ACTIVITY 6 Properties of the Fourier Transform

6.A Familiarization with FT of Different Patterns

Below are some simple patterns commonly used in image processing and their Fourier transform. It can be seen that the Fourier transform is unique to each pattern. Complex patterns may have Fourier transform as combination of the Fourier transform of these simple patterns.

Figure 1. click the image for a better view

6.B Anamorphic Property of the Fourier Transform

The anamorphic property of Fourier transforms is being investigated here by getting the Fourier transform of sinusoid of different frequencies and at different directions. Figure 2 shows the variation in the Fourier transform when the frequency of the sinusoid is varied. The increase in frequency is illustrated in the decrease in spacing between the lines and the width of the lines itself (white lines). Notice that the image of the sinusoid can also be represented as equally spaced slits. The Fourier transform is, therefore, composed of the Fourier transform of two slits (see Figure 1). Two rectangular spots are prominent in the Fourier transform. The spacing between these two spots increases as the frequency of the sinusoid is increased. These also become longer when the frequency is increased. It can also be observed that the direction of the Fourier transform is perpendicular to the direction of the lines. In this case, the sinusoid is composed of horizontal lines and so, the Fourier transform is in the vertical direction.

Figure 2. click the image for a better view

Figure 3 illustrates the effect of a constant bias to a sinusoid. It is evident from the images that adding a constant bias would just give a zero frequency in the Fourier transform, i.e., at the center. No matter how large the constant bias is, it will just be translated as a zero-frequency in the Fourier domain. To get the actual frequency of the sinusoid, one can cover/filter the zero-frequency using a filter mask that has zero value at the center and a value of 1 at other pixel locations. Filtering the zero-frequency leaves only the actual frequencies in the Fourier transform. If the bias added isnonconstant and unknown, a Fourier transform containing only the actual frequencies can still be produced by creating a filter mask which has a value of 1 only at pixel locations having the information about the actual frequencies. However, this is only possible if the actual frequencies are known, i.e., involving the common patterns such as those presented above.

Figure 3. click the image for a better view

Rotating the sinusoid can also be reflected in its Fourier transform as shown in Figure 4. But the direction of rotation in the Fourier transform is always opposite or perpendicular to the direction of rotation in the image. In the figure below, the sinusoid is rotated with respect to the horizontal but the rotation in the Fourier transform is with respect to the vertical. The angle of rotation is the same for both domains.

Figure 4. click the image for a better view

The Fourier transform of a combination or an addition of sinusoid with different frequencies and with different angle of rotations is just the superposition of the Fourier transform of each component. As predicted, the Fourier transform in Figure 6 is the superposition of the Fourier transform the 8 sinusoid that constitute the image. If the sinusoid are multiplied, then a Fourier transform such as that in Figure 5 will be the result.

Figure 5. click the image for a better view

Figure 6. click on the image for a better view

In summary, different patterns produce different patterns of Fourier transform. A combination of these patterns results in a superposition of their Fourier transform. The anamorphic property of Fourier transform is the characteristic difference in spacings due to the change in spacing in the image. It also explains the perpendicularity in the direction of the image and the Fourier transform lines. Any rotation in the image results in a rotation in the Fourier transform.

I give myself a grade of 10 for this activity because I was able to do everything that has to be done. I would like to thank Thirdy and all those who have helped me finish this acitvity.