Entry 13: Properties of the 2D Fourier Transform

In this activity, we explored the properties of the 2D Fourier Transform (FT).  The main goal of this activity is to have a nice grasp of the ideas and the actions of FT to a subject image.

Two sections were dedicated in this activity.  The first about familiarization of FT of different  2D patterns while the second section was about anamorphic property of the FT.

Figure 1: Fourier Transform pairs of basic patterns.

In Fig. 1, the FT of the basic figures in the top most layer are shown in the bottom.

The corresponding images are called FT pairs.  It is also useful to know that the FT of the image's FT will yield the original image.

The second section about the anamorphic property of FT when applied on 2D images was then studied.  When we say anamorphic property, it refers to producing different optical imaging effects along mutually perpendicular radii.

Figure 2: Anamorphic property of Fourier Transform.  Shown here are the resulting Fourier Transforms of each sinusoidal images of varying parameters and conditions.

In Fig. 2, we see the FT's of different sinusoidal patterns.  On the first column, the FT pair of a sinusoid along the x-axis with frequency equal to 1 Hz is shown.  The second column displays the FT pair of sinusoid along the x-axis with frequency equal to 4 Hz. Observe that the FT of the sinusoid with frequency equals 1 are symmetric dots about the origin of the fx-axis located 1 unit away while the FT of the sinusoid with frequency equals 4 is depicted by symmetric dots about the origin along the fx-axis located 4 units away.  From this observation, we can infer that the signal's frequency are projected in the frequency space after the application of FT.

Figure 3:  Effects of DC bias and bias with frequency on a sinusoidal signal and their corresponding Fourier Transforms.

In Fig. 3, we show the effects of DC bias and bias with frequency to the Fourier Transform of each original sinusoidal image.  The FT of the DC-biased sinusoidal signal is characterized by a pair of symmetric dots indicating the principal frequency of the signal and a dot in the center.  The dot in the center is the constant carrier signal "DC".  This behavior depicts the mathematical form of the FT of a constant value which is equal to a Dirac delta at 0.

In this activity I would give myself a 10/10 for a completeness.

Source:

    Activity Sheet for Activity 7 - Dr. Maricor Soriano