# 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 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 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 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 Apr 5, 2023 at 9:03

I think you did the Fourier Transform (FT) correctly, the first plot is just shifted incorrectly. I'm not familiar with Mathematica's FT but in MATLAB this would be computed as fftshift(fft(data)). What you have shown is what you would see if you only plotted fft(data).

That should address the first three questions. The answer to question 4 is yes, your units would be mm^-1. You transformed from the spatial domain to the frequency domain, that is, spatial frequency.