Activity 6: Properties of the 2D Fourier Transform

Fourier Theorem states that any function can be expressed as infinite sum of sinusoids of different frequencies. In imaging, this translates to finding the spatial frequencies of an image. In 2D FT, rotation of the image results to rotation of the resulting FT, we will investigate this property in a while but let us first familiarize ourselves with FT of different patterns [1]

Activity 6.A: Familiarization with FT of different 2D patterns

Below is the FT of different patterns. A quick way to check if the result is correct is to imagine placing a mask(aperture) in front of a light source and observe the image in a screen far away. The resulting FT should have the somewhat the same symmetry as the object.

Observe the difference with the square above, i.e. similar to subtracting terms in your diffraction. In Fresnel diffraction, this amounts to removing zones in your image.

Textbook example of two slits along the x-axis (see Hecht, section diffraction on Fraunhoffer diffraction)

The FT of two dots is a sinousoid. This is textbook example of Young's double slit experiment.

Activity 6.B: Anamorphic Property of the Fourier Transform









Sinusoid of frequency 4 generated using sin(2*pi*f*x).

Sinusoid of frequency 8. The dots are more widely separated. Imagine placing a frequency axis on the y-axis above. The higher the frequency of the sinusoid, the farther apart the dots.

In digital imaging, there is no negative values, adding a bias to an image results to the presence of zero order. This is exactly what happens when we add bias to the image of the sinusoids above.

Sinusoid of frequency 4 with bias. The center dot is the zero-order frequency. In electronics, we can imagine a DC bias or in imaging, the image is offset by the background.

Suppose then that we took a picture of an interferogram, we can obtain the actual frequencies of the object by adding a constant bias to the image (i.e., min value). Or we just obtain the FT and neglect the zero order frequency. However, if the resulting interferogram has non-constant bias then we can add a filter to obtain the desired frequencies. For example, suppose that the image was added with very low frequency sinusoids. We can obtain the frequencies by obtaining the FT of the image and multiply the resulting FT with a high pass filter (i.e, to remove the low frequencies) and then obtain the inverse FT to get the final image of the object. Depending on the bias, we can adjust the filter to obtain the desired frequencies.

In 2D FT, rotating the sinusoids result to a rotated FT.

Rotated sinusoid with its corresponding FT. Observe that the resulting FT is also rotated in the same direction as the object.

Fourier transform of combination of sinusoids. Left: multiplication of two corrugated roofs, in the X and Y direction and right: FT of the object, 4 dots of the same distance signifying same frequency in the X and Y direction.

We added a rotated sinusoid to the corrugated image above using different frequency (f=16 and rotated for 90 degrees). Observe the presence of the 4 dots similar above and the addition of two dots.

We added a different rotated sinusoid using different frequency (f=12 and rotated for 30 degrees).

Added rotated sinusoid of f=8 and rotated for 60 degrees.

Combination of all the above rotated sinusoids. Here we can see that the result is just the addition of all the FT of the different rotated sinusoids. This is because FT is a linear transform, i.e.,

let X=A+B+C then F{X} = F{A} + F{B} + F{C}

Here, we investigated the different properties of the 2D Fourier Transform, in particular

• rotation of the object results to a rotated FT
• FT is a linear transform
• FT obtains the spatial frequencies of the image

For this activity, I give myself a grade of 10 for doing the required objectives.

I would like to acknowledge miguel and martin for useful discussions and master for knowing the spelling of Hecht =).

References:
[1] Activity 6 Manual