When does a screen have "too much" resolution?

(Note: This entry is a translation from the original post in Spanish, written back in Oct/2013. Re-posting it in English because I'll be writing some things shortly also in English, about the current trends in display technologies and the consumer TV market, OLED vs. LCD, HDR, wider Color Gamuts, and so on. That post will likely refer to things covered here, so wanted to have this text already in English.)

4K resolution, also called "4K Ultra HDTV" or "Quad HD" is the resolution offered by the latest-generation TVs. This resolution is equivalent to approximately 4x the resolution of Full HD or 1080 (see the image above.) 4K is 3840 x 2160 (or even 4096 x 2160 pixels), a little more than eight million pixels in total. Quite a few pixels! But is it really useful to have such a high resolution on our televisions? That depends on several things, and that is the topic of this post. The idea is to inform you so that you won't spend a fortune on something that possibly you will not be able to enjoy or take advantage of, in spite of what salesmen or even other consumers would want you to believe.

Consider first a cell phone, such as an iPhone 5. Its screen size being only 4 "(diagonal), and its resolution 1136 x 640 pixels. Note that this resolution is relatively" low "in the sense that it's not even the minimum HD (which is 1280 x 720.) But this "low" resolution in such a small screen results in a very high pixel density: 326 ppi (pixels per inch,) that's about 12.83 pixels per millimeter. In other words, the pixel size on this iPhone, assuming they are square, is only 0.078 mm per side (less than 8% of 1 mm the side of each pixel.)

As a marketing strategy, Apple gave a rather picturesque name to this pixel density of the iPhone: they called it retina display. The reason was that, in principle, our eyes, or our vision in general, cannot distinguish those pixels if we place the iPhone at a distance of at least one foot (30 cm) from our eyes. And without resolving the pixels, the image would then appear completely smooth rather than pixelated. What Apple is telling us here may actually be true or false, and that depends on our visual acuity.

In a previous post we saw that a person with "normal" or 20/20 visual acuity can distinguish between two parallel lines separated by one minute of arc, or arcminute. An arcminute is just 1/60th of a degree (and a degree is just 1/360th of a full circle.) An arcminute is thus a fairly narrow angle. How narrow? If we plot an angle of one arcminute starting from our eyes, the separation of the sides of that angle at a distance of six meters would be just 1.75 mm. (Remember, this is calculated using the circumference formula: [2 * pi * R / 360] / 60 = 1.75, where R is the radius of the circle, which in this case would be 6000 mm = 6m.)

About 30 cm away, the separation of the sides of an angle of one arcminute would be just 0.087 mm. Less than 9% of a millimeter. Ah! But there you go! Above we saw that the side of each pixel of the iPhone 5 has a length less than 8% of a millimeter, so in this case, pixels are a little smaller than what a "normal" visual acuity can resolve at a distance of 30 cm. That's the key! That's why in principle we can't resolve those pixels at that distance. Apple then did tell us the truth about the retina display, at least when a visual acuity no better than "normal" is assumed.

If you bring the iPhone close enough to our eyes, then you would distinguish the pixels, even if you have normal vision. (A 30-year-old can focus even at 15 cm, and a child can focus even at less than 7 cm.) And if we have a visual acuity higher than normal, then we would be able to resolve the pixels of the iPhone even at 30 cm.

We see that resolving or not the pixels of a screen with a particular resolution will depend on several things. Those things are precisely the terms highlighted above in bold, namely:

1) Pixel size (which is derived from the screen size and its resolution)
2) Distance between our eyes and the screen
3) Our visual acuity

The final effect on our eyes will depend on these three factors. We can assume that our visual acuity is already the best we can muster (using glasses if we need to, for example,) so overall we cannot improve factor #3. Then we can only modify factors #1 and #2. Modifying #1 means a different screen size, or a different resolution, or both. Modifying #2 means changing the distance between the screen and our eyes.

Clearly, if we can distinguish the pixels on a given screen, then either the resolution is too low for that distance, or we are too close to the screen given its resolution. The fact is that if we start moving our eyes away from the screen, at some point we will reach a distance at which we can no longer resolve the pixels. Only then, given that screen and distance, and our visual acuity, we could say that that resolution is "satisfactory".

But then again, when do we have too much resolution?
(Remember, this is the key question concerning this post.)

We will have too much resolution "A" when, for the same screen size, there is at least one lower resolution "B" that will also **not** let us resolve its pixels at the same viewing distance.

That is because, if resolution A is greater than B, but both resolutions at distance X on screens with the same size do not allow us to resolve their respective pixels, then at that distance the images of A and B are completely indistinguishable (in terms of resolution) to our eyes, no matter how finer resolution A is with respect to B. For that viewing distance, for that screen size, and for our visual acuity, resolution A would therefore be excessive and technically useless over and above B.

