Resolution (visual acuity) and screen discreteness

[fog37]

Hello Forum,

Resolution is an angular distance (measured in fractions of an 1 degree angle). The lower the resolution the better. It represents the ability to distinguish two objects that are very closer to each other.

For an optical system, the resolution increases when the diameter D of the lens increases (or if the wavelength of the used light decreases).

1) If the system has multiple lenses, will the aperture pupil determine the resolution or the size of the aperture stop?

Because of the unavoidable diffraction, two object points will be imaged by a circular lens as Airy disks. If the two discs are too large they will overlap and look like a single blobby image: we will not be able to distinguish the two objects separately.

The two point images (Airy discs) form on the imaging screen that is often assumed to be continuous. What if the imaging screen is (like the human retina) discrete an formed by a finite number of photoreceptors with a certain size and shape?

Does the size of each receptor need to be smaller, larger or equal to the diameter of the Airy disks?

Thanks,
Fog37

 

[Drakkith]

First, the higher the resolution the better, as higher resolution means you can distinguish objects closer to each other than a lower resolution.

If the system has multiple lenses, it is usually the diameter of the aperture stop that determines the resolution. For example, my telescope has an aperture stop that is 8 inches in diameter, while the mirror is about 9 inches in diameter. The resolution is the same as an 8 inch mirror that doesn't have an aperture stop. (The reason for the difference is that having the mirror larger than the aperture stop helps with vignetting)

If you are worried about resolution, your detector should have a pixel size approximately 2-3 times smaller than your airy disk. See here: http://starizona.com/acb/ccd/advtheorynyq.aspx

 

[Andy Resnick]

<snip>
The two point images (Airy discs) form on the imaging screen that is often assumed to be continuous. What if the imaging screen is (like the human retina) discrete an formed by a finite number of photoreceptors with a certain size and shape?

Discrete (sampled) imaging systems can be more difficult to analyze because they are not linear shift-invariant systems: small shifts in object position don't result in small shifts of the image. Often, the pixel size is designed to be much smaller than the Airy disc, and as Drakkith notes, in that limit 'normal' rules apply. The opposite limit (pixel size much larger than the Airy disc) is not much relevant.

In between, a variety of imaging artifacts can occur, most notably 'aliasing' which is the introduction of spurious high-frequency components to the image caused by the mismatch between sample frequency and image frequency, similar to Moire' patterns. The best (IMO) book to understand this is Vollmerhausen and Driggers' "Analysis of Sampled Imaging Systems" (SPIE press)

Note that the retina is not anything like a simple pixelated array- there are multiple processing layers located in the retina that operate on the signal prior to the optic nerve.

 

[fog37]

So, in essence, for the simple case of a single lens and a viewing screen with a finite amounts of receptors on it, the resolution is primarily determined by the diameter D of the lens and wavelength of light and later by the number of receptors on the screen.

If there are too few receptors we run into the problem of undersampling that leads to artifacts which compromise the overall resolution.

fog37

 

[PJC01]

Hello Fog37,

Thank you for bringing up the topic of resolution and screen discreteness. I would like to provide some clarification and additional information on these concepts.

Firstly, resolution is not necessarily better when it is lower. In fact, a higher resolution allows for more detailed and clear images. Resolution is defined as the ability to distinguish between two objects that are very close together, and it is measured in units of length per unit of angle. The higher the resolution, the smaller the distance between two objects can be before they appear as one to the observer.

Secondly, the statement that resolution increases when the diameter of the lens increases is not entirely accurate. While a larger lens may allow for more light to enter the optical system, which can improve image quality, it does not directly affect resolution. Resolution is primarily determined by the size of the aperture stop, which controls the amount of light and information that enters the system.

To address your question about multiple lenses, the aperture pupil, or the opening through which light passes, will determine the resolution. Each lens in the system will have its own aperture pupil, and the smallest one will determine the overall resolution of the system.

Now, onto the issue of screen discreteness. In an imaging system, the image is formed on a continuous screen, such as a camera sensor or the retina of the eye. However, as you mentioned, these screens are not truly continuous but are made up of a finite number of discrete elements, such as pixels or photoreceptors. The size and shape of these elements can affect the perceived resolution of the image.

In the case of the human retina, the size of each photoreceptor is smaller than the diameter of the Airy disks formed by the lens. This allows for individual photoreceptors to detect and distinguish between two closely spaced objects, improving the overall resolution of the image. However, if the photoreceptors were larger than the Airy disks, the image would appear less detailed and objects may appear to blend together.

In conclusion, the size and shape of the elements on a discrete imaging screen can affect the perceived resolution of an image. In order to maintain a high resolution, the size of these elements should be smaller than the diameter of the Airy disks formed by the lens. I hope this helps to clarify the relationship between resolution and screen discreteness. Thank you for your question.

Best,