Activity 7 - Properties of the 2D Fourier Tranform

Hello guys.. I have done the new activity. It is about how Fourier Transform (FT) works or simply, FT's properties.

1. Familiarization with FT with Different Patterns
   
   In this part of the activity, we were tasked to create several patterns and then apply FT in each of the patterns and of course, observe what happened.
   
   
   1. Square
      
      Figure 1 shows the pattern(top row) I created with two different sizes with their FT (bottom row). The top-left image has an area of 0.64 and the top-right image has 0.16. Applying FT on both images resulted to the image on the bottom row of Figure 1. As you may observe, the larger the area of the square produces smaller pattern in the Fourier plane.

   

   Figure 1. Square with size of 0.8 (top left) and 0.4 (top right) with their corresponding Fourier Transform

   2. Annulus
      
      Next pattern is an annulus which is like a donut. Figure 2 shows the object(top row) and their corresponding FT(bottom row). The thickness of the annulus(top-left) is 0.2 while the thickness of the other annulus(top-right) is 0.4. The same observation will be seen with the Square in part A1.

   

   Figure 2. Annulus with thickness of 0.2 (top left) and 0.4 (top right) with their corresponding Fourier Transform

   3. Square Annulus
      
      This pattern is quite tricky to create but obviously, it is not that hard. Figure 3 shows the object(top row) and their FT(bottom row). The top-left image has an aperture area of 0.36 while the top-right image has an aperture area of 0.04. It is observable that the FT of a smaller aperture area produces sharper image compared to that of a bigger aperture area.

   

   Figure 3. Square Annulus with aperture area of 0.36 (top left) and 0.04 (top right) with their corresponding Fourier Transform

   4. Two Slits along the x-direction
      
      This pattern is the same with a famous experiment, Young's double slit experiment. Figure 4 shows two parts, with a separation distance of (a)0.1 and (b)0.3. Diffraction patterns are seen in their FT which differs between the slit width and the separation distance of the two slits. When their separation distance are the same and differs in slit width(a,b), more order of diffraction is seen as the slit width increases(a - bottom left to a - bottom right). When their separation distances differs and the same slit width(a - bottom left and b - bottom left), still, more order of diffraction is seen as separation distance increases.

   

   Figure 4. Two slits separated by (a) 0.1 and (b) 0.3 and slit width of (a,b - left) 0.05 and (a,b - right) 0.15 with their corresponding Fourier Transform

   5. Two Dots Symmetric about the Origin
      
      I encountered some difficulty in creating this dot but fortunately, I managed to create them. Figure 5 shows the objects(top row) and their FT(bottom row). Both all 3 objects are separated by 1.0 with radii, both 0.1 for the 1^st image, 0.1 and 0.4 for the middle image, and both 0.4 for the 3^rd image. It can be seen the pattern in the Fourier plane decreases in the number of diffraction order as the dot size increases.
      

   

   Figure 5. Two dots with sizes of 0.1 (top leftmost), 0.1 and 0.4 (top middle), and 0.4 (top rightmost) with their corresponding Fourier Transform

2. Anamorphic Property of Fourier Transform
   
   There are many things that are needed to do in this activity but those things are not really that hard but needed a lot of time to do it.
   
   First in the tasks is to create sinusoid of different frequencies and applying FT on each pattern which is shown in figure 6. It can be seen that the pattern separates as the frequency increases.
   

   

   Figure 6. Sinusoid with frequency (top row, leftmost) 4, (top row, middle) 8, and (top row, rightmost) 16 with their corresponding FT

   
   After that, a constant bias is added to the sinusoid in order to simulate a real image that has no negative values. Figure 7 shows this simulated real image with its FT. Comparing with figure 6, it can be seen that there is an additional information at the center of the image. When a non-constant bias is added to the sinusoid, it is easy to separate the two frequencies by getting its FT and locate the frequency where the maximum value is.
   
   

   

   Figure 7. Biased sinusoid with frequency (top row, leftmost) 4, (top row, middle) 8, and (top row, rightmost) 16 with their corresponding FT

   
   Figure 8 shows the rotation of the sinusoid and their corresponding FT. It can be seen in the FT figure that the whole figure rotate.
   

   

   Figure 8. Rotated sinusoid with frequency 10 and θ = (top row, leftmost) 30°, (top row, middle) 60°, and (top row, rightmost) 90° with their corresponding FT

   
   Figure 9 shows the (a)combination of two sinusoid along the x- and y-directions together with their corresponding (b)FT. When two sinusoid were added to one another, its FT form a diamond structure where only the edges are seen(b - top row). When two sinusoid were multiplied to one another, its FT form a square structure where only the edges are seen(b - bottom row). After obtaining the sinusoid, a rotated sinusoid were added. Figure 10 shows the (a)combination and their (b)FT. When the rotated sinusoid was added, only the sinusoid that was rotating rotates from 30° to 90°.
   

   

   Figure 9. Addition (a,top row) and multiplication (a, bottom row) of sinusoid with frequency (left) 4 and 8, and (right) 10 and their corresponding (b)FT.

   
   

   

   Figure 10. Multiplication of sinusoid of frequency 10 and added to a rotated sinusoid with the same frequency at and angle θ = (top row, left) 30°, (top row, right) 60°, (bottom row, left) 90°, (bottom row, right) 120° with their corresponding FT.

Finally, once again, I have completed this activity. It was not that hard but need some time to work on it. I will rate myself with 12 points. 5 points for the Technical part, 5 points for the Quality part and 2 points for the Initiative part because I compared the patterns in different sizes.

Thank you. :0