Creations & Distortions: Activity 6 - Properties and Applications of the 2D Fourier Transform

Hi there visitor! This activity is just a continuation and somehow a more advanced version of Activity 5 wherein we tend to explore more on the deeper properties and characteristics of the Fourier Transform.

First of all I would not start my blog with a certain life lesson or realization due time limitations. Well, I guess the main point here is that sometimes, no matter how Fast we think we are in Transforming, we must realize that there will be certain moments that we need to accept our limitations and do our best in the midst of the unwanted circumstances.

ANAMORPHIC PROPERTY OF THE FOURIER TRANSFORM

The Anamorphic properties of the FT just tells us the inverse properties of the FT wherein if a certain dimension longer at the digital space, it will tend to be shorter in the Fourier space, and vice versa. We can see this in Figure 1 below wherein the corresponding patterns on the left, with their FT on the right.

Figure 1. (Top to bottom) Tall and wide rectangles, and two dots symmetrically separated from the center, with their corresponding Fourier transforms.

The theoretical expectation is met upon examining the shift in the dominant, longer dimension in the Fourier space which can be obviously implied in the long and wide rectangles. The two dots tend to have an FT that is a sinusoidal pattern along the x-axis. Anamorphism can be seen upon observing Figure 2 that shows a decrease in the line widths of the FT (bottom) upon increasing the gap between the two dots (top).

Figure 2. Two dots (top) and its corresponding FT (bottom) showing a decrease in the line widths in Fourier space upon increasing the dot separation in digital space.

Here's a cool GIF for further visualization of Anamorphism (please take note Aliasing due to very high frequencies of the sinusoid):

ROTATION PROPERTY

Here we explore the rotation property of the FT upon the introduction of rotation parameters and sinusoid addition and overlap. The summary of this part can be observed in Figure 3 which shows (from left to right) the FT of a sinusoid showing peak frequencies (one negative-y and one positive-y: the reason why there are 2 dots), the resulting FT upon adding a constant bias, the FT upon the introduction of a rotational parameter, the FT of combined sinusoids, and the FT of rotated and combined sinusoids.

Figure 3. The sinusoidal patters(top) and their corresponding FTs (bottom).

Shown below are GIFs for better visualization.

Sinusoid of increasing frequency with its corresponding FT

Sinusoid with increasing constant bias with its corresponding FT

Sinusoid rotated clockwise with its corresponding FT

Combined sinusoids with one in the x and the other in the y direction with its corresponding FT

Combined and rotated sinusoid with rotational functions and their corresponding FT

Combined and rotated sinusoid with more complicated rotational functions and their corresponding FT

CONVOLUTION THEOREM

Here we revisit and go deeper in the convolution theorem of Fourier Transforms. Figure 4 shows the summary of what we did in this part. (From top to bottom) First, the FT of two Dirac deltas (peak ponts) were obtained then the said points were replaced by first a circle, a square and a gaussian bell curve. The sizes of these patterns were studies and the results were found to still exhibit anamorphism and for very small sizes, they tend to represent almost the FT of the Dirac Delta.