Activity 7: Enhancement in the Frequency Domain

7A. Convolution Theorem

In this part, we were tasked to observe the changes in the modulus of the FT of the images as the size of the apertures are increased.

Figure1. 2-dot images and their FTs

Using two one-pixel dots, Fig.1(upper left), a grating-like FT was observed. As the radii of the dots increase, the FT becomes circle with decreasing radius located at the center of the two dots. this is expected since the FT is in inverse space.

Figure 2. square images and their FTs

Same was observed using squares. As the width of the squares increases, the width of their FTs decreases. Again, this is expected.

Also, for Gaussian dots, same was observed in Fig.1. As the variance of the Gaussian is increased - which results to an increase on the width of the Gaussian dots, their corresponding FTs is decreased.

Figure3. Guassian dot images at different variances and their FTs

We also take a look on how the FT of these Gaussian dot images differ from that of the inverted images (max becomes min; min becomes max). Here we plot in Fig.4 the images of the real part and imaginary part of the original and inverse Gaussian dots for comparison.

Figure 4. (upper row) images for the real part of original and inverse of 2 Gaussian dots; (lower row) images for the imaginary part of the original and inverse of 2 Gaussian dots.

We observe from the figure that only the real part changed upon inversion. The imaginary part remained the same.

7B. Fingerprints: Ridge Enhancements

In this part of the activity, we are to remove the blotches present in the image. Here we use the FT, we use a mask to remove the frequencies that result to the blotches and retain the information needed. We then take the inverse FT, we obtain a better image.

Figure 5. Removal of blotches present in the image in top left. A mask in bottom left was multiplied to the FT of the image (top right) to obtain a reconstructed image in bottom right.

As observed in the reconstructed image in Fig. 5 (bottom right), we essentially remove the blotches from the original image. Although the area surrounding the blotches darkened. This can be further improved if a better mask is used.

7C. Lunar Landing scanned Picture: Line removal

In this part of the activity, we were tasked to remove the vertical lines present in image below. Here, we use the Fourier transform to take information of the image in the frequency domain. We remove the frequencies that cause the vertical lines. From the previous exercise, we have learned that the frequencies of vertical lines are contained on the horizontal axis of the FT. Therefore, removal of frequencies lying on the horizontal axis removes the vertical lines.

Figure 6. Steps done (from upper left to lower right) to remove the vertical lines in the image.

From the FT in Fig.6(upper right), a mask was created Fig.6lower left) that will remove certain frequencies on the FT. The mask was then multiplied to the FT. Its FT was taken to recover the better image. In Fig.5, we see that the vertical lines were removed by removing the corresponding frequencies. Using FT, we were able to obtain a better image.

7D. Canvas Weave Modeling and Removal

In this part, we use again the technique demonstrated in 7C to remove the weave cross-hatches on the image in Fig.6(upper left). Since, we are dealing with cross-hatches (intertwined vertical and horizontal lines), we must take the frequencies on the horizontal and vertical axis of the FT domain. We might also consider that the cross-hatches are a result of the product of sinusoids (which has also been observed in Activity 3 Fig.5) and hence we also remove that constitute these sinusoids.

Figure 7. Steps of weave cross-hatches removal

Using the mask in Fig.7(lower left), a reconstructed image in Fig.7(lower right) was obtained. Here we see a decrease of cross-hatch visibility.

Figure 8. Weave pattern extracted from the image.

Furthermore, we can see in Fig. 8 a weave pattern extracted from the image. This was done by inverting the mask and multiplying it to the FT of the image and taking the inverse FT.

Although the technique produced promising results, our limit on reconstructing the image (or in other words our limit in creating better reconstructions) is dependent on the visibility of the peaks seen in the frequency domain. We can only block the frequencies that we see.

In this activity, I give myself a grade of 10 for doing a good job.

I thank Cindy Esporlas very much for helping me to create a Gaussian dot. I thank also Master and Thirdy for their help.