Activity 6: Properties of the 2D Fourier Transform

In this activity, the properties of the 2D Fourier Transform (FT) were investigated. For the first part, the FT of a square, donut, square annulus, two slits, and two dots were obtained. The following images show the images with its corresponding FT's.
The second part is to create a 2d sinusoids with different frequencies and obtain their FT's. Results showed that as you increase the frequency, the spacing between the dots in the FT's also increases. This is because at higher frequencies, the spacing between the sinusoids decreases. This is equivalent to an increase in the spacing of the delta function (or in this case, the dots) in the frequency domain. Note that the FT of a sinusoid is a delta function.

On the next part of the activity, we investigated the effect of adding a constant bias on the sinusoid. From the results, it can be seen that adding a negative or positive value of bias does not change the FT as long as the magnitudes are equal. Note that frequency = 4 for all was used.
When the sinusoids are rotated from 0-120, their FTs also rotates, which is in accrodance to the Introduction part of the manual. [1] (frequency = 4)
The product of two sinusoids is a checkerboard with circles or ellipses. Circles are obtained when both the coefficients for X and Y are the same, otherwise, an ellipse. The FTs produced are dots arranged in a square (for circles) and in rectangles (for ellipses) . Notice also that the distribution of the dots has the same shape as the component of the checkerboard. What follows are the images of the generated sinusoids and their corresponding FT's.
For the last part of the activity, the FTs of combinations of the sinusoids with varying rotating angle and the product of two sinusoids were first predicted. Superposition of the waves were basically done. From the linearity property of FT, the FT of the sum of two waves is just the FT of the first wave plus the FT of the second wave. Here, we expect to get the same result as the previous only this time they are rotated. Results verified this linearity property.
For this activity, I give myself an 8/10. Thank you to Raffy for introducing the 'mesh()' function. It was fun to look at the sinusoids in 3D! :D Thank you also to Ma'am Jing for the code :)

**Scilab code can be found in Reference 1.
Note that to see better results, open the image on a new tab :)

Reference:
Dr. Maricor Soriano. AP 186 Activity 6 Manual