# how formula for saturation component is working in RGB-HSI conversion?

As shown in attached snapshot,chap6 of gonzalez 3rd edition

How the highlighted expression for S in eq 6.2-3 is working? and what will be range of values of S? By range,i mean,maximum and minimum values of S. and by working of equation , i mean how the second term of equation and specially the part [min(R,G,B)] is working and typically what will be the values here? Is it possible that all R,B and G have '1' value at same time here in this case? • The range is given by that formula; it's not clear what you're asking? May 9, 2020 at 12:03
• OK, so where's the problem calculating the range? you're not telling which ranges R,G and B have, but I'm pretty sure you're able to consider S a function of e.g. R with G, B fixed, and at least start with an approach here? Finding the maximum for the two cases "R is smaller than G and B" and "R is larger than G or B" is really not that hard with high-school math. May 9, 2020 at 13:14

Let's say they are all at their maximum values of 1. $$min(1,1,1)$$ is equal to 1, so the equation will simplify to $$S = 1 - (3*1)/3 = 0$$. This makes sense because if you have all three colors at full then the resulting image will appear white, black or gray. This could only be the case with no saturation. So 0 is the minimum value for S.
Now lets do an example where one or two of the colors are off, say red and green. Lets also say blue is some non zero value because is all three channel values were off we wouldn't have an image to begin with. $$min(0,0,1)$$ is equal to 0, so the equation will simplify to $$S = 1 - (3*0)/1 = 1$$. This lines up with the idea that as long as one channel is off, you can't make muted colors like white, black, gray, or gray scale version of other colors. So 1 would be the maximum value that S could be.