How is resolution defined?

Question

So I'm looking at a 1 meter square object in my digital camera. What size of pixel is considered fine enough that the object is considered to be resolved? Obviously a pixel size of one meter squared will 'notice' the object, but it will likely affect four pixels in varying degrees, so we can hardly call it an image. A pixel size of 10cm will at least let me see that I'm looking at a square, but is that considered fine enough?

Answer 1

How is resolution defined?

Differently in different contexts, and even differently by different people within the same context. Some definitions are objective, expressed as an equation or a computational procedure, others may be more subjective.

We can always start with the grandparent of all resolution definitions, which is The Rayleigh criterion

Two airy disks at various spacings: (top right) twice the distance to the first minimum, (middle) exactly the distance to the first minimum (the Rayleigh criterion), and (bottom left) half the distance. This image uses a nonlinear color scale (specifically, the fourth root) in order to better show the minima and maxima. (cropped and rotated from source)

The middle image doesn't look as resolved as it should because the author chose to use the fourth root of brightness instead of a linear scale, but you can see a dark band between the two "stars" in the middle one even in this image.

These patterns are what diffraction from a circular aperture causes, either your pupil or a telescope or your camera's lens. It doesn't take into account your square pixels though, so it doesn't help.

What size of pixel is considered fine enough that the object is considered to be resolved?

I'd say that some form of

Beauty is in the eye of the beholder

can be used here. Astronomers push things to the max. Loosely speaking they can measure the apparent size of a "resolved" object if it's 6 pixels wide for example as long as all the other stars are no more than 4 pixels wide, but they'll do a thorough computer analysis first including optical blurring and simulations shifting the positions of the stars fractions of a pixel in every direction first to make sure that object is definitely bigger than all the unresolved stars in the field.

above: from @PeterIrwin's answer to Has Hubble ever been used to try to image a near Earth asteroid?

Resolved? Not?

Here's another case in point. It's from this answer to How big would a QR code have to be on my roof for a satellite to be able to scan it given today's allowable resolution? in Space Exploration SE.

It's the result of a simulation of a 6 meter pixel "QR code" (not exactly) on Earth's surface seen from a 9 cm aperture telescope in orbit 575 kilometers above Earth with the sensor's pixels having a resolution at Earth of 3 meters. I've used an Airy function to simulate the image, the same thing that generates those concentric rings in the first image here.

Resolved? Not?

Perhaps 'barely" or "almost" or "mostly, except for some parts..."? We can recognize that it might be QR-code-ish but from our eye can't

You can see that I was naughty and rotated the image sensor's axis by 45° on purpose just to make it more interesting.

If you gave the right image to a computer program, it could probably recover the data that was encoded into the original "QR code" most of the time, but it might be much harder for a person to figure out.

Punch line

It's up to you really, based on your application. For amateur astrophotographers or lay people like me, I'd say that if other folks look at a tiny few-pixel image of the Moon and recognize it as probably the Moon without being told, then it's resolved. So maybe 10 pixels wide? Is it the Moon? If I told you it was, would it be believable? If I asked you what it was without any hints, would Moon be your first guess?

Resolved? Not?

Now if we had a thousand Moons and you had to identify which one? then you might need to go to maybe 16 pixels wide.

From barcodeart.com/artwork/portraits/pixel_presiden:

Resolved? Not?

Leon Harmon - 1973

In November 1973, a researcher at Bell Labs named Leon Harmon wrote an article for Scientific American titled, "The Recognition of Faces." It includes several "block portrait" illustrations, most notably this one of Abraham Lincoln. He created the portraits with some prehistoric computer equipped with a "flying-spot scanner." Harmon used these pixelated portraits to test human perception and automatic pattern recognition. The article actually doesn't have the word "pixel" in it, but certainly introduced a new way of seeing.

Salvador Dali - 1976

A few years after Harmon's article, Salvador Dali completed this painting titled, "Gala contemplating the Mediterranean Sea, which at 30 meters becomes the portrait of Abraham Lincoln (Homage to Rothko)." Not only did Dali appropriate Harmon's portrait of Lincoln into the overall composition, but Dali also reincorporated a smaller grayscale version into a single tile.

Answer 2

An example of a single, square object is not enough to explain resolution in this context. Resolution is defined in terms of being able to resolve details in an image -- and a single, square object contains no such details. Instead, it is defined in terms of being able to tell apart separate features/objects -- if the resolution is not high enough, separate features melt together into one, and you cannot tell whether it is a single feature, or multiple features blurred together. Using your words and thinking process, you were really close to the true definition, though. Instead of a single, square object, please just think about two square objects sitting at some distance apart from each other; and whether or not you could tell there are indeed two separate objects. If we progressively decrease the resolution of an image, at some point you would not be able to differentiate between them because they would melt together and be seen as a single object.

It is not really about the ratio of object size vs pixel size. Object size vs pixel size is rather about our visual comfort, where the individual pixels are small enough that we do not notice the grid of the screen. However, even if we made a giant screen made of 10 m × 10 m pixels, we still would be able to see detailed images if the resolution was high enough. The screen would be, for example, an 8 km × 6.4 km juggernaut miracle of engineering that would be useless if viewed from the distance we usually sit away from a computer screen or TV, but walking just a few kilometers back away from it would let us see it as a whole and thus see the detailed image.

Please look at this TV resolution chart from 1956. It helps to measure the resolution of TV screen. It is made of various fine-detail features, like thin lines spaced apart in progressively smaller distances.

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