CINDY 186: Activity 7 | Enhancement in the Frequency Domain

The Fourier Transform properties had been familiarized already in the previous activities. With this knowledge we use it to enhance images not by changing the image itself but its Fourier Transform. Now we examine the FTs of different patterns such as two dots with increasing radius where the smallest dots are two separate pixels as shown in Figure 1.

The FT of the two pixels is a series of lines that looks like a sinusoid. We can remember from previous activities that the FT of a sinusoid are two small dots along the axis of propagation of the sinusoid. Taking the FT of the two separate pixels is like doing the inverse FT of the FT of a sinusoid. The FTs of the two small circles on the other hand is like the product of the FT of the two pixels and a circle like that from Activity 5, and it is observed that increasing the size of the spots decreases the size of the resulting FT.

Figure 1. Two dots with increasing radius and their respective FTs.

We know that a pixel is square in shape. If we enlarge these squares the FT will change from that of a sinusoid pattern to like that of the two small circles only instead of the product of the FT two dots and a circle, the patterns looks like the product of two dots and a scquare. Increasing the size of the squares also decreases the size of the FT pattern just like in Figure 1. The images for the squares and their FTs are shown in Figure 2.

Figure 2. Two squares with increasing size and their respective FTs.

Now Figure 3 shows the two gaussian of varying variance and their FT while Figure 4 shows the Gaussian dots and the inverted counterparts and the real and imaginary parts of their FTs. It may not be visible but there is a small faint set of vertical lines shaped in a circle centered at the bright spot.

Figure 3. Two Gaussians with increasing variance and their respective FTs.

Figure 4. Two Gaussians with increasing variance and their respective FTs' real (middle row) and imaginary (last row) parts.

Using the concepts above we apply them in enhancing images in the frequency domain. We perform enhancement on three images namely a fingerprint for ridge enhancement, lunar landing scanned pictures for line removal, and a digital painting for canvas weave modeling and removal.

RIDGE ENHANCEMENT - Fingerprint

Figure 5. To be enhanced fingerprint image and its FT.

Enhancing this fingerprint such that the ridges are defined is the main goal for this part of the activity. Looking at the FT of the fingerprint we can see areas that are the brightest and could possibly be the FT pattern for the ridges. Basing from the FT we then create a mask to obtain the ridges of the fingerprint alone.

Mask 1: Thresholding

LINE REMOVAL - Lunar landing scanned pictures