6th Activity: Enhancement in the Frequency Domain

The goal of this activity is to improve images using filters and Fourier transforms.

A. Convolution Theorem

Two dots with one pixel each was created along the x-axis, symmetrical about the center. The Fourier tranform of this was taken as shown in Figure 1. It can be seen that the obtained Fourier transform of the dots are lines along the y-axis.

 Figure 1(a) two dots along x-axis symmetric about center; (b) Its Fourier transform 

     Next, two circles with radius = 0.2 was made symmetric about the center along x-axis. The 3D plot of the circles and Fourier transforms are shown in Figure 2 (b) & (c). In Figure 2 (d) & (e), it can be seen that there is a bright spot surrounded by fringes. There are also vertical lines along y-axis. 

Figure 2. (a)  two circles along x-axis symmetric about center ; (b) its 3D plot; (c) Its Fourier transform in 3D;
(d) & (e) Its Fourier transform

     Figure 3 shows circles with varying radii and their corresponding Fourier transforms. It can be observed that as the radius of the circle increases, the bright spot in the Fourier transform decreases.

Figure 3. Fourier transform of circles along x-axis symmetric about center; (a) circle with r = 0.2; (b) circle with r = 0.5;       (c) circle with r = 0.8 

     This time, the circles were replaced with squares. The results are shown in Figure 4.
   

Figure 4. (a)  two sqaures along x-axis symmetric about center ; (b) its 3D plot; (c) Its Fourier transform in 3D;
(d) & (e) Its Fourier transform

      As the width of the square is increased, the Fourier transform shrinks.

Figure 5. Fourier transform of squares along x-axis symmetric about center; (a) square with side = 0.1; (b)  square with side = 0.25; (c)  square with side = 0.8

     An image containing two gaussian was created and is shown with its 3D plot in Figure 6 (a) & (b). Figure 6 (c) & (d) shows the Fourier transform of the generated image.

Figure 6. (a)  two gaussians along x-axis symmetric about center; (b) its mesh; (c) 3D plot of its Fourier transform; (d) Its Fourier transform

      As shown in Figure 7, as the value of σ is increased, the Fourier transform is reduced. 

Figure 7. Fourier transforms of gaussians with σ= 0.01, 0.1, and 0.9, respectively

     A 200x200 array of zeros was then generated with ten random locations of 1. A 5x5 pattern was also created. The Fourier transform of both arrays were taken and multiplied to each other. Then, the Inverse Fourier transform of it was obtained and shown in Figure 8. This process is called convolution.

Figure 8. Convolution of two arrays

     Another array was created with 1's at equally spaced locations in the x and y axis. It is shown at the left of Figure 9. Its Fourier transform is also shown beside it. It can be seen that its Fourier transform is composed of vertical and horizontal lines that are equally spaced. As the spacing between them is increased, the number of vertical and horizontal lines also increased accordingly. 

B. Lunar Scanned Pictures: Line removal

     An image shown in Figure 10 was provided to us. It was then converted into grayscale image. Then the Fourier transform of the image was obtained and shown at the right of Figure 10. The Fourier transform showed a bright spot at the center with a cross pattern.  

Figure 10. (left) Craters in the moon ; (right) Its Fourier transform

     It was shown in Figure 1 of Part A of this activity that the Fourier tranform of dots along the x-axis is composed of parallel lines along the y-axis. Since the image to be enhanced have lines along the x-axis, the filter that must be made must consist of series of dots along the y-axis. The created filter mask is shown in Figure 10.  

Figure 10. Filter for enhancing image of the moon

     Multiplying this filter to the Fourier transform of the original image, the original image was now enhanced as shown in Figure 11.  It can be seen in Figure 11 that the horizontal lines in the image was removed using the filter created. Thus the filter used was effective in enhancing the original image.

Figure 11. (left) Original image of the moon; (right) Enhanced image 

     C. Canvas Weave Modeling and Removal

     Same procedure were done to enhance the image shown in Figure 12.    

Figure 12. Detail of "Frederiksborg", oil on canvas by Dr. Vincent Daria

     It was then converted into grayscale image and its Fourier transform was obtained as seen in Figure 13.

Figure 13. Fourier transform of the image

     Using GIMP, a filter mask was created by drawing circles at locations corresponding to the bright spots in the frequency domain. There is a low signal to noise ratio at the black region so it was covered to remove the weave pattern.

Figure 14. Mask filter for the canvas

     Figure 15 shows the result after filtering the original image. It can be seen that the weave pattern was indeed removed.

Figure 15. Enhance image of the canvas

     In order to visualize the canvas weave pattern, the enhanced image was subtracted from the original image. Figure 16 shows the obtained weave pattern.


   

Self-evaluation:

I will give myself 10/10 for completing the job and presenting the results clearly.