Understand Fourier Transform of Images

We’ve covered Fourier Transform in [1] and [2] while we use only examples of 1D. In this post we are going to see what 2D Fourier Transform looks like.

First, we look at a 2D image with one direction sinusoid waves (left) and its Fourier Transform (right).

I added coordinates to help you understand the illustration. Let’s assume the $x$ -axis in the image corresponds to $i$ -axis in frequency domain; $y$ -axis in the image corresponds to $j$ -axis in the frequency domain. Because there is never fluctuation along the $x$ -axis, there should only be $i=0$ frequency. However, along the $y$ -axis there is regular sinusoid change, therefore there should be $j>0$ frequency. As a combination, we see two dots (two dirac delta functions) on the line $i=0$ .

To reinforce our understanding, let’s look at three more pictures. The first two have vertical bars with different spatial frequencies. As you might guess, the first picture with sparser bars have a smaller frequency than the second picture. Their frequencies are only on the line $j=0$ because there is no signal variation on the $y$ -axis.

The third picture is generated by $sin(x+y)$ (https://www.wolframalpha.com/input/?i=+plot+sin%28x%2By%29). And it has regular variations of waves along both the $x$ and $y$ -axis. Thus, we see the two white dots in the frequency domains are on a diagonal.

Next, we examine a real-world photo, a famous picture titled “cameraman”. We mark one leg of the tripod in th image and its corresponding frequency band. This leg is almost perpendicular to the $x$ -axis. Therefore, it introduces high-frequency variation (sharp edges) along the direction of $x$ -axis. Meanwhile it introduces less variation on the direction of $y$ -axis. On the frequency domain, you can see the corresponding line has a slope more towards the $i$ -axis than the $j$ -axis.