A7 - Properties of the 2D Fourier Transform

The Fourier Theorem states that any signal or image can be represented by a superposition of sinusoids. Getting the Fourier Transform of a signal returns the spatial frequency distribution of the signal. In this activity, the rotation of the FT's upon rotation of the sinusoids was also observed.

Familiarization with FT of different 2D patterns

The image on the left is the object, the image on the right is the FT.

a. Square


b. Annulus


c. Square Annulus


d. Two slits along the x-axis


e. Two dots along the x-axis


Anamorphic pattern of the Fourier Transform

The following are sinusoids with increasing frequency. Their FT's are seen at the right

f = 4


f = 50


f = 100


We could observe that as the frequency increases, the dirac deltas that we see as dots move farther away from each other.

Adding a bias

Adding a constant bias:


Adding a constant bias shows a dot in the center representing that the constant has a zero frequency.

Adding a sinusoid with low frequency:


Adding a low frequency sinusoid as bias shows another pair nearer to the center in the FT.

Supposing that these images represent a real image of an interferogram in a Young's Double Slit experiment. We can get the actual frequencies by easy getting the Fourier Transform. Once this is done, we could now differentiate the bias and the signal. We could get the value of the actual frequencies by choosing the higher frequency and we could also reconstruct the signal by high pass filtering.

Rotating the sinusoid

theta = 30


Rotating the sinusoid results to an FT that is rotated also.

Combination of sinusoids in X and in Y



Combining sinusoids in X and Y result to an FT with dimensions along X and Y also.

Adding several rotated sinusoids



Even though the addition of different rotated sinusoids result to a chaotic signal, the FT is easily predicted by just adding the FTs of the individual sinusoids. I did this and I was able to correctly predict the FT of this noisy signal.

For this activity, I give myself a grade of 10 for understanding the basic properties of the Fourier transform.