Act 6: Properties of the 2D Fourier Transform

This activity was divided into two parts. First we were to apply FFT to different 2D patterns and then we were to apply it to various sinusoid forms.

A. Familiarization with FT of different 2D Patterns

In this part, we were to apply the 2D FT to the following patterns (square, annulus, square annulus, slit, dots, respectively):


Applying the FT to these images yields the corresponding outputs:

What happens is that the initial images act as apertures. The second set of images are the output of the apertures.

B. Anamorphic property of the Fourier Transform
This time, we are to apply FT to this image of frequency 4:

which was produced by

z = sin(2*%pi*f*X);

and its various distortions.
For the initial FT, the result was:

Varying the frequency (f = 1, 2, 3, 5, 6) results to

This time, we apply a constant bias of 1, 3, 6, and 9 to the original sinusoid by and apply FT. This results to

Next, we rotate the sinusoid by an angle of 30, 45, 60 and 90. This results to


This time we make use of combination of sinusoids in X and Y. The used combinations are as follows:

1st: z = sin(2*%pi*f*X).*sin(0.5*%pi*f*Y);
2nd: z = sin(2*%pi*f*X).*sin(2*%pi*f*Y);
3rd: z = sin(0.5*%pi*f*X).*sin(0.5*%pi*f*Y);

The output is as follows:


Finally, we add several combinations of sinusoids. The following combinations of varying frequencies were used:

1st:
z = sin(2*%pi*f*X).*sin(2*%pi*f*Y);
a = sin(2*%pi*f*(X*sin(theta)+Y*cos(theta)));
z = z+a;
2nd:
z = sin(2*%pi*f*X).*sin(2*%pi*f*Y);
a = sin(2*%pi*f*(X*sin(theta)+Y*cos(theta)));
b = sin(4*%pi*f*(X*sin(theta)+Y*cos(theta)));
z = z+a+b;
3rd:
z = sin(2*%pi*f*X).*sin(2*%pi*f*Y);
a = sin(2*%pi*f*(X*sin(theta)+Y*cos(theta)));
b = sin(4*%pi*f*(X*sin(theta)+Y*cos(theta)));
c = sin(0.5*%pi*f*(X*sin(theta)+Y*cos(theta)));
z = z+a+b+c;
4th:
z = sin(0.5*%pi*f*X).*sin(0.5*%pi*f*Y);
a = sin(2*%pi*f*(X*sin(theta)+Y*cos(theta)));
b = sin(4*%pi*f*(X*sin(theta)+Y*cos(theta)));
c = sin(0.25*%pi*f*(X*sin(theta)+Y*cos(theta)));
z = z+a+b+c;

The output should be the similar to the previous, except that there should be a corresponding rotation. The output is as follows

As shown from the above, the results are in agreement with my prediction.

I will grade myself 10/10 in this activity for completing all the steps in this activity.

I would like to thank my adjacent seatmates Earl and Rommel for their advice and assistance in this activity.