Airy disks and pixel pitch on circle of confusions?

Question

I read in this paper by Toshiba, which is in regards to optics in machine vision, that

In machine vision, which processes each pixel of an image sensor at high brightness levels, the permissible circle of confusion (δ) is calculated based on the pixel pitch (Ppix) or the diameter of the Airy disk (DAiry) that represents a limit to the optical resolution of an image created by a lens. In the case of monochrome cameras, the larger of these values is used as δ.

Nevertheless, I've also read many answers in regards to other questions on pixel size and Coc that pixel size tends to not be the limiting factor but rather other factors. Per the accepted answer to the question, Clarification for Effect of Pixel Size on Depth of Field:

Actually, this isn't right. The CoC criterion is the largest blur that will be perceived by the viewer as a point. At low resolutions, this may be limited by pixel size, but generally in real world use other factors are dominant ­— display size, distance of viewer, etc.

As well as the fact that circle of confusions tend to be much larger than even a grid of 2x2 cells on a sensor. (C.f.: the accepted answer to How do depth of field and the circle of confusion relate to pixel size on the sensor?)

I'm sure this might also relates to a demo I saw on Steve Yedlin's site, where he displayed that the difference between a 2k and 4k output is perceptually nothing, as the average viewer in average viewing conditions was not able to resolve the extra pixels.

To me it seems like that the first statement of circle of confusion criterion being equal to the size of either the airy disk /pixel pitch contradicts the other statement, where it pixel size doesn't determine the coc criterion.

What am I not seeing?

Side Note: I've already written quite a few questions and have gotten even more incredible answers, although I tend to take some time to truly grasp them. Nevertheless, if anyone knows any great reliable resources to learn a lot of these things, it would be much appreicated. I understand that searching on the internet may be the best tool, but obviously there is the dilemma of if the particular source is reliable resources.

Answer 1

Pixel pitch does not determine the standard CoC limit, and it is not a factor considered in the DoF calculations.

That's because the accepted CoC limit for normal viewing by an average individual is far less demanding. The CoC standard requires less than 2MP of image resolution for viewing any size displayed image... every camera far exceeds that requirement. I.e. the limit is human perception (~20/40 vision viewing an image from a distance equal to the image diagonal (~45˚ HFOV)). Note that most images viewed on your computer are at ≤2MP.

But the airy disk most certainly relates to pixel pitch in terms of recorded resolution. If the airy disk is much smaller than a pixel, it cannot be resolved. As the airy disk becomes larger than the pixel, contrast is reduced (which is a large portion of our perception of sharpness/resolution). When it becomes ~ 2x the size of the pixel, recorded resolution drops (depending on contrast requirement). And when it becomes larger than the CoC limit, visible resolution decreases in the output image (when viewed according to the standard).

The scenario where the airy disk exactly matches the pixel pitch is less than ideal as well; because it results in aliasing errors when the two do not line up (which is more probable than not).

And for viewing conditions other than standard the standard CoC is not applicable. I.e. for a large format print to be viewed at short distance in a gallery, you might want to use a much smaller CoC limit (or for machine vision).

You might like to read this paper: Do Sensors Outresolve Lenses' Capabilities?

Answer 2

I am not sure if you are trying to juxtapose claim about machine learning and claim about human vision. These are very different scenarios and clearly the CoC is defined differently. Even though the term is the same, the meaning is different.

For a machine there is no loss of information because of displaying/printing and viewing with an organic eye.

Answer 3

The CoC is traditionally based on what will be visible in the final picture. It's not affected by pixel size, because frankly you can't see individual pixels on any reasonably sized print taken with a modern resolution camera. It's not based on the Airy disk, because if you're stopped down enough for it to be a factor you're probably not going to be happy with the overall sharpness of the picture anyway.

Everything changes when you're talking about machine vision though. The machine is always pixel peeping, it can sense detail at the pixel level. And the odds that your Airy disk will be larger than a pixel are pretty good. So yes, it might make sense to choose a CoC based on the larger of the two factors. But even so you should consider the needs of your vision system overall and determine the maximum detail resolution that it actually requires.

Answer 4

Check out Photographic Lenses by C.B Neblette: Paraphrased: The circle of confusion size is based on the diameter of the largest circle which will appear as a point from a given viewing distance. Basis – The average observer can resolve lines viewed from 1/3000 the distance. Some examples, a 1-inch coin viewed from 3,000 inches (250 feet). A 25mm coin from 75 meters. A 3-foot wagon wheel viewed from 1.7 miles.

In terms of a displayed photograph: The above is too stringent for pictorial photography as photographs have lowered contrast and generally viewed under indoor lighting. Generally accepted is 1/1000 of the viewing distance = 1/100 of an inch viewed from 10 inches. That’s 1/4 mm viewed from 10 inches (250mm). If the displayed image is viewed from 20 inches (500mm), then the circle size is 2/100 inches = 1/2 mm.

We can apply this to an 8x10 inch print made from a 35mm camera image. To make the 8x10 requires 8X magnification. Thus, the circle size at the image plane must be no larger than 1/800 inches = 0.032mm.

If the camera used is an APS-C format, the crop factor is 1.5. For this lash-up, the magnification to make the 8x10 is 8 X 1.5 = 12X. Give this layout, the circle size is 0.032 ÷ 1.5 = 0.02mm at the image plane.

Most depth-of-field tables are based on an 8X10 inch displayed image scrutinized from 10 inches = 0.25mm circle diameter.