There are four pieces involved when blending:
- Source color
- Coverage information (alpha)
- Destination color
- A working color space
We can all agree on what coverage is. You simply model each sample as a square, or rectangle, or circle, and decide on how much of that area is covered by the foreground. Therefore, it is fractional and unitless (because it’s a ratio). It can never be greater than 1 or less than 0. It is associated with the foreground because it represents the geometry in the foreground. You may have more than one value per sample (for example, if you are interested in each sub-pixel individually).
The working color space is the color space that our blending computations are performed in. This color space must be a linear color space. This means that, if you have a number which represents once channel of a color, and you double it, its distance from zero must also exactly double.
sRGB is not a linear color space. However, we can come up with a conceptual “linearized sRGB” color space which uses the same primaries as sRGB and same 0-point and 1-point, but uses linear interpolation between 0 and 1. Converting from sRGB into this new color space is simply raising each value to the 2.2 power. (The conversion is actually a little more complicated than that - it uses a peace-wise function - but we’re only discussing the conceptual model here.)
So, the first step is to convert all our colors into this working color space. Then, each blending operation is performed by taking a weighted average of the color primaries’ values, using the coverage information as a weight. Successive blending operations are performed back-to-front.
The formula for this weighted average is:
Source Primary * Source Alpha + Destination Primary * (1 - Source Alpha)
You can see that if the alpha is 0, the result is equal to the destination, and if the alpha is 1, the result is equal to the source. This is simply a linear interpolation between the two.
Now, it turns out that the requirement of rendering items from back-to-front is greatly constraining. It means that if we have a whole bunch of items to blend together, we can’t precompute the result of blending certain items together, and then blend those with a background. This is because the formula above is not associative.
However, a very similar formula is associative:
Source Primary + Destination Primary * (1 - Source Alpha)
The only difference between this formula and the original is the replacement of “Source Primary * Source Alpha” with “Source Primary.”
Well, let’s come up with a new concept, called a “premultiplied color.” This is the same thing as a regular color, except the values in the primaries’ channels have already been multiplied by the alpha of the color. This is possible because the color primaries’ values and the alpha channel have the same lifetime, so we can perform the multiplication at the time when this object is created.
Well, we can see that if we use these objects in the associative formula, we get the same answer as before (because the new “Source Primary” is equal to the old “Source Primary” times the Source Alpha; this multiplication is performed inside the “Premultiplied Color” object). However, we get the benefit of using an associative formula.
Therefore, with premultiplied colors, you can blend in any order. It’s worth noting that using premultiplied colors is not a requirement - if you blend out-of-order with premultiplied colors, you will get the exact same result as if you had blended back-to-front with non-premultiplied colors. This also means that you can start blending out-of-order, but if you notice that you have blended all the deepest items, you can transition to non-premultiplied blending halfway through all your blending operations. The answer will be the same.
By contrast, using a linear working color space is a hard requirement. If you don’t do this, your math will yield values which are meaningless. Once you’re done with all your blending, you usually want to write out the output in a well-known colorspace (such as sRGB), which means you usually have to un-linearize the result just before output.
Because of this, linearization / unlinearization should be the first and last steps. Premultiplication and unpremultiplication should be the second and second-to-last steps (if they are used at all). Premultiplication is optional and you can even unpremultiply halfway through your calculations if some conditions are met.
Note that linearizing / unlinearizing sRGB can have some pretty dramatic results. For example, if you blend pure black and pure white (technically “sRGB black” and “sRGB white”) with 50% alpha, you end up with your (resulting sRGB) primaries having values of 74%, nowhere near the 50% you would get if you performed the same calculation (incorrectly) in the non-linear sRGB space.