In this activity, some properties of the 2D Fourier Transform (FT) will be demonstrated. According to the Fourier theorem, any signal can be decomposed into superposition of sinusoids. Fourier Transform is a linear transformation, that is, rotation of the sinusoids would also result into rotation in the frequency domain.
A. Familiarization of FT of different 2D patterns
Using Paint, images of a square, annulus (donut), square annulus, two slits along the x-axis that is symmetric about the center spanning the whole y-axis, and two dots symmetric about the center were created. Then, these images were loaded into Scilab using the gray_imread() function. The function fft2() was applied, then the absolute value was obtained using abs(). The function fftshift() was applied to the absolute value to center the FT. Figure 1 shows the 2D images and their corresponding FT.
Figure 1. Different 2D patterns (left) and their corresponding FT (right).
Notice that if the 2D patterns are apertures, their corresponding FT is the interference pattern of the monochromatic light that passes through it.
B. Anamorphic Property of the Fourier Transform
In Scilab, a 2D sinusoid was created with varying frequencies. From figures 2-4, the frequency is increased. Notice that as the frequency is increased, the peaks in the Fourier domain goes farther from the center. From this, we can say that lower frequencies are near the center, and the higher frequencies are farther away. This knowledge is helpful in noise filtering. If one knows whether the needed data is in low frequency or high frequency, he/she could detect if the data is full of noise.
Figure 2. A sinusoid with frequency equal to 4 (left) and its FT (right).
Figure 3. A sinusoid with frequency equal to 8 (left) and its FT (right).
Figure 4. A sinusoid with frequency equal to 12 (left) and its FT (right).
Next, a real image was simulated by adding a constant bias on the sinusoid. Figure 5 shows the image of the FT. Notice the addition of a peak at the center. This peak corresponds to the added bias on the sinusoid.
Figure 5. FT of a sinusoid with bias equal to 0.5 and frequency of 12.
Next, the sinusoid patterns were rotated, and their corresponding FT were obtained. Notice that the rotation of the sinusoids resulted also in the rotation in the Fourier domain.
Figure 6. A sinusoid pattern rotated at 30 degrees (left) and its FT (right).
Figure 7. A sinusoid pattern rotated at 60 degrees (left) and its FT (right).
Figure 8. A sinusoid pattern rotated at 90 degrees (left) and its FT (right).
What will happen to the FT if we created different patterns of sinusoids? The prediction is that, since the FT is a linear transformation, the corresponding FT will the the superposition of the individual patterns. It is seen from figures 2-4 that the FT of a corrugated roof running at the x-axis are two dots symmetric along the x-axis. The pattern created is a product of two corrugated roofs, one running at the x-axis and the other at the y-axis. As expected, the FT is the combination of the individual FTs. A sinusoid rotated at 30 degrees was added to the pattern at figure 9. Its FT is expected to be composed of four symmetric dots, and two tilted dots like in figure 6 (left). As seen in figure 10, the FT is indeed what expected.
Figure 9. A product of two corrugated roofs, one running on x-axis and the other
at the y-axis (right), and the corresponding FT (left).
Figure 10. A product of two corrugated roofs, one running on x-axis and the other
at the y-axis (right), added with a sinusoid rotated at 30 degrees, and the corresponding FT (left).
I would like to thank Ma'am Jing and Joseph Raphael Bunao for enlightening me on some points. I would give myself a score of 9/10. I produced all the required outputs, but I was not able to answer one question about finding the actual frequencies on an interferogram image.
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