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<title>Kevin Boone: Snake oil: 4K displays on smartphones</title>
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<h1>Snake oil: 4K displays on smartphones</h1>
<p>
<img class="article-top-image" src="img/snakeoil.png"
alt="snake oil"/>
The Sony Xperia 1 IV has a 6.5-inch "4K" display with a quoted resolution
of 643 pixels per inch (PPI). I don't know if this is the highest screen
resolution currently available on a portable device, but it must be close.
Many current smartphones have QHD screens
-- resolutions of 500 PPI or more -- and the resolution race shows
no signs of slowing down.
</p>
<P>
Increasing the screen resolution of a portable device has significant
disadvantages. Not only are higher-resolution screens more expensive,
they use more battery power themselves, and require more CPU resource
to manage, which in turn uses more battery power. Given these
disadvantages, the increasing resolution would have to be associated
with a significant optical improvement.
</p>
<P>
But is it? Or is this just another marketing gimmick? Is there <i>really</i>
a need for 4K screens on smartphones?
</p>
<P>
It turns out that we can kind-of answer that question. The answer
is 'Um... depends; probably not'. To begin answering it, however,
we have to understand
the notion of visual acuity. We will also have to face up to
some basic algebra and trigonometry.
</p>
<h2>Visual acuity</h2>
<p>
To justify a higher screen resolution, it must be the case that the
existing, lower resolutions allow the user to see individual
screen pixels. Once we get past the point where pixels are indistinguishable
from one another,
further increases in resolution are wasted. Common sense suggests
that the <i>is</i> such a point; we can't keep increasing resolution
forever, and still see gains in display quality. But where is that point?
</p>
<p>
<i>Visual acuity</i> is the measure of the eye's ability to distinguish
two points of light that are close together. That is, it's a measure of the
eye's resolving power. For a person with perfectly healthy eyes,
actuity is fundamentally limited by the number of light-sensitive cells
in the retina: for two visible objects to be distinguished, they must,
as a minimum, stimulate different retinal cells. Acuity is fundamentally
an <i>angular</i> measure -- what matters is the angle between the two
objects that is made at the lens of the eye, which depends on how
far away they are. This is just common
sense -- objects that are further away are harder to tell apart,
because the angle they make at the eye is smaller.
</p>
<p>
For a display screen, this situation is shown in the diagram below.
A single pixel of size
<i>D<sub>p</sub></i>
is on a screen some distance
<i>D<sub>v</sub></i> from the viewer. The angle this pixel makes
at the lens of the eye is
<i>θ<sub>p</sub></i>. This is the same angle that would be made
between two pixels whose separation is
<i>D<sub>p</sub></i>.
</p>
<p align="center">
<img src="img/4k_1.png" width="650px"
alt="diagram showing pixel resolution"/>
</p>
<p>
In the interests of full disclosure, I should point out that the
diagram above, and the analysis that follows, assumes that the
eye is looking directly at the screen region of interest. Pixels
that are off this optical axis are a little further from the eye,
just by the principles of geometry. However, it's reasonable to assume
that the pixels of interst are the ones the eyes are actually
looking directly at.
</p>
<p>
From basic trigonometry, so long as
<i>D<sub>v</sub></i>
is much larger than
<i>D<sub>p</sub></i>
(and it always will be, unless you're looking at the screen with a
magnifying glass),
</p>
<p align="center">
<i>θ<sub>p</sub></i> =
<i>D<sub>p</sub></i> /
<i>D<sub>v</sub></i>,
</p>
<p>
so long as the angle
<i>θ<sub>p</sub></i>
is expressed in radians. We learned this in high school -- surely you
remember?
</p>
<p>
If we know how large a screen pixel is, and how far the screen is from the
eye, we can work out
<i>θ<sub>p</sub></i>. But how large must this angle be, for the
pixel to be seen as a separate object from other, nearby pixels?
</p>
<p>
For opticians, "normal" vision (sometimes expressed as "20/20") corresponds
to the ability to distinguish objects that are separated by an angle
<i>θ<sub>p</sub></i>
of one minute of arc. One minute is one sixtieth of a degree. This figure,
however, comes from the practice of testing whether a person
needs vision-correcting
lenses. <i>With</i> perfect correction (e.g., well-fitted spectacles),
it turns out that limit of human visual acuity is about 0.6 minutes. Of course,
there are some people with better acuity than this, but that's a typical
figure. 0.6 minutes corresponds to 0.6/60 degrees, or 0.000175 radians.
It's unlikely that many people have visual acuity radically better than
this, because we all have similar retinas.
</p>
<p>
So, returning to our diagram, two nearby pixels will be distinguishable
in all cases where
</p>
<p align="center">
<i>D<sub>p</sub></i> /
<i>D<sub>v</sub></i> > 0.000175.
</p>
<h2>Apply acuity limits to screen design</h2>
<p>
Of course, we don't normally measure screen resolution in terms of
angles -- this measure would change with
viewing distance. Usually we measure resolution in terms of
<i>pixels per inch</i> (PPI). Usually this figure is quoted in the most
generous way, by measuring across the screen diagonal. The diagonal
size of a pixel is about 40% larger than the height or width so, at the
limit, we might be able to distinguish pixels that are diagonally
separated even if we can't distinguish them when the are side-by-side.
