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In doing research on vision, I have learned that "20/20" vision corresponds to a visual acuity of being able to resolve details 1 arcminute in size, that most people have around 20/15 vision, and that due to the limits of physiology basically nobody has vision better than 20/10 vision. This is an upper limit of resolving details about 0.5 arcminutes in size.

According to Wikipedia the moon is around 30 arcminutes wide when seen by the naked eye.

Put these together, and it seems to say that when looking at the moon with the naked eye nobody can see more details than would be visible in a 60×60 image of the moon
Moon on black background, 60×60 pixels
and that the average person can't see any more details than in a 40×40 version
Moon on black background, 40×40 pixels

Those seem so small on my monitor. Can that really be all the detail that I can see on the moon with the naked eye?

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4 Answers 4

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It doesn't seem so far-fetched to me. Sure, you might be off by a few pixels, due to differences between the human eye and a computer monitor, but the order of magnitude seems about right — the detail in your images, viewed closely, more or less matches what I see when I look at the full moon.

Of course, you could fairly easily test it yourself: go outside on a dark night, when the moon is full, and see if you can spot with your naked eye any details that are not visible (even under magnification) in the image scaled to match your eyesight. I suspect you might be able to see some extra detail (especially near the terminator, if the moon is not perfectly full), but not very much.


For a more objective test, we could try to look for early maps or sketches of the moon made by astronomers before the invention of the telescope, which should presumably represent the limit of what the naked human eye could resolve. (You needed to have good eyesight to be an astronomer in those days.)

Alas, it turns out that, while the invention of the telescope in the early 1600s brought on a veritable flood of lunar drawings, with every astronomer starting from Galileo himself rushing to look at the moon through a telescope and sketch what they saw, very few astronomical (as opposed to purely artistic) drawings of the moon are known from before that period. Apparently, while those early astronomers were busy compiling remarkably accurate star charts and tracking planetary motions with the naked eye, nobody really though it important to draw an accurate picture of the moon — after all, if you wanted to know what the moon looked like, all you had to do was look at it yourself.

Perhaps this behavior may be partly explained by the prevailing philosophical opinions at the time, which, influenced by Aristotle, held the heavens to be the realm of order and perfection, as opposed to earthly corruption and imperfection. The clearly visible "spots" on the face of the moon, therefore, were mainly regarded as something of a philosophical embarrassment — not something to be studied or catalogued, but merely something to be explained away.

In fact, the first and last known "map of the moon" drawn purely based on naked-eye observations was drawn by William Gilbert (1540–1603) and included in his posthumously published work De Mundo Nostro Sublunari. It is quite remarkable how little detail his map actually includes, even compared to a tiny 40 by 40 pixel image as shown above:

William Gilbert's map of the moon The moon, scaled down to 40 px radius and back up to 320 px
Left: William Gilbert's map of the moon, from The Galileo Project; Right: a photograph of the full moon, scaled down to 40 pixels across and back up to 320 px.

Indeed, even the sketches of the moon published by Galileo Galilei in his famous Sidereus Nuncius in 1610, notable for being based on his telescopic observations, are not much better; they show little detail except near the terminator, and the few details there are appear to be inaccurate bordering on fanciful. They are, perhaps, better regarded as "artist's impressions" than as accurate astronomical depictions:

Galileo's sketches of the moon from Sidereus Nuncius (1610)
Galileo's sketches of the moon, based on early telescopic observations, from Sidereus Nuncius (1610), via Wikimedia Commons. Few, if any, of the depicted details can be confidently matched to actual lunar features.

