The weighted method is well-known and considered good.
I have used a fast method in the past which is not averaging: Just take the green channel. It is brutal, but it’s fast and a surprisingly good proxy for luminance.
I really can’t tell the difference. Switching between the two grayscale images I can tell, but going from the color to grayscale, I can’t tell which formula is being used.
Nice quick look at at perception and color. I was wondering if Omar would address the relative color balance across the 3 color channels and he did!