Activity 6 - Properties of the 2D Fourier Transform

Fourier Transform of Different 2D patterns

For the 2 dots, the FT is a 2D sinusoidal pattern. This means that there are many sinusoidal functions of different frequencies summed up to make the 2 dots. Conversely, if we consider the 2 dots to be the frequency, taking its FT would be like taking the inverse FT. This would result in a sinusoidal pattern whose frequency are the 2 dots. Similarly, the FT of the slits is a 1D sinusoidal pattern. For the annulus, the FT is just like the FT of a circle, minus certain frequencies for the hole of the annulus. This is why the FT of the annulus is also annular. The same explanation holds for the square and square annulus.

Sinusoids of different frequencies

PATTERN / FOURIER TRANSFORM

frequency = 4

frequency = 10

frequency = 20

The FT of the sinusoidal patterns are the positions on the x-axis of the image whose x-coordinate is the frequency of the sinusoid. We can see from the images above that as the frequency was increased, the peaks on the 2d pattern become closer, and the dots on the FT image move farther away. This is because the vertical line bisecting the square is the y-axis, and the horizontal line bisecting the square is the x-axis. Higher frequencies are farther away from the origin.

Adding a Bias

To simulate a real image, we add a bias to the sinusoid. Taking its FT, we get:

PATTERN / FOURIER TRANSFORM

frequency = 20; bias = 0.5

The bias we added was actually a DC signal of zero frequency added to the sinusoid signal of frequency= 20. In the FT image above, the middle peak corresponds to frequency = 0 for the DC bias, and the other two peaks for the frequency of the sinusoid.

Suppose in a Young's double slit experiment, we obtained an image of an interferogram, and we want to know the actual frequencies. In this real life situation, it is inevitable to have background signals which we want to separate. One way to do this is to read the ambient signals and subtract this to the actual signals before taking the Fourier transform. Or one could use a filter if the frequencies of the background signals are known.

Rotation of sinusoids

The sinusoids were rotated and the Fourier transforms were taken.

PATTERN / FOURIER TRANSFORM

theta = pi/6

theta = 2pi/6

theta = 4pi/6

theta = 5pi/6

Rotating the sinusoids resulted in rotation in the corresponding FT, but in the opposite direction.

Combination of sinusoids in X and Y direction

The sinusoid generated with both X and Y components and its FT is shown below.

PATTERN / FOURIER TRANSFORM

frequency = 10

The resulting image is a cross-hatch pattern, and its FT shows four peaks, the coordinates of which is the frequency of the original image. The x-coordinate of each point is 10 units away from the y-axis, and the y-coordinate of each point is 10 units away from the x-axis. This is because each point in the signal has both X and Y component, generated by the code:

sineSignal = sin(2*%pi*f*X).*sin(2*%pi*f*Y);

This is different from just adding two perpendicular sinusoids of the same frequency, given by:

sineSignal = sin(2*%pi*f*X) + sin(2*%pi*f*Y);

Shown below is the image of the sum of two perpendicular sinusoids and its FT.

The FT shows the frequency of the sinusoid in the x direction and the frequency of the sinusoid in the y direction separately.

Below is an image of multiple sinusoids added to the cross-hatch sinusoids, created with its FT forming a sun-like pattern in mind. Taking its FT, it is indeed a sun-like pattern.

I give myself 10 points for this activity, because everything was done correctly, and I understood every step in the process.

Reference:
A6 - Properties of the 2D Fourier Transform (by Dr. Maricor Soriano)