Properties of the 2D Fourier Transform

In this entry, the properties and concepts of the 2D Fourier Transform are investigated. As a summary of the last entry, the fourier transform (FT) of a signal basically gives the spatial frequency distribution of that signal. According to the manual presented by Dr. Soriano, one unique thing about the 2D FT, in comparison with the 1D FT, is that rotation of an image (considered as a 2D signal) translates to the rotation of its corresponding FT.

I. Familiarizing with the different FT of several 2D patterns.

Recall the concepts and basic properties of Fourier Transform. Note that all the patterns and images presented are made using scilab. Thier FT's are derived using the algorithm developed by Cooley and Tukey and is implemented in Scilab using the fft2() function.

Figure 1: Square and its corresponding Fourier Transform


Figure 2: Annulus (donut) and its corresponding Fourier Transform


Figure 3: Square Annulus and its corresponding Fourier Transform


Figure 4: Two slits along the x-axis that spans the whole y-axis and its corresponding Fourier Transform


Figure 5: Two symmetrically spaced dots and its Fourier Transform

Figures 1 to 5 show the basic and common patterns and shapes with their corresponding Fourier Transform. Analytically solving for the Fourier Transform of these patterns verifies the Fourier Transform of these patterns derived using the algorithm presented by Cooley and Tukey.

Familiarizing the FTs of these basic 2D patterns is essential because due its linear transformation property, any relatively complex 2D pattern can theoretically be broken down to basic patterns whose FT can be easily derived.

II.Anamorphic Property of the Fourier Transform
Consider a 2D sinusoid along the X direction, as shown in Figure 6.

Figure 6: 2D sinusoid along the X direction with frequency equal to 4


Figure 7: Fourier Transform of the sinusoid in Figure 6

Figure 7 shows the Fourier Transform of the sinusoid in Figure 6. Based on Figure 7, the FT of a sinusoid is a pair of dots symmetric about the center and is along the X axis, as expected theoretically. Now consider a sinusoid with increasing frequency, Figure 8 shows that for an increasing frequency, the number of sinusoid present in the image also increases. Figure 9, on the other hand, relates to the effect on the FT of the sinusoid for an increasing frequency. Figure 9 shows that for an increasing frequency, the distance between the two dots also increases.

Figure 8: 2D sinusoid with increasing frequency


Figure 9: Effect on the FT of the sinusoid for an increasing frequency

Before proceeding any further, it must be noted that real images cannot display negative values and therefore must be offset to obtain real positive values. Sinusoid images (including the FT's) that are already presented and also those that will be presented have offsets so that all their corresponding values will be shown.

Now consider the rotation of sinusoid functions, illustrated in figure 10. Figure 11, the FT of these rotated sinusoids, shows that the dots also rotate corresponding to the rotation of the sinusoid.

Figure 10: rotated sinusoid


Figure 11: Fourier Transform of the rotated sinusoid. The images, rotation of the dots, correspond to the rotation of the sinusoid in Figure 10.

Now consider the combination of two sinusoids, one sinusoid is along the x-axis while the other is along the y-axis, as shown in figure 12. Figure 13 shows that the FT of this combination is 2 pairs of dots, one pair is about the x-axis with the other pair about the y-axis. Figure 13 also shows that both pairs are symmetric about the center of the image.

Figure 12: Combination of two sinusoid function. One is along the x-axis and the other is along the y-axis.


Figure 13: The Fourier Transform of the combination of sinusoids

Based on Figure 11, rotating the sinusoid also rotates the FT (the dots). From this postulate, it can be predicted that rotating the combination of sinusoids also correspond to the rotation of the dots.

Figure 14: Rotation of the combined sinusoids


Figure 15: FT of the rotated sinusoid combinations.

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Technical Correctness: 4/5 (due to late posting)
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