# How is resolution defined?

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?

• Did you read en.wikipedia.org/wiki/Optical_resolution - especially the 'spatial resolution' paragraph? What is unclear once you read it? Oct 17 at 16:00
• @planetmaker the question presents a specific example of the shape of an object and the ratio of object to pixel size. Simulation of the complete imaging system, determining response functions (e.g. point spread function) and use of image deconvolution are barely even mentioned much less discussed in that article, but techniques like those are going to be necessary to form a good answer to this question as it is currently written.
– uhoh
Oct 17 at 22:27
• To be honest I was hoping for an answer that wouldn't take a PhD in optics to understand! 'Resolution for dummies' ;-) Oct 18 at 0:15
• TLDR: The resolution of an astronomical telescope ususally is expressed as an angle. If the angle between two bright points (e.g., stars) in the sky is less than the resolution of the instrument, they will appear as a single point. But if the angle is greater than the resolution, then they will appear as two distinct points. Oct 22 at 21:15

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.

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.

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.