In activity 6, we have already defined what is Fourier transformation. As a recap, Fourier transformation is a linear transformation that converts a signal to its temporal or spatial frequency space. It essentially gives you the frequency distribution of a signal, it is a histogram of frequencies. It also states that any signal can be represented by superposition of sinusoids having frequencies that are present in the frequency histogram. In 2D FT, a rotation of the signal also gives rotation to its Fourier transform.
Familiarization with FT of different 2D patterns
Different 2-Dimensional object are generated in Scilab and Fourier transformation was applied to each patterns. A square, an annulus, a square annulus, a two slits along the x-axis that are symmetric at the center, and two dots along the x-axis symmetric about the center, were the patterns used for the Fourier transformation.
Figure 1. A generated square pattern (left) and its corresponding Fourier transform (right).
Figure 2. An generated annulus pattern (left) and its corresponding Fourier transform (right).
Figure 3. A generated square annulus pattern (left) and its corresponding Fourier transform (right).
Figure 4. A generated two slit pattern (left) and its corresponding Fourier transform (right).
Figure 5. A generated two dot pattern (left) and its corresponding Fourier transform (right).
From figures 1 to 5, we can see the different patterns and its corresponding Fourier transform. In figure 1, the center part of the Fourier transform resembles its pre-transformed image. In figure 2, we can see that the Fourier transform of the annulus is also an annnulus but it is repeating. In figure 3, the Fourier transform is a combination of the Fourier transform in figures 1 and 2. In figure 4, the Fourier transform is a sinusoid enveloped in a sinc function. Lastly, in figure 5, the two dots' Fourier transform was sine function.
Anamorphic Property of the Fourier Transform
Now let us try to play with the Fourier transform by transforming same signal forms (in this case, a sinusoid along the x-axis) with different properties. Let us try to transform sinusoid signals with different frequencies.
Figure 6. A generated sinusoid with frequency of 2 (left) and its corresponding Fourier transform (right).
Figure 7. A generated sinusoid with frequency of 4 (left) and its corresponding Fourier transform (right).
Figure 8. A generated sinusoid with frequency of 6 (left) and its corresponding Fourier transform (right).
Figure 9. A generated sinusoid with frequency of 8 (left) and its corresponding Fourier transform (right).
Figure 10. A generated sinusoid with frequency of 10 (left) and its corresponding Fourier transform (right).
From figure 6 to 10, it is expected, from the previous part of the activity, that the Fourier transform of a sinusoidal signal is a two dot along the axis where the sinusoid is. Also, for all of the figures, it can be seen that as the frequency of the wave increases, the the sinusoid compress while on the other hand the two dots spreads. This can be explained by knowing that the Fourier transform is a histogram of the frequencies in the sinusoid signal. Since the sinusoid signal has only one frequency in it, the histogram displays it as dot (meaning single valued), one represents the real frequency and the other is the imaginary frequency.
Now what if instead of generating a sine signal, lets read as an image, meaning, let us first generate a sine signal and save it as an image then use Fourier transformation on it.
Figure 11. An image of a sinusoid with frequency of 10 (left) and its corresponding Fourier transform (right).
The Fourier transform in figure 11 showed a dot at the center as compared to the Fourier transform in figure 10. This dot represents the constant bias added to the signal when the sinusoid was saved as an image. Since images do not have negative values, the values of the sinusoid signal was converted to values between 0 to 1. So when we have a real image of an interferogram and we want to digitally process it to obtain the frequencies, we must first filter the constant bias added to it.
Now let us see what is the effect of rotating the sinusoidal signal to its Fourier transform. We will add an angle of rotation by transforming the axis (X' = Xcos(a), Y' = Ysin(a)).
Figure 12. A generated sinusoid rotated at an angle of 30 degrees (left) and its corresponding Fourier transform (right).
Figure 13. A generated sinusoid rotated at an angle of 60 degrees (left) and its corresponding Fourier transform (right).
Figure 14. A generated sinusoid rotated at an angle of 90 degrees (left) and its corresponding Fourier transform (right).
As we can see from the results of figures 12 to 14, the Fourier transforms were rotated as its corresponding sinusoids were rotated. This shows that if the axis in real space was rotated, the inverse space axis are also rotated.
Now let us combine two sinusoids, one that is running along the x-axis and the other one running along the y-axis.
Figure 15. An egg-carton sinusoid signal with frequency of 10 (left) and its Fourier transform (right).
We can see that the Fourier transform resembles the Fourier transform of both sinusoid that runs along the x-axis and a sinusoid that runs along the y-axis. So what if we take the Fourier transform of a signal that has various form of sinusoids?
Figure 16. A signal with an egg-carton sinusoid; a an egg-carton sinusoid that is rotated at -60deg, -45deg, -30deg, 30deg, 45deg, and 60deg (left) and its Fourier Transform.
In this activity, it was observed that morphing a signal also morphs its Fourier transform, but the morphing is related on how the signal was morphed. It was also showed that applying Fourier transform on a signal gives the frequencies that are present in it, even if the signal is complicated, Fourier transform can break it down to sinusoids. In this activity, I would give myself a grade of 10.
Reference:
Dr. Soriano. Applied Physics 186 activity handouts: A7 - Properties of the 2D Fourier Transform.
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