The colorful world of tomatoes and lasers: keep it cool (and simple)
We started with a spectral power distribution of a light source – or reflected light from an object, e.g. bacon, lettuce, tomato – we intertwined it with the human visual response for each of the eye’s three receptors – called color matching functions – and we got three numbers: X, Y and Z.
Now, let’s look at how these numbers relate to what we humans readily perceive: brightness and color.
The lumen farm: all energies are equal - but some energies are more equal than others
As it turns out, the middle number Y is very important: the green curve in the middle of Figure 5 in the previous chapter (included below again) – also called V(), is conveniently chosen so that it measures our brightness sensitivity to each wavelength. So the tristimulus value Y is linked to the visual perception of ‘brightness’ of the object or image.
Figure 1 – The three color matching functions (CMFs) according to CIE 1931
Depending on the type of ‘power’ we consider from a light source – directional or ‘total’, across all angles and directions – using the process laid out above, we can derive either a quantity called ‘luminance’, measured in cd/m2, nits or foot-Lamberts (directional), or a quantity called ‘luminous flux’, measured in lumens – a very important projector metric!
Now note the position and shape of this green curve. We humans are most sensitive to light at 555nm: meaning that 1W of light power would produce maximum lumens at this wavelength (683 lumens to be exact). But go down – or up in wavelength, and the ‘lumen production’ decreases. 1W of blue or red light can produce only about 100 lm or less, depending on the wavelength, and you won’t be able to see 1W of IR or UV light at all, that is… zero lumens.
Why do we care when talking about projectors? Because power is power, whether you can see it or not.
So, in pumping watts of energy through the projector optics, lots of it will be absorbed or scattered and will heat up the optics but not all the watts that are projected produce a lot of lumens. In other words, we don’t only need projector lumens, but we need ‘smart’ (efficient) lumens – produced from the least amount of watts possible!
Keep it cool!
The dark side of the moon
Having a way to get from a myriad of colors to only three numbers is great.
The next step is to make this even simpler.
And what is simpler than 3 numbers? 2 numbers.
Remember that we used one of the three numbers to represent luminance (the ‘Y’). What about the other two? Math comes in handy here: starting from the tristimulus values X, Y and Z, we can derive two numbers called x and y (dubbed color chromaticities), in this way: x=X/(X+Y+Z) and y=Y/(X+Y+Z).
Conveniently, this transformation kind of warps the pure wavelength spectrum (Figure 2) onto a sort of ‘horseshoe’ – the black outline around the colored chart, called the ‘spectral locus’. But instead of one colored line, we now have a line enclosing a whole 2D surface. The spectral locus colors have maximum saturation, just like the wavelengths in Figure 2.
Inside this horseshoe you can find the mixed colors like yellow, orange, cyan, magenta (purple), or white, which have less saturation – the further a color is from the locus and closer to white, the less saturation. Outside the horseshoe is the forbidden realm. There is no color in this universe that can go out of this enclosure – those (x, y) values just do not represent any humanly ‘visible’ color. The whole (x, y) chart is called the ‘chromaticity chart’.
Figure 2 – An illustration of an x, y chromaticity chart. Note that you only see the colors that your display or paper allows you to see: in reality the colors towards the edges of this chart are much more saturated and pure. The wavelengths of the pure colors (in nm) are shown along the horseshoe line.
This graph is the most widely used ‘color chart’ in common industry literature. However it’s the most troublesome one as well. The problem arises when we try to express color differences (distance from one (x1, y1) color value to another (x2, y2) color value. It turns out that the CIE 1931 chart is completely non-uniform – the same numerical (x, y) difference can be either hugely noticeable or not seen at all, depending on the color region. So in 1976, color scientists proposed a new chart, called the u’v’ chromaticity chart. It’s also based on the CIE 1931 XYZ tristimulus values, however the ‘warping’ to the chromaticity values u’v’ happens. Figure 3 shows the details of this color space.
