Note: This paper provides additional information to the main PenTile RGBW topic in our Origin and Evolution of Computer Displays paper. A related document that may be of interest is our Color Vision paper.

Well, what a surprise – fancy seeing you here! So you're puzzled as well, are you? Well, worry not, because (as I pen these words) I spent a sleepless night trying to wrap my brain around this before all suddenly became clear. Let's start by reconsidering the first example from our main topic/paper, which involves fully lighting up the traditional RGB pixel in row 2 column 2 (the 100% values associated with the RGB pixel in the diagram below mean that each of its sub-pixels are fully activated and are propagating the maximum amount of backlight they can):

Before we proceed further, let's make a few assumptions (we'll revisit these assumptions in a little while). First, let's suppose that the aperture ratio of all of the RGB and RGBW sub-pixels is 1 (that is, we're assuming that their opaque periphery is negligible or non-existent). Also, let's assume that all of the RGB and RGBW sub-pixels propagate the backlight with no losses associated with the various colored filters.

Finally, let's assume that the cross-sectional area of each traditional RGB sub-pixel is 1 "unit", while – as discussed in our main topic/paper – the cross-sectional area of each PenTile RGBW sub-pixel is 1.5 "units" (for the purpose of these discussions we don’t care exactly how big these "units" are; our only concern is the relative sizes of the sub-pixels). On this basis, we might say that the traditional RGB pixel being fully on could be represented by the following equation:

That is, if each of the red (R), green (G), and blue (B) sub-pixels are turned on 100%, and each has a cross-sectional area of 1 "unit", then each is transmitting 100 "things" of backlight, which equates to 300 "things" of white light.

Now, let's assume that we wish to achieve an equivalent quantity of white light "things" using the sub-pixels in the PenTile matrix. (Why would we wish to do this? Actually we wouldn't – this is another point we'll return to in a little while.) Thus, we need to determine by what percentage we wish the various RGBW sub-pixels to be activated.

So how should we go about this? Well, first of all we know that we wish to achieve a total of 300 "things" of white light. Next, we know that we have 1 red (R) sub-pixel and one green (G) sub-pixel in the middle of the group, and we have four blue (B) sub-pixels to the north, south, east, and west of the group.

We also know that in order to form white light from red, green, and blue light we need equal quantities of the red, green, and blue. Thus, if we say that the four blue sub-pixels each have a value of U% (where 'U' stands for "unknown"), then the red and green sub-pixels will each have a value of 4U%. Finally, let's make the assumption that the four white sub-pixels will be activated by the same amount as the four blue sub-pixels:

Pulling all of this together, the following equation describes the case where the RGBW sub-pixels will achieve the same white-light output level as the traditional RGB sub-pixels:

From this we see that 'U' = 12.5%. And, of course, if we plug this value for 'U' back into our previous diagram, we end up where we started in the main topic/paper. That is, with the red and green sub-pixels in the middle of the group activated by 50%, while the four groups of blue and white sub-pixels surrounding the group are each activated by 12.5%:

As we previously noted, however, we wouldn’t actually want to use the sub-pixels in the PenTile matrix to achieve the equivalent output of the traditional RGB pixel. What would be the point of that? In the real world, we would double all of the values associated with the RGBW sub-pixels; that is, the red and green sub-pixels in the center of the group would both be activated by 100%, while the four groups of blue and white sub-pixels surrounding the group would each be activated by 25%.

In this case, if we continued to use the original number of backlight diodes, our display (of this one equivalent pixel) would be twice as bright as that of the original RGB pixel. Alternatively, if we were to halve the number of backlight diodes, our display (of this one equivalent pixel) would have the same brightness as the original RGB pixel while consuming only half of the power.

OK, before we proceed, just to make sure that we've all fully wrapped our tired old brains around this, consider our original illustration comparing the lighting up of the traditional RGB pixel in row 3 column 3 and the corresponding RGBW sub-pixels as shown below.

Your mission (should you decide to accept it) is to replicate the process we just went through to prove that the percentage values associated with these RGBW sub-pixels are correct (assuming we wish to achieve the same total number of white light "things" as for the traditional RGB pixel).

And finally (he said with a long, heartfelt sigh), let's consider what happens if we decide to fully activate both of the traditional RGB pixels in row 2 column 2 and row 3 column 2. In this case – as we indicated in our original topic/paper – we could achieve the same effect with the PenTile matrix by taking the values associated with the RGBW sub-pixels from the individual cases presented above and adding them together as follows:

Just a moment! Is someone trying to pull the wool over our eyes? The above certainly allows us to use the RGBW sub-pixels to achieve the same number of total white "things" as the two original RGB-sub-pixels. However, the four RGBW sub-pixels in the center of the group are shown as being activated by 62.5%. Doesn't this mean that there's no way we can double this value to achieve twice the brilliance or halve the number of backlight diodes?

And you wonder why I have sleepless nights trying to make sense of this stuff. The problem is that everything we've done thus far has involved simplifications. There were no tricks involved, because right at the beginning of this addendum we said: "Let's start off by making a few assumptions..."

This is like watching a magician perform his art, where "The quickness of the hand deceives the eye!" (Did you see what just happened to your wristwatch? Are you sure you're still wearing undergarments?). For example, we said: "Let's assume that all of the RGB and RGBW sub-pixels propagate the backlight with no losses associated with the various colored filters." In reality, the white (transparent) sub-pixels in the PenTile matrix experience significantly lower losses than the red, green, and blue sub-pixels. This boosts the efficiency of the PenTile matrix as a whole; in turn, this means that the various RGBW sub-pixels don’t need to be activated to quite the extent we portrayed above.

But the BIG kicker is where we said: "Let's suppose that the aperture ratio of all of the RGB and RGBW sub-pixels is 1 (that is, we're assuming that their opaque periphery is negligible or non-existent)." Ooooh, we are such scamps! In, reality, the aperture ratio is extremely significant. Also, it can vary considerably depending on the particular LCD technology. Having said this, in the case of a 2.4-inch VGA panel, the aperture ratio of the traditional RGB sub-pixels will be in the order of 0.35 (that's only 35%, which means that 65% of the available light is being blocked, even when the sub-pixels are fully activated). By comparison, in the case of an equivalent display panel based on PenTile matrix technology, the aperture ratio of the RGBW sub-pixels will be around 0.50 (that's 50%, which means that a significantly higher proportion of the light is getting through).

This is a HUGE difference. If we were to take these aperture ratios into account in our equations above, we would discover two things: first, our equations would become much more complicated (especially when we remember that the final value of each sub-pixel is "tweaked" as a function of surrounding sub-pixels as determined by Clairvoyante's sophisticated sub-pixel rendering algorithms); and second, it is possible to achieve twice the efficiency (that's twice the brightness or half the power consumption) using the capriciously cunning PenTile matrix technology from Clairvoyante.