VII

2D Fourier Transforms (2D FT) require a deeper understanding of the underlying mathematical arrangements that happen in order to appreciate its capabilities. More imagination and knowledge are needed to properly visualize the transform's effects on images.

Note: Most of the Fourier Transforms shown here are zoomed and not it their scale. This was done to enhance the clarity of the discussion.

7A. 2D FT Familiarization

Figure 1. L-R Various patterns: Square, annulus, square annulus, 2 slits & 2 dots (top row) 

and their respective Fourier Transforms (bottom row)

Fig. 1 exemplifies the quite complicated nature of visualizing FTs of 2D patterns if one doesn't have a firm grasp of its concept. From the square pattern, one realizes that the focused FT in the middle demonstrates that the frequency is focused in the middle, with the trailing  frequencies demonstrate the x and y dependence, it being a square. The annulus also demonstrates this, however, the 2D FT is also circular. We get the sense that these arise when there are sharp changes of value (edges) in the image. Speaking in terms of frequency, this is a sharp change. From the square annulus, we also observe the same pattern but with more pronounced black segments, due to the interruption in the middle.

The slits and dots are quite straightforward. The line can be treated as a sinusoid with a frequency that can be observed is projected on its FT. The dots show their spherical nature in its FT.

7B. Anamorphic Property of Fourier Transforms

Figure 2. L-R Fringes of varying frequency: f = 4, 8, 16 & 32 (top row) 

and their respective Fourier Transforms (bottom row)

Fig. 2 shows a more simpler approach to understand the Fourier space. We see an immediate trend: as the frequency is increased, the spacing between the values becomes greater. This is now an exemplary example that the scales of the Fourier space is frequency. The presence of the 2 dots is an artifact of the dimension of our image: it is symmetric. Also worth noticing is that the symmetry is confined in one axis, since the corrugated roofs are in one dimension.

Figure 3. L-R: Fringe pattern with f =16 with a bias constant bias of 2 and its FT.

Fig. 3 is the result of adding a constant bias to the function that generates our image. We see that this resulted to a point between the 2 spots in its FT. This indicates the shift of the function from the origin, and thus the symmetry was broken. 

To find the actual frequencies in an interferogram image, it is then immediate that FT is the solution. For non-constant biases, we can find the frequencies by noting that what we will see should be symmetric on a diagonal.

Figure 4. L-R: Fringe pattern with f =16, shift of 30 degrees and its FT 

Fig. 4 illustrates that a shift in the actual image also reflects as a shift of the bright spots in inverse space.

Figure 5. L-R: Fringe pattern with f =16 on x & y axis and its FT 

Fig. 5 further cements the idea of the symmetry of the FT of straightforward sinusoidal functions. The FT can be interpreted by looking at its quadrant 1. Treating the center as the origin, the bright spot is then easily viewed as having x & y components. Then with that, we see that it is also symmetric in those axes, thus producing the 4 spot FT image.

Lastly, I add 3 rotated sinusoids with (same f, theta=30,45,50) to Fig. 5. Before I see the FT, I tried to predict it. The  four spots on the FT of Fig. 5 should remain. The addition of the 3 sinusoids should add a total of 9 spots: each per quadrant (symmetry) with an angle equal to theta measured from the origin.

Figure 5. FT of a 2D corrugated roof added with 3 rotated sinusoids

Disappointment. Apparently, addition doesn't yield a symmetry in the Fourier space since it only adds, it doesn't "convolve" functions such as multiplication.

  Self-Assessment: 9/10