# Repeating alpha over/under operation multiple times

I have two RGBA colors in linear [0..1] space, src and dst and I want to alpha blender src over/under dst n times. Of course, I could call the respective operation n times using the result from the previous one (which is what I am currently doing), but I want to optimize it into a single operation. I think just calculating the correct alpha on src should be sufficient, but I may be wrong. Here's what I currently use:

public static final Color alphaOver(Color dst, Color src) {
float src1A = 1 - src.a;
float outA = src.a + dst.a * src1A;

if (outA == 0)
return Color.TRANSPARENT;
return new Color(
outA,
(src.r * src.a + dst.r * dst.a * src1A) / outA,
(src.g * src.a + dst.g * dst.a * src1A) / outA,
(src.b * src.a + dst.b * dst.a * src1A) / outA);
}

public static final Color alphaUnder(Color dst, Color src) {
return alphaOver(src, dst);
}

public static final Color alphaOver(Color dst, Color src, int times) {
Color ret = dst;
for (int i = 0; i < times; i++)
ret = alphaOver(ret, src);
return ret;
}

public static final Color alphaUnder(Color dst, Color src, int times) {
Color ret = dst;
for (int i = 0; i < times; i++)
ret = alphaUnder(ret, src);
return ret;
}


How can I optimize the last two functions?

So taking source over as an example, the math works out to:

dst = a * src + (1 - a) * dst

Taking Nathan Reed's suggestion of replacing (1 - a) with b, that gives us:

dst = a * src + b * dst

If we perform the next iteration, it becomes:

dst = a * src + ab * src + b^2 * dst

Doing it again, we get:

dst = a * src + ab * src + ab^2 * src + b^3 * dst

And then:

dst = a * src + ab * src + ab^2 * src + ab^3 * src + b^4 * dst

The pattern appears to be:

int exponent;
for (exponent = 0; exponent < numIterations; exponent++)
{
newDst += a * pow(b, exponent) * src;
}
newDst += pow(b, exponent) * dst;

• Are you sure to plug the result of the first alpha over into src of the next one? As I understand it, I would put it into dst instead. This lines up with some quick tests of the final equation with black src, white dst and an a>0. Overlaying black over white should darken the color but  + dst prevents this. – piegames Sep 7 '18 at 20:43
• I agree that substituting into dst is the correct thing to do here. If you imagine doing n separate blends on top of each other, the src would be the same for all, and result of each blend would be the following blend's dst. I think you should be able to simplify the result to a single blend using alpha of 1 - (1 - a)^n though. (Try defining a new variable b = 1 - a, and repeating the derivation you used for the first part of the answer.) – Nathan Reed Sep 8 '18 at 2:37
• Great idea! I've updated it with your suggestion. Let me know how it works. – user1118321 Sep 8 '18 at 5:30
• How about dst := (1 - b)*src + b*dst, and then rewrite that as src + b*(dst - src). Then it works out similarly to your original derivation. :) – Nathan Reed Sep 8 '18 at 5:48
• Ah, I see what you mean. I'll give that a try in the morning. I'm getting sleepy now and would probably screw it up. – user1118321 Sep 8 '18 at 6:18