Figure 3 – An illustration of a (u’, v’) chromaticity chart. This one is more uniform, so that the same distance between any two u’v’ points anywhere in the chart is roughly perceived as the same visible ‘color difference’. We will only use this one from now on. It’s 40 years old, so it’s about time it reaches maturity…
What’s the catch? Because these charts miss the third dimension – brightness – you cannot see the dark colors like brown, dark green, grey, or black. They are, so to speak, ‘behind’ or below the chromaticity chart. So consider this chart as just the top view of the bright side – it also has a shadow, dark side below it. The complete color space is actually a three-dimensional thing, but we people love simple stuff – why else would you be reading this article? And the full moon is a circle, not a sphere, right?
So now we can finally relate language to (color) math. If somebody tells us “this color has (x,y) chromaticity values of (0.235, 0.710) and a brightness of 500 cd/m2”, we know that he’s talking about green, and a very saturated and bright one!
The bright side of projectors
What happens if we mix colors? Do we need to go back and measure the SPD of the new color, and calculate the X, Y, Z components from it? Fortunately not.
There is this theory called ‘Grassmann’s law’. To calculate the X, Y, Z values of a color consisting of two separate colors, you just have to add the X, Y, Z tristimulus values of the two mixed colors (so X=X1+X2, Y=Y1+Y2, and Z=Z1+Z2). So picking any two colors on the chromaticity chart, you can create all colors that lie on a line between them, by mixing the two colors in different proportions (intensities). And then apply the math to calculate the (x, y) or (u’, v’) values – and so you can predict the color of the mixed light!
What a wonderful tool! It also means that you can occupy a whole surface – a triangle within the (x, y) chart and produce all the colors that are within the triangle with just three different colors – typically red, green, and blue. And by mixing the red, green, and blue colors, we get (a kind of) white. I certainly hope you share my enthusiasm…
This ‘triangle’ on the chromaticity chart including its ‘dark side’ is called the ‘color gamut’ or ‘gamut triangle’, and the three colors at the triangle’s corners are called the RGB ‘primaries’.
How does a projector create colors?
Using a lamp or a laser light source, and suitable color filtering – if necessary – the projector can produce three distinct ‘primaries’: red, green and blue. These primaries are characterized by their brightness Y and chromaticity (x, y). Mixing all three of the primaries at their maximum power will produce the ‘native white’ of the projector.
This white might not necessarily be the white that we need – it could be too bluish or too greenish – so by applying electronic corrections, or by dimming the power of the distinct red, gree, and blue lasers in the case of an RGB laser projector, we can influence the r-g-b balance and thus shift the white point to where it needs to be. And by mixing the primaries in a certain proportion, we can produce all colors within the native projector color gamut – even those at the dark side! See figure 4 to see how we can make a khaki color.
Figure 4 – Color arithmetic for video displays. Mixing red, green, and blue in different proportions we can produce all colors within reach of the display’s color gamut, including khaki.
And due to the metameric principle, we are sure that – if we produce a color that has a certain X, Y, Z value, it will look the same as that very color from nature that has the same X, Y, Z value but probably a widely different spectral power distribution.
As you might start noticing now, the choice of primaries – or color gamuts – has always been a hot topic in the display industry. On the one hand, you want to go as wide as possible in order to represent reality faithfully – or even exceed it, creative minds have no limits. On the other hand, you have the technological limitations, price, availability, and other practicalities and tradeoffs.
About the author
Goran Stojmenovik is Senior Product Manager within Barco’s projection division and is currently working on laser projection for the cinema and other Barco markets. With focus on image quality as well as user experience, Goran has managed different products in Barco since early 2005. Initially he was responsible for professional LCD monitors and software solutions for various Barco professional markets (control rooms, broadcast and post-production). In September 2011 Goran started at Barco digital cinema where he worked on introducing dedicated projectors for post-production as well as on remote service solutions for cinema (CineCare Web). Before joining Barco, Goran Stojmenovik acquired a PhD degree in Engineering Physics at the Ghent University, Belgium. He is based in Belgium.