**Intro**

The Contrast Transfer Function (CTF) is a measure of contrast between adjacent objects. In theory, the CTF is described as the convolution between the point-spread function (PSF) of the imaging system, and the step-function object that is being imaged:

The CTF is a function of step-function spacing, or spatial frequency. The CTF of a diffraction limited imaging system is:

In the above picture, the CTF is the blue curve, and the pink curve is the Modulation Transfer Function (MTF) - from an object that has a sine wave pattern.

Experimentally, one measures a CTF by measuring the contrast of some pattern of known spatial frequency. We use the MRS-5, which looks like this:

A point on the CTF curve is then $CTF\left(f_{spatial}\right)=\frac{Contrast\left(f_{spatial}\right)}{Contrast\left(f_{0}\right)}$

where $f_{0}=1/3\mathit{um}$ in our case.

**How well does the CTF correspond to what we hope to see?**

Though the CTF is a good measure of the quality of an imaging system, the limiting CTF resolution resolution doesn't correspond to the limiting resolution of atoms in a lattice. The reason for this is obvious - when taking a CTF, the object being imaged is extended, while atoms in the lattice are point sources. Below, we compare intensity distribution from the image of a "3-hatch pattern" seen in the MRS-5, to that of 3 point sources, both at a spatial frequency of 1/700nm. We assume that our imaging system has a resolution of 400nm, and the units of the x-axis is microns:

where the blue curve is the "3-hatch pattern" intensity, and the pink curve is the 3 point source intensity. If now we also find the intensity distribution for the 3um hatch pattern (pink curve), while also including the 700nm hatch-pattern intensity from above for reference (blue curve):

we can compare a CTF from the hatch pattern and 3 point sources at 700nm:

Object |
CTF @ 700nm |

hatch | .60 |

point-source | .97 |

**Comparing CTF to MTF to PTF**

We can plot the CTF, MTF and PTF (Point Transfer Function - the contrast of 3 point sources) next to each other to directly compare them:

We can see that while the CTF and MTF gradually decrease as the spatial frequency of the pattern increases, the PTF has almost perfect contrast until the point is reached where the pattern separation becames of order the width of the point-spread function. This is seen directly below, where the intensity distribution arising from the image of three point sources is plotted as a function of $x$, the spatial period:

**Converting CTF to PTF**