Basics
To explore the basics of the Fourier Transform, it is performed on a variety of shapes such as the following:
The columns represent different shapes, while the rows represent the object, the center-shifted FFT and the FFT of the FFT, in that particular order. The results will be tackled one shape at a time.
A
The FFT of the letter A looks like a 6-pointed star, with a bright center. We can also think of it as three lines that intersect at the center. Looking at it more closely, the lines appear to be the normals/perpendiculars of the three main lines that make up the letter A. Thus, the FFT of a line will lie along a line that is perpendicular to the object line.
The FFT of the FFT is a vertically inverted version of the object A. The explanation for this is that the FFT of an object is what the object would look like at the focus if its image is passed through an infinitely large converging lens. Thus, the FFT of the FFT is as if the object image has passed through two lens, thus becoming inverted.
Circle
The FFT of the larger circle small compared to the FFT of the smaller circle, implying that the FFT of a smaller object will result in a larger FFT. This is because you need increasingly higher harmonics to represent a thinner/smaller object.
The FFT of the FFT of the circle appears like the object, since inverting it will have no effect.
Double Slit
The FFT of a double slit lies along the line perpendicular to the slits. However, if we look at it from the side, we get…
which appears to be a periodic function with a Gaussian envelope.
The FFT of the FFT is the same as the original double slit, as inversion again has no effect.
Sine Wave
If we also take a line scan along the middle of the FFT of the sine wave, we get…
Which appear as 3 narrow peaks. The center peak is called the DC term, and is due to the bias needed for the sine wave to have positive values. The peaks to the left and right correspond to the frequency of the sine wave.
Taking the FFT of the FFT of the sine wave, it can be observed that it has been inverted horizontally. Its vertical inversion cannot be seen, although it must have been inverted in that way as well)
Grating (i.e. Square Wave)
Note that the FFT of the square wave or grating is very similar to that of the sine wave, aside from the smaller peaks to the left and right. These higher harmonics are needed to capture the rapidly changing edge of the square wave.
The FFT of the FFT of the square wave has also been inverted horizontally.
Gaussian Circular aperture
Last but not the least, the 2D Gaussian bell is tackled. Note that its FFT also has a guassian distribution, and that a smaller gaussian object leads to a larger FFT. This is similar to the case of the circle.
The FFT of the FFT of the Gaussian yields the object once again, since it is radially symmetric.
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