# What is the cause of differences that are too small to see?

Consider two identical pieces of paper.

Scenario 1: On both something is drawn in black ink. If the difference between the areas covered in black ink is sufficiently small, I cannot see the difference between the two drawings.

Scenario 2: Now imagine that the same parts are covered with ink, but their color differs. Again, if the difference is sufficiently small, I cannot see the difference.

Firstly: what causes this? I am aware of the terminology of just-noticeable differences/limina, but do not understand precisely what causes them. Is there some discretization of continuous signals going on that causes nearby signals' to be treated the same way?

Secondly: one person may have better vision than another, but are there physical constraints that are the same for any two persons and that imply that if the difference in scenario 1 or 2 is sufficiently small, no two persons will be able to see a difference?

Addenda: I phrased the question informally by not defining what a small difference' is, trusting/hoping that this does not lead to confusion. If desired, such statements can be made precise by adopting appropriate distance functions. In scenario 1, one way to measure distance between two areas covered in ink could be to use the Hausdorff distance; likewise, one way to measure the distance between colors in the second scenario could depend on the difference in responsiveness of the three types of cone cells.

## 1 Answer

The spatial resolving capacity of the visual system is called visual acuity, namely the ability of the eye to see fine detail. There are various ways to measure and specify visual acuity, depending on the type of acuity task used.

Visual acuity is limited by

• diffraction,
• aberrations and
• photoreceptor density in the eye.

The photoreceptor density allows a resolution that by far exceeds the limitations imposed by the other two.

Diffraction of light is an important factor limiting acuity. This is caused by optical limitations of the eye, mainly the light scattering by the cornea and other structures (Fig. 1). Raleigh’s criterion is used to calculate the resolution of the eye for stimuli that are degraded by the optics of the eye: two points or lines are just resolved if the peak of the point spread function lies on the first trough of the other point spread function (Fig. 2).

Aberrations also are an important source of image degradation on the retina. These include the well known spherical aberrations that lead to near- and far sightedness, as the retinal image becomes out of focus. These can be corrected well with glasses and lenses (Berrio et al., 2010).

Other sources that affect acuity are:

• The size of the pupil; Large pupils reduce diffraction, but aggravate aberrations of the eye. A small pupil reduces optical aberrations, but resolution will be diffraction limited. Therefore, a mid-size pupil of about 3 - 5 mm is optimal as a compromise between diffraction and aberration limitarions
• Illumination: photopic vision allows the cones to operate and that yields higher acuities than rod-mediated vision (Kallioniatis & Luu (2012).
• For the same reasons the area of the retina stimulated affects acuity - the cones are clustered in the center of the retina, called the fovea. Foveal acuity is much higher than peripheral vision (Kallioniatis & Luu (2012).
• Time of exposure of the target causes adaptation. If the cones are adapted to the timulus, they may become less responsive and acuity drops (Kallioniatis & Luu (2012)
• Eye movements degrade visual acuity as a stable retinal image is needed (Kallioniatis & Luu (2012).

Fig. 1. Resolution limit eye. source: Kallioniatis & Luu (2012).

Fig. 2. Reigheigh criterion. source: Kallioniatis & Luu (2012)