Let's elaborate a bit more.

Imagine we put many iPhone 5 screens together to build a single large 60" screen. That would require a lot of iPhones, in fact 15 x 15 == 225 iPhones. And do the math: the resolution you would get with that amount of screens (at 1136 x 640 per little screen) would be a wooping total of 17025 x 9600 pixels! That is more than 18 times higher than 4K. But ask yourself: would that be perhaps somewhat excessive and unnecessary? Well, given normal vision, we already saw that we cannot resolve pixels on any of those iPhones when our eyes are just 30 cm away. How much further from resolving them wouldn't we be, when this 60" screen has pixels of the exact same size as those on the iPhones, and we are to see them now from let's say three meters, so 10x times farther away?

In fact, a "normal" vision already **can not** resolve the pixels on a 60" screen with the "so much lower" 1920 x 1080 resolution (Full HD) from three meters away. Just getting closer to less than 2.38 m (7.8 feet) would allow you to begin resolving those pixels (this can be calculated similarly to what was already explained above.) So at distances beyond 2.38 m, no "normal" vision will reap any "benefits" from this Super Ultra Ridiculous resolution 18+ times higher than 4K on a 60 "screen, compared to a modest screen of the same size, with a simple 1080p resolution. Our eyes at that viewing distance simply cannot see the difference between these two resolutions.

That is a hyper-exaggerated example, but I hope the idea comes across. A resolution can be absolutely excessive and completely useless to our eyes compared to some other much lower resolution, depending on our visual acuity, and the viewing distance.

Now back to 4K.

A 60 "screen with a 4K resolution has quite small pixels. In fact, four of its pixels can fit inside one pixel from a 60" screen with 1080p resolution. At what distance can we resolve those 4K-60" pixels? Actually only at less than 1.19 meters (or ~3.9 feet; again, normal vision.) So you sit at 1.19 meters or farther away from that screen and you won't see any pixelated images; perfect! However, don't go sit beyond 2.38 m (7.8 feet) away from that screen, because then you will have paid that higher 4K resolution for nothing. As we saw above, beyond 2.38 meters you already wouldn't be able to resolve the much larger pixels on a 4 times lower 1080p resolution screen of the same 60" size. So if you are considering sitting beyond 2.38 meters away from a 60-inch TV, then it makes little sense to have it be 4K over 1080p, because a 1080p screen will look just as well at that distance (you won't even be able to resolve the pixels on the 1080p screen from that distance.)

What is more, if you sit beyond 3.57 m away (11.7 feet,) then it doesn't even make much sense to have a 60" 1080p TV, because at that distance you can no longer resolve the pixels in the 720p resolution (HD rather than Full HD) for that screen size. So all other things being equal, at 3.57 meters or more, a 60" 720p screen will look just as good (without pixelation) as a 1080p, and as a 4K the same size. Again, all this is assuming normal vision.

Of course, we would need to calculate things for each screen size, resolution, and every viewing distance possible to see if the combination works and makes sense or can be recommended for our particular needs. But I don't need to do that, because others have done it already (click to visit and enlarge):

One way to use this graph: first choose the viewing distance you are considering. For example, if it's three meters (~10 feet), then locate the value 10 feet on the vertical axis to the left, and draw a horizontal line across the entire graph at that height. Then check screen sizes on the horizontal axis below, and draw a vertical line from your screen size of interest, and see where it intersects that horizontal line you drew right before. Let's say, if you are considering 60" at 10 feet, the intersection between the two lines would fall near the tip of the red triangular area. Depending on the color where the intersection falls (blue, green red, or purple), a given combination will make sense or not according to the descriptions associated with that color (text blobs on the right, both for triangular areas and for the boundary lines between them.)

In our example, the intersection is on the red area, and the description for the red area tells us that the benefits of 1080p would be noticeable. That means, from 60" viewed at 10 feet we are ok with 1080p. But it also tells us, it would not be a combination that would let us benefit from 4K; we would need a larger screen, or a shorter viewing distance, or both, to move the intersection towards the purple area in order to do so.

This graph allows us then to respond fairly easily to the question on the title of this post: when does a screen have too much resolution? Answer: when the intersection between the viewing distance and the screen size falls outside (most likely above) the color associated with that screen's resolution. Note, for example, that only when the intersection falls below the red line, only then we would observe benefits from a 4K resolution.

