrightness you can try both a gamma and a rough-n-ready formula using the different factors (click on Custom and enter your figures) and if you find something coming closer to the Photoshop gamma than I've so far devised, then I'd love to hear about it.

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