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Consider a very small point of light. If the visual system had perfect optics the image of this point on the retina would be identical to the original point of light. So if the relative intensity of this point of light were plotted as a function of distance, on the retina, such a plot would look like the dashed, vertical, green line. However, the eye's optics are not perfect so the relative intensity of the point of light is distributed across the retina as shown by the red curve. This curve is called the "point spread function" (PSF).

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Those who have not studied calculus may find the concept of integration a bit strange. In principle it is really quite easy. Integration is used to find the area enclosed by a curve, the length of a curve or the volume enclosed by a surface.
The area of a rectangle is found by multiplying its length by its width. The area under the above curve can be approximated with a series of rectangles. The area of each rectangle is easily determined and the one adds up the areas of all the rectangles to obtain an approximation of the area under the curve. As the number of rectangles under the curve increases the measurement accuracy increases.
When using calculus the actual computations are different, but the principle is the same.

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Modulation Transfer Function
Another concept that may be new to neophyte vision people is that of modulation transfer function.
Most lenses including the human lens are not perfect optical systems. As a result when visual stimuli are passed through them they undergo a certain degree of degradation. The question is how can this degradation be evaluated? Before we can answer this question we need to define "modulation."
Recall in a square wave grating there were dark bars and light bars. We can measure the amount of light coming from each. The maximum amount of light will come from the light and the minimum from the dark bars. If the light is measured in terms of luminance (L) we can define modulation according to the following equation:
modulation = (Lmax - Lmin ) / (Lmax + Lmin)
where Lmax is the maximum luminance of the grating and Lmin is the minimum. When modulation is defined in terms of light it is frequently referred to as Michelson contrast. Indeed when one takes the ratio of the illumination from the light and dark bars one is measuring contrast.
Now, let's assume that you have a square wave grating of a specific frequency (v) and modulation (contrast) and this stimulus is passed through a lens. The modulation of the image can now be measured.
The modulation transfer function (MTF) is defined as the modulation, Mi, of the image divided by the modulation of the stimulus ( the object), Mo, as shown in the following equation.
MTF(v) = Mi / M0
A lens system may behave differently depending on the nature of the optical information that passes through it. For example, lens systems vary as a function of the spatial frequency of the stimuli that they handle. You undoubtedly noticed, above, that MTF has spatial frequency (v) as a parameter. Click on image modulation as a function of spatial frequency to see a graphical illustration of how the transfer function of a lens effects the image modulation.

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