 Convolution Pa More
In this section, instead of convolving a geometric shape, we convolve a 3×3 matrix of 1’s and 0’s. The pattern chosen (the “X”) is seen below. This is convolved with a 200×200 array with 10 randomly placed 1’s (middle in figure below). The effect can be seen on the right of the figure. As can be seen, the X’s in the convolution appear in the same location that the 1’s were in. Hence, the convolution replaced each 1 (dirac delta approximation) by the X pattern.
Next, we observe the Fourier Transform of dots. As we know, the FFT of a sinusoid are two dots located symmetrically from the center. In the same way, the FFT of two dots located symmetrically is a sinusoid. So taking the FFT of multiple dots along the x and y axes, we expect to obtain a summation of many sinusoids. Below, we see the results when the dots are spaced 10, 20 and 40 pixels apart. We see that the FFT of dots along x and y yield a grid of horizontal and vertical lines. This makes sense because a line of dots is similar to a slit, and the FFT of a single vertical slit results in horizontal lines.
A noticeable feature is that as the dot spacing increases, the grid spacing becomes smaller. This is another example of anamorphism. What’s also nice is that even though dots are being removed from the left image, more lines appear in the image on the right. This just shows that the FFT is more complex, and that destructive interference of sine waves should not be underestimated.
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