Diffraction limit and CTF with small pixel detectors

Many integrators use large resolution cameras with very small pixels without taking care of the actual lens performances. The resolution of a lens is typically expressed by its MTF (modulation transfer function) graph, which shows showing the response of the lens when a sinusoidal pattern is imaged. However, the CTF (Contrast Transfer Function) is a more interesting parameter, because it gives the contrast achieved when a black and white striped pattern is imaged, thus simulating the behaviour of a lens when imaging an object edge.

If "t" is the thickness of a white or black stripe in object space, the related spatial frequency w (usually expressed in line pairs/mm) is computed as

w = 1/2t.

For any given value of w the contrasts is computed as:

CTF(w) = (Iw - Ib) / (Iw + Ib)

where Iw and Ib are the maximum intensities (or the "grey levels") you can measure on the image plane, for white and black stripes respectively.
CTF is limited by diffraction and the limit lowers with increasing F/#s: for a given spatial frequency w, the CTF increases when the working F/# gets smaller.
At the same time, CTF also depends on the wavelength range: the shorter the wavelength, the higher the CTF. Expressing the CTF as a function of these parameters yields the following:

CTF = CTF (w , WFN, lambda)

where

w = spatial frequency in linepairs/mm

WFN = working F/#

lambda = wavelength (in millimeters)

the "cut-off frequency" is defined as the w value for which

CTF = 0

which occurs when

w = 1/(WFN * lambda)

For instance, a TC Series lens with working F/# 8 and operating with in green light (lambda = 0,000587 mm) has a cut-off frequency of:

w(cutoff) = 1/(8 * 0,000587) = 213 lp/mm

which corresponds to a pixel size of about 1/(2*213) = 2,3 micron.

Theoretically, one wouldn't like a lens yielding a very small contrast (CTF) at the pixel spatial frequency; however it looks like a small pixel size is helpful in reducing noise and better defining the profile of an object.

For this reason, although the increase in resolution is less than proportional than the pixel size (because the CTF curves are decreasing when the spatial frequency increases) there are still some good reasons to use small pixels. In addition, the edge detection is carried in two dimensions (therefore decreasing the pixel size a little strongly increases the number of pixels over a certain image area and probably makes the edge detection more efficient).