</p>
<p>
If we express the diagonal pixel size
<i>D<sub>p</sub></i>
in inches, then the display DPI is just
</p>
<p align="center">
PPI = 1 / <i>D<sub>p</sub></i>,
</p>
<p>
and, with a little basic algebra, we can rewrite our previous,
limiting expression as
</p>
<p align="center">
PPI > 5730 / <i>D<sub>p</sub></i>.
</p>
<p>
What this expression says is that for two pixels to be distinguished,
the display PPI must exceed 5730 divided by the viewing distance in
inches. This is a fundamental principle of optics and the
physiology of the human eye, and applies to all types of screen.
Nothing about this is subjective to any significant degree. All
we need to know, to work out what screen PPI is optimal, is how
far from the eye the screen is.
</p>
<h2>What does this mean in practice?</h2>
<p>
Let's consider the PPI figures for some specific screen resolutions
and sizes. We are, as I mentioned earlier, interested in
the <i>diagonal</i> PPI here, which we can work out from the
vertical and horizontal resolutions using Pythagoras' theorem
(assuming the pixels are square). Of course, we might not need
to work it out -- it's the figure that manufacturers generally
quote for their products. The table below shows some
PPI figures for different display resolutions, assuming a
screen (diagonal) size of either 6" or 10".
The table also shows the furthest viewing
distance at which a person with perfect eyesight would be able
to see individual pixels.
</p>
<pre class="codeblock">
Type Resolution PPI, 6" screen Max view distance (inches)
---- ---------- -------------- --------------------------
FHD 1920x1080 367 16
QHD 2560x1040 490 12
4K 4096x2160 772 7
Type Resolution PPI, 10" screen Max view distance (inches)
---- ---------- --------------- --------------------------
FHD 1920x1080 220 26
QHD 2560x1040 294 19
4K 4096x2160 463 12
</pre>
<p>
So, for a 6", full HD (FHD) smartphone screen, the limit of human vision allows
pixels to be distinguished when the screen is closer than about
sixteen inches from the eye. When the resolution is increased to QHD
that distance is about twelve inches.
By coincidence, that distance is about how far away I hold my
phone when I'm looking at the screen -- I'm sure most people hold their
phones in roughly the same way.
</p>
<p>
What this means is that, for a person with <i>perfect</i> eyesight,
it's <i>just about</i> possible to justify using a QHD screen, in preference
to FHD or lower. By perfect eyesight I mean that the user should
have perfect focus (or perfect spectacles), with no floaters or cataracts,
and no retinal irregularities. My eyesight is far from perfect --
even with my eyeglasses.
</p>
<p>
With this reasoning, there's simply no reason to use a 4K screen,
unless the user has perfect eyesight, and will be holding the
screen 7" away or closer. Most people can't even focus at this
disance.
</p>
<p>
The situation is only a little different for tablets or laptop computers.
Their screens are larger than smartphones, so the same screen geometry
produces a smaller PPI figure. But laptops and tablets are usually
used at a greater viewing distance, so these factors somewhat
cancel one another.
For a tablet or laptop to benefit
(even theoretically) from a 4K screen, it would need to be used
no more than about 12 inches from the eye.
</p>
<h2>Is it as simple as that?</h2>
<p>
The human eye/brain system is not a camera. In some ways our vision
is better than theoretical predictions suggest. This is (in part)
because vision is not static -- we don't hold our eyes still like
a camera. Our perceived image is built up from many sensory impressions
made from slightly different parts of the scene, and with different
focus. For most of us, our perception come from integrating the
images from two eyes, which see the scene from slightly different
angles.
</p>
<p>
It's therefore possible that our visual acuity is better, in some
circumstance, than predicted
from geometry and physiology. For example, we humans are weirdly
good at determining
when two fine lines are parallel. Our ability to do this
exceeds the theoretical limit of optical resolution, and the reason
is not entirely clear.
</p>
<p>
On the other hand, the widely-accepted figures for visual acuity are
derived from tests on high-contrast, static images. It's almost certain
that our visual acuity is lower on multicolour, moving images.
On the whole, we don't scrutinize our smartphone displays for whether
we can see the pixels -- the subjective experience of image quality
depends on what we're actually looking at.
</p>
<p>
Moreover, image quality isn't determined solely, or even mostly, by
spatial resolution. We also have to consider brightness, colour accuracy
and gamut,
update smoothness, contrast, and other things. It isn't obvious
that increasing raw resolution won't have deleterious effects on these
other factors.
</p>
<h2>Conclusion</h2>
<p>
So -- 4K screens on phones: snake oil or not? It isn't that long ago
that a 720p screen on a smartphone was an innovation. Those of us who
had such screens when they first became available
were generally impressed by the image quality, and
didn't hanker for something better. But that doesn't mean that we
didn't appreciate 1080p screen when they came along.
</p>
<p>
But the step from 1080p to 4K is different -- we're now
in an domain where almost nobody will be able to perceive the improved
resolution at anything close to normal viewing distances.
There really doesn't seem to be any benefit, in the majority of
usage scenarios.
</p>
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