Equations to convert xyY to RGB

Evidently 6 years of college engineering has left me mathematically inept. Does anyone have equations to convert to RGB from xyY? I tried deriving them using the xyY->XYZ and XYZ->RGB equations from Bruce Lindbloom and Poynton's sites, but there is either some step I'm missing or I'm just plain stupid. If you have them, please note if they are normalized or for a specific range.

I need these in reference to AVSHD. There have been some reports that the saturation windows are incorrect, and I'm trying to check them. The RGB triplets for the windows was given to us by the HCFR guys best I remember. The patterns seem to hold Y constant while only varying saturation. I wanted to take a known xyY coordinate (e.g. 25% Red @ x = 0.4764, y = .3295), determine the constant Y that the HCFR guys came up with, then use all 3 (xyY) to compute the RGB triplet and see if it matched the triplet we used to author the patterns.

Equations need to be for D65 Rec. 709.

Thanks for the help Tom. If you don't mind, let me get clarification on a couple things.

1) The first matrix, am I to dot (compute the dot product) of that matrix with my xyY coordinate, and that will give me XYZ coordinates, correct? I got confused with it because it had RGB out to the left. Just to make sure I have it correct, the first equation would be:

X = (X0*x) + (Y0*y) +(Z0*Y)

Seems like I have that wrong just looking at it. Please let me know. The second matrix is the same one Poynton has, just inverted, so I at least had that part close.

2) Please see if the following seems to be a logical approach for validating 50% red:

RGB values we used for 50% red sat pattern: 190,95,95 (video range 16-235)

The goal is to verify that this RGB produces the correct chromacity for 50% red @ some unknown, arbitrary Y as chosen by the HCFR guys. I have to solve for the unknown Y first.

Take the R equation with x,y being known for 50% red saturation. Normalize the R component (190) by the video range (219), e.g. 190/219. Now R, x, y, are all known leaving Y as the only unknown. Solve for Y, which should be constant from here on.

Use the known x,y,Y (Y obtained from step above) to compute G and B for 50% red. If everythig is kosher, G and B should come out to 95,95 to match our original values above. If not, we at least know something is wrong.

Does this seem like a logical approach to validate the RGB? I'm almost positive that the encoding is correct as a reference decoder spits out the correct values.

P.S. - I don't know why it was chosen to hold Y constant, unless chromacity starts to interact with Y for some displays. I'm sure there was a good reason, those HCFR guys are pretty smart.

I think we are talking past each other too. The matrices you provided are just in a little different format than what I'm used to seeing/using, but no problem. Let's see if this helps. Let's break it down onto 3 equations from the 3x3 matrix, going in reverse:

1) From the second matrix (XYZ->RGB) we should have the following 3 equations:

R = (3.24156*X) - (1.53767*Y) - (0.49870*Z)
G = -(0.96920*X) + (1.87589*Y) + (0.04155*Z)
B = (0.05562*X) - (0.20396*Y) + (1.05686*Z)

This should give us normalized (0-1) RGB values that will have to be scaled to whatever range we are using. We can agree on this part, right?

2) Now for part 2. I don't know the Y value, only x and y for the pattern I wish to check (50% Red, 190,95,95). Evidently this Y was selected by the HCFR guys. Because of this, I can't really go straight from (xyY->XYZ) using the first matrix you provided. What I can do is use the set of equations you provided and substitute in the RGB equations above:

Equations for xyY->XYZ
X = x(Y)/y
Y = Y
Z = (1-x-y)Y/y

Substituting into equations from step 1:

R = (3.24156*x(Y)/y) - (1.53767*Y) - (0.49870*(1-x-y)Y/y)
G = -(0.96920*x(Y)/y) + (1.87589*Y) + (0.04155**(1-x-y)Y/y)
B = (0.05562*x(Y)/y) - (0.20396*Y) + (1.05686**(1-x-y)Y/y)

Does everything look ok so far?

Now I need to solve for Y because, as stated above, I don't know it. Just looking at the first equation for R:

R = (3.24156*x(Y)/y) - (1.53767*Y) - (0.49870*(1-x-y)Y/y)

I know R,x,y but Y is unknown. One equation and one unknown so I can solve for Y. R for this equation will be the normalized value from the pattern we are checking (190/219) = 0.86758.

Using this information we can solve for Y. Now, we want to make sure that 190, 95, 95 is indeed the correct triplet for 50% red sat. To verify, calculate G and B by using the known x,y and the solved for Y:

G = -(0.96920*x(Y)/y) + (1.87589*Y) + (0.04155**(1-x-y)Y/y)
B = (0.05562*x(Y)/y) - (0.20396*Y) + (1.05686**(1-x-y)Y/y)

G and B should work out to the normalized value of (95/219) = 0.43379. If they don't, then the RGB triplet of 195,95,95 is wrong for 50% sat red.