Much more accurate drawings of the moon, also based on early telescopic observations, were produced around the same time by Thomas Harriott (1560–1621), but his work remained unpublished until long after his death. Harriott's map actually starts to approach, and in some respects exceeds, the detail level of even the 60 pixel photograph above, showing e.g. the shapes of the maria relatively accurately. It is, however, to be noted that it is presumably based on extensive observations using a telescope, over several lunar cycles (allowing e.g. craters the be more clearly seen when they're close to the terminator):

Thomas Harriott's lunar map, c. 1609 The moon, scaled down to 60 px radius and back up to 320 px
Left: Thomas Harriott's lunar map, c. 1609, based on early telescopic observations, via Wikimedia Commons; Right: same photograph of the full moon as above, scaled down to 60 pixels across and back up to 320 px.

Based on this historical digression, we may thus conclude that the 40 pixel image of the moon, as shown in the question above, indeed does fairly accurately represent the level of detail visible to an unaided observer, while the 60 pixel image even matches the detail level visible to an observer using a primitive telescope from the early 1600s.

Sources and further reading:

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Great answer, ideas, and research! –  Phrogz May 12 at 2:58
    
An excellent answer to the original question and very convincing comparisons, thanks. –  Patru May 12 at 7:58

Yes and no.

Yes, it's true that the apparent size of the Moon is 30 arcmin. It's true that the visual acuity of most people is 1 arcmin. So it's true that if you take the angular size of the smallest detail you can see on the Moon, and you put a bunch of those lined up straight in a row, you could span a Moon diameter with only a few dozen of them. In that sense, you are correct.

However, when you try to reproduce the situation on a computer screen, the comparison breaks down. First off, the eye doesn't see in "pixels". Like most optical systems, there's a point-spread function, that takes very tiny details and smears them up to a larger spot. The resolution of the eye is not the pixel size, but the size of the bell curve coming out of the point-spread function, and that has soft edges and is round, and it's everywhere and it's not fixed.

You assimilate the size of that larger spot with the size of a pixel on a digital screen, in your comparison. But that's not the same. The pixel grid in those thumbnails is fixed, so whatever falls between pixels is lost forever. Aliasing intervenes and creates artifacts that are not there in the original image. The dynamic range of the monitor is not the same like the dynamic range of the eye (the eye is much better). Color and brightness levels on the monitor are discrete, whereas the eye sees them as a continuum. Finally, the visual center in your brain is like a powerful computer that applies intelligent correction algorithms to the live image.

The list goes on and on. The bottom line is - all these effects combine and allow you to perceive a live image that is slightly more rich than those dead, frozen thumbnails that you posted. Not a whole lot better, but a little bit better. It's not like the eye can "work around" limitations, but it's more like you lose too much when you shrink a large image into a tiny fixed pixel grid on a computer screen.

It's very hard to reproduce reality on a computer screen. A much better way would be to take a 2000px by 2000px image of the Moon, put it on a big super HD monitor, and move it back to the point where the apparent size of that image is 30 arcmin. I know that doesn't sound satisfactory in the context of your original query, but it's a much better simulation.


Similar problems appear whenever you try to map the resolution of any continuous optical system (like a telescope) to a fixed digital grid (like a camera).

Let's say you're using a sensor with a pixel size of 4 microns. Let's say your telescope has a linear resolution in prime focus equal to 4 microns. You might be tempted to say - great, the sensor matches the telescope, right?

Well, not really. When that happens, you actually lose a bit of resolution. The image is good, but it's a little bit softer than it really should. See below an image of the Moon I took a while ago, with a system having exactly the parameters indicated above.

You can tell it's a bit soft, it's not really down to the pixel. Turbulence also plays a role, but part of the problem is that linear resolution is equal to pixel size.

Click the image below and open in new tab; if your browser shrinks it again to fit the window, left-click the big image to expand to full size - you must do this to see the full resolution image and notice the effects I'm talking about. The fuzziness is not visible on this small version here:

One way around that phenomenon, as an example, is to blow up the image in the telescope with a barlow until the linear resolution in prime focus is much greater than the camera pixel size, maybe 4x bigger. You do all your processing, and then you shrink it back, if you like, and you'll get a sharper image. Combine it with stacking multiple frames, and the overall quality can get pretty close to 100% the theoretical performance of the telescope.


