# Calculate Modulation Transfer Function from Line Spread Function

I am trying to measure the MTF of a radiology imaging system from a set of CT scans of a phantom. The imaging system is primarily designed for stationary scans so the CT images were low in spatial resolution. I am not particularly familiar with optics and I cannot come to a physical understanding of the results I produced. I am hoping that you could help me. Here is what I had done:

I have measured a line profile and calculated its modulation transfer function (MTF) according to the following: $$MTF(f)=|\int LSF(x)e^{-2\pi ifx}dx|$$

The profile was taken using Image J in a similar fashion as depicted below: The line spread function was calculated after basic baseline removal:

Fourier transform of the profile was performed in Mathematica, MTF was then normalised to 100% and plotted:

There are several things in this plot that I do not understand.

1. the first data point has value < 1
2. the plot is symmetric about a point. Only half of the plot is useful?
3. why does MTF reach zero twice before the point of symmetry? Is this because of the asymmetry in LSF?
4. LSF(x) was a plot of intensity (grey value) varying with position (mm), would the spatial frequency in MTF(f) have units of mm^-1?

When measuring MTF using 14 line pairs (14 line profiles drawn, max&min intensity values were tabulated to produce a plot of %amplitude retained by the system), the following was produced:

and this was a much nicer curve! The problem perhaps isn't in the imaging system, but in the samples involved. I am hoping that someone could tell me why my approach involving LSF(x) had produced such a bizarre graph?

Many thanks

• When performing a discrete Fourier transform there is an effect called aliasing (you can check a Wikipedia page on it, otherwise a good reference on the topic is [Papoulis "The Fourier Integral and Its Applications", 1962], there is an example, a problem describing DFFT aligning. Right now I do not have an access to the book to give you the exact page). Anyway it means that a Fourier spectrum will be periodic with $2\pi/X_s$ where is $X_s$ is a sampling "distance". This is why your spectrum is symmetric over a vertical line Commented Apr 5, 2023 at 8:53
• Normally you could take a half of a spectrum on left, and another which is on right - to shift it to half of whole scale, in order to have it on the tight of 0 axis Commented Apr 5, 2023 at 8:55
• The point close to 0 is wierd, true, this might be, however, due to a not enough sampling, hard to tell Commented Apr 5, 2023 at 8:57
• BTW, your function is similar to $sinc(x)$, and the Fourier transform of this function is well known "window". Thus, at least qualitatively, the result of a DFFT seems to be correct Commented Apr 5, 2023 at 9:03