I'm sure you will be surprised to realize how closely you have to sit from the screens (despite their large sizes) in order to truly reap the benefits offered by each resolution over the previous lower one. For example, beyond three or more meters away (10 feet or more,) a 50" 1080p screen hardly makes sense against a 720p, at least not for a "normal" vision. A 4K 60" screen would start to make sense only if you plan to see it sitting within midway between 5 and 10 feet, so about 7.5 feet, which is exactly the same distance we mentioned before: less than 2.38 m. But that would only be the distance at which normal vision "begins to notice the benefits" of 4K compared to 1080p. To really enjoy those benefits you would need to sit even closer to this large 60". Such close distances to large screens may be impractical, inconvenient, or simply uncomfortable for you. Or it may be the case that for your viewing distance of interest, the recommended ideal combination of screen size and resolution end up beyond budget.

By the way, we have only discussed here viewing distance with respect to resolving pixels. But there is something else which is very important to take into account when choosing the ideal viewing distance for a particular screen, especially when it comes to watching movies, and that is the horizontal angle that the screen covers on our field of view. SMPTE's recommendation is about 30 degrees, and any value between 28 ° and 40 ° complies with the THX recommendation (and certification.) In general, for home theaters, it is advisable to use at least 20°. Many consider this more important than resolving or not resolving pixels, because sufficient visual field coverage increases the effect of "immersion" into the film, whether it's pixelated or not.

Here's an example that could be used as some sort of reference. For a 50" 1080p (Full HD) display, any viewing distance between 1.98 and 2.97 m (basically two to three meters, or 6.5 - 9.7 feet) matches what would be the three key criteria for optimal viewing of movies:

1) We are far enough to not resolve the pixels on the screen (with normal vision)
2) We are in the range of distances where we are effectively taking advantage of our screen's higher resolution (in this example 1080p) over the immediately lower resolution (720p)
3) We are at a distance that allows the screen width to horizontally cover between 20º and 40º of our visual field 

For example, with a 4K - 60" screen, we would comply with #1 beyond 1.19 m, but we would need to sit no closer than 1.82 m to comply with #3. And as we saw earlier, we should not sit beyond 2.38 m to comply with #2. So it's important to realize how narrow the ideal range of optimal viewing distances gets for higher resolution screens. For a 60" 4K screen, it's between 1.82 and 2.38 m (between 6 - 7.8 feet). If we sit outside that range, we violate one or more of those three criteria above, and it would be best to change some of the variables at play: either the screen size, the resolution, or the simplest, our viewing distance.

For a giant 100" 4K screen, the ideal range complying with all three criteria would be between only three and four meters (about 10 - 13 feet.) In fact, a little narrower: between 3.04 and 3.96 m. But depending on your visual acuity, at 3.96 m (~13 feet) you are already risking not seeing any benefit from 4K over 1080p from a 100" screen. Better sit at the lower end of that range, just a little over three meters (10-12 feet). So yes, believe it or not, if your vision is normal, ideally you would sit just slightly beyond three meters away (~10 feet) from a giant 100" 4K screen.

In conclusion, there are cases in which resolution can be too high; effectively, unnecessarily, and uselessly too high, and it makes little sense to pay more for something that doesn't offer perceptible improvements over something cheaper. If you are looking for a TV or monitor, don't let all the fuzz from salesmen and even from other consumers fool you, about the alleged "remarkable and incredible benefits" any higher resolution is supposed to offer above lower resolutions. Take into account how far your eyes will be from that screen at your particular room and setup (that's possibly the most important thing,) and try to use the chart above (or the above formulas) and the three criteria mentioned here, to determine the combinations of resolution, screen size, and viewing distance that are truly convenient or even optimal for your particular needs and budget.

Additional Information:
- 1080p Does Matter - Here's When (Screen Size vs. Viewing Distance vs. Resolution)
- Resolving the iPhone resolution
- Optimum HDTV viewing distance (Wikipedia)
- 4K resolution (Wikipedia)

PS. The 4K resolution I mentioned as 4096 x 2160 is the one from Digital Cinema Initiatives (DCI.) 4K UHD (or Quad HD in the first image) is exactly equivalent to 4 times 1920 x 1080 (Full HD), or 3840 x 2160. In any case, they are very similar.

PS2. Refined some calculations on the post, and here taking the opportunity to explain an additional formula that might be useful. If you do not have the density of pixels per inch or per centimeter from the screen manual, you can get the pixel size by dividing the screen height by the number of vertical pixels on the screen. For HD that number is 720, for Full HD, 1080, and for 4K, 2160. To get the height of a screen, simply measure it, or from the diagonal use Pythagoras, knowing that the ratio of the screen is 16:9, that means the base of the screen is always equal to 1.777x the height. With this data and the diagonal D, we can easily calculate H the screen height: H = sqrt (D * D / 4.16). To convert that height from inches to millimeters, multiply by 25.4. For example, a 60" screen has a height of 747.2 mm, and pixels for resolutions of 720, 1080 and 4K on that screen would be 1.038 mm, 0.692 mm and 0.346 mm per side respectively.