TLDR: Continuous optical systems, and discrete grids of pixels, are very different things and cannot be easily compared.

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Very nice and clear explanation. Wish I could give you more than +1. –  Tonny May 9 at 20:26
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Why stop at a 2000×2000 image? Why not make a 4k×4k image and move that farther away? At some point the extra pixels are adding zero perceived detail to the observer. While 120×120 may add subtle details over 60×60, does 240×240 add actual observable details beyond 240×240? I'm guessing not. You're right that the eye is not a digital system, but there are discrete cones gathering light, and Nyquist does have a say in how much information they can actually pull in at some point. –  Phrogz May 10 at 14:22
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This is wrong. According to the Nyquist sampling theorem‌​, to model a waveform with a frequency cutoff, you should sample at twice the cutoff and then low-pass filter the reconstructed result. In other words, it's a 120x120 image blurred according to the ideal point spread function in the illustration. –  Blackbody Blacklight May 11 at 15:02
    
@BlackbodyBlacklight Thank you for the details. It's been so long since Nyquist and I shook hands that I had forgotten about the "twice the frequency" bit. (Though, that may be what takes the upper limit from 1 arcminute to 0.5 arcminutes.) Anyhow, my point in invoking Nyquist's name wasn't that 60 pixels is the correct limit, but rather that there is some limit (presumably lower than 2000). –  Phrogz May 12 at 2:58

When you gaze at the moon "live", you are not seeing a still image. You're seeing a "video": your retina is gathering multiple images over time. Those pixels have to be taken into account; they amount to extra pixels.

Suppose that 60x60 pixel images are taken of a scene using a tripod-mounted camera which slightly jitters. From the multiple images, a higher-resolution image could be reconstructed.

Have you ever noticed how a sharp-looking video can appear blurry when paused or stepped frame by frame?

As an aside, another thing to remember is, that a pixel is not a unit of information; not unless you specify how many bits encode a pixel. Suppose you sample 60x60 points, but with continuous amplitude resolution, and zero noise. The 60x60 pixel image then contains infinite information (though, of course, its ability to resolve adjacent details is still limited).

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This is an excellent point. Even if your eyes aren't moving, the atmospheric shifts are certainly lensing in different details. –  Phrogz May 10 at 14:18
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The "gathering of multiple images" are saccades. Each is a single high resolution snapshot that the brain composites into a single image. For every perceived instant of image, you take over a dozen snapshots. –  TechZen May 10 at 17:00
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Pausing a video will reveal either VHS or digital compression artifacts. "Sub-pixel" eye vibrations would already be accounted for in any visual acuity test. Taking advantage of atmospheric lensing, or moments of good seeing, is the domain of adaptive optics and I wouldn't assume the brain is capable of that sort of processing. –  Blackbody Blacklight May 11 at 14:59
    
@Phrogz - the "atmospheric shifts" are called seeing. Seeing is never a limiting (or enhancing) factor for naked eye observations. The only visible effect that way is the twinkling of the stars, but that's it. –  Florin Andrei May 13 at 0:12

After all these astronomic answers, I will add a computer one.

Pixels are not the same on all monitors. Take a 1990's monitor and take the latest smartphone screen, the 60 pixels won't be the same.

How did you calculate the pixel size according to the vision accuracy ?

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You are right, how you view those pixels matter if you want it to look roughly the same as the moon. You would need to see the 60 pixels on a screen at around 100-120ppd, for example a 27" monitor seen from 6 feet away, or a 50" HDTV seen from 12 feet away. Try my calculator. (Note: does not work in IE, and the SVG diagram currently looks bad in Firefox. Use Chrome or Safari for best results.) The question was, though, not how to make it look just like the moon, but how much detail there is when you see the moon with a naked eye. –  Phrogz May 13 at 12:48

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