Fourier transform of a picture?

Hi, I'm trying to create a model that simulates a lens with fourier optics. Studying fourier optics I have this formula:

Now, when d=f we have a simple fourier transform. Now I think that matlab doesn't put f (that states for focal lenght) in the exponential. How can I change the factors in the exponential?

Answer 1

[Bjorn Gustavsson]

Note that the actual Fourier transform is the second row of your equation, while the first factor you can see as a matrix with normalization coefficients that varies over the Fourier-plane (main contribution here is to include a spatial variation of the phase due to optical path-lengths differences). Just calculate the FFT of your image, then calculate the normalization-coefficient-matrix, take care to calculate u and v correctly scaled, and use sensible values for f, k and lambda. That should be manageable. The f in the Fourier-transform is just a scaling-factor - accounting for the scaling of the u and v coordinates - those are given by the optical system and your image-size.

[Alberto Paniate]

Thanks for the answer. Yeah I know that the fourier transform is the second row. If I have understood correctly, I should create new variables: x= 2pi/(lambda*f)*u and same thing for v .In this way we obtain the same structure of fft in matlab

[Alberto Paniate]

@BjornGustavsson