Hopefully I have explained my method a little better this time; sorry for all the confusion the first time. Do you see anything wrong with what I'm doing?

We are taking the display out of it altogether. We know the theoretical chromacity for 50% red sat... the question is whether or not the RGB triplet we are using to author the pattern is correct (195,95,95) to represent that chromacity.

In a sense, yes. We are just dealing with the theoretical math, so everything is perfect.

You are right about 8 bit RGB, but it would be true for any discrete system. The error just becomes smaller as you increase the bit depth. Pretty much every pattern we produce has some rounding.

The Y stuff isn't important as far as I'm concerned (maybe it is to alluringreality, but I doubt it). We were basically using what we were given to match HCFR.

However, I don't think we want to go changing things on the disc until we can verify that they are incorrect. I'm not knocking your measurements, but we have to find if there is a problem first, and then where the problem is in order to fix it.

I'm just trying to check our RGB values using the theoretical math.

I'm pretty sure that Lindbloom (who I also played around with) uses PC levels. Maybe I'm thinking about it wrong, but 100,50,50 is different for PC level 0-255 than video level 16-235.

In other words, when you use Lindbloom's calculator, you are getting an x,y output based on 0-255.

Maybe I'm crazy. At this point I have looked at it too long and need to take a break.

EDIT: Nevermind, just saw you can use it with normalized 0-1. I got the same xy you got with 1, .5, .5 for RGB (Scale is not checked).

Tom,
I just tried normalizing the 50% red we are using (190,95,95) and put those numbers into Lindblooms calculator. Basically use 190/219 for R, and 95/219 for Green and Blue (Scale box unchecked).

The resultant xyY is much closer to the Accupel/Lindbloom numbers you give, indicating that (190,95,95) may actually be a correct triplet for 50% red sat. It appears that all of us may be calculating the target x,y incorrectly. I was doing it the same way you were, by just finding the 25% distance along the saturation line. I have no idea why that would be the wrong way to do it, but it looks like it is Accupel/Lindbloom/HCFR against us... I hate to bet against us, but...

Casey

I don't either. Are you going to find out for us?

I think the issue that Dan brought up is probably the reason the HCFR guys chose to keep it constant. I'm sure they knew exactly what the xy targets should be and how to get the correct RGB, so holding Y constant for them may have been trivial.

At this point I'm stumped on the xy coordinates. It seems from the HCFR help file that we are doing the same thing that they are. I feel fairly sure that the different methods we are using (Lindbloom/Accupel/Our calcs) are all using the same white point and primaries to at least 3 digits precision, so I don't know how we are all calculating them differently.

Dan,
Check your calculated targets against the targets Tom posted in post 17. I believe they are different. We are trying to figure out why. Bruce Lindbloom's calculator gives those xy as the correct targets for Red.

Dan,
Tom's Accupel and Bruce Lindblooms calculator both suggest the following targets for Red:

75% x0.574, y0.330
50% x0.456, y0.329
25% x0.366, y0.329

You can check out Lindblooms calculator on his site. If you input 1, 0.5, 0.5 (which is what Tom is using with his Accupel) in the RGB box (leave scale unchecked) then you will get the 50% value above. Likewise, if you take the 190, 95, 95 RGB triplet used to master AVSHD 50% Red Sat pattern and normalize it (e.g. 190/219) and plug that into the calculator, you will get the same numbers. This means that 190,95,95 that is used for AVSHD gives the same xy as RGB= 100%, 50%, 50%, indicating that AVSHD is correct.

However, the chromacities don't line up with what we are calculating. I think Tom and I are calculating the same numbers you are. The only way we are really going to figure this out is to use the equations that govern xyY->RGB and hand check the values.

If you look at one of my previous threads, I derived a set of equations that should go straight from xyY-RGB. One way to check things out is to set RGB = 1, 0.5, 0.5 in those equations and then solve the 3x3 matrix for x,y, and Y to get the 50% Red Sat targets.

Here are the equations if you want to solve them for us and see what you get:

If solving those gets too hairy you can always do it in two steps (this might be a lot easier now that I look at it). Revert to the XYZ<->RGB equations first to calculate the XYZ, then convert XYZ->xyY. Either way it is a system of 3 equations/3 unknowns.

XYZ<->RGB:
XYZ<->xyY:

I don't know at this point Dan. I have a feeling gamma corection has something to do with it, and that is why you see a non-linearity. Tom just posted some good info that we can experiment with.