Figure 1. Different image patterns: (a) circle, (c) annulus, (e) square, (g) horizontal double slit, (i) vertical double slit and (k) two dirac deltas symmetric with the y-axis and their corresponding Fourier transform (b, d, f, h, j and l, respectively)

The circle and the annulus have a similar Fourier Transform, which are both circular in nature. The square has a square-ish FT which is indicative of its shape. More shapes (or apertures in this case) have unique FTs. Next are the FTs of the slits. Again, both FTs are indicative of its original image. Last is the FT of two dots symmetric along the y-axis. We can see that it has a somewhat sinusoidal pattern along the x-direction.

Note that we may be exchanging terms FT and FFT. FT means Fourier Transform while FFT means Fast Fourier Transform. The main difference is well, the latter is, uhm, fast. Yeah, I’m not joking! FFT uses the 2^x where x is a number to optimize FT, making it fast. Most programs use FFT to save time instead of directly evaluating FT by itself. :)

For the next part of the activity, we were asked to find the FT of two dirac deltas, two circles and two squares. Figure 2 shows the results. The first column shows the original images, the second column shows the 3 Dimensional look of the original image, then we display is FFT in 3D and 2D. The dots show a fan-like FT with sinusoidal ridges. As for the circles, it is similar to the FT of that of in Figure 1b. Conversely, the FFT of the squares is similar to that of Figure 1f. This is again because certain apertures have a unique corresponding FFT pattern.~~~~