# A Image FFT and lens MTF

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1. Apr 25, 2017

### Neels

Hi, I am trying to determine the final image on a detector having a 600x600 mm viewed object and the known MTF of a lens. So when I FFT the object, thus creating frequencies -128/600 : 128/600 for a 256x256 image, how does the multiplication of the transformed object and the MTF "keep track" of the frequencies or the size of the input object? Because I can change the object to 60x60 mm and that would not affect the multiplication. How then do I evaluate the correctness of the final IFFTed image? thanks

2. Apr 25, 2017

### Andy Resnick

If I understand you correctly, shrinking the object scale will result in larger spatial frequencies- the FT has a scale factor essentially corresponding to object size (or feature size, if you prefer). Because the MTF is constant (in your case), the smaller the object, the more attenuated those larger spatial frequencies are, resulting in more pronounced blurring.

3. Apr 25, 2017

### Neels

Thanks Andy - yes, I understand that there will be more pronounced blurring, but if I FFT the object image I don't know how to handle the frequency information when the transform is multiplied with the lens MTF or OTF.

4. Apr 25, 2017

### Andy Resnick

Goodman's book has all the information you need. In brief, calling 'h' the point spread function and H the OTF (|H| is the MTF), 'o' the object field and 'O' the transform of object field, 'i' the image *intensity* and 'I' the transform of i, and letting '×'/'⊗' represent the convolution/autocorrelation integral, there are two cases, depending on whether or not the imaging system is phase-sensitive:

Incoherent imaging: i = |h|2×|o|2; I = [H⊗H][O⊗O]
Coherent imaging: i = |h×o|2; I = HO⊗HO

Does that help?

5. May 2, 2017

### Neels

Yes, but I guess my issue is more about the frequency values that apply - I have a measured lens MTF with a cut-off at 20 lp/mm. The object I am viewing is 600x600 mm - so do I need to sample (i.e. use nr of pixels) on the object to get the same order of max frequency before I perform the ⊗?

6. May 2, 2017