Activity 6. Properties of the 2D Fourier Transform

Disclaimer: partially done.

6.A Familiarization with FT of different 2D patterns

For this part of the activity the Fourier transform of the different patterns: square, square annulus, annulus (doughnut), Two slits along x-axis symmetric about the center and two dots along x-axis symmetric about the center were simulated.

Two-slits

square

two dots

square annulus

annulus(doughnut)

B Anamorphic property of the Fourier Transform

For this part of the activity a 2D sinusoid was generated. Similar to the previous part of the activity, its Fourier transform was calculated.The Fourier transform of a sinusoid is a dirac delta peaking at both positive and negative values of its frequency. The sinusoid generated is along the x axis with frequency equal to 4. Thus its resulting Fourier transform is an image peaking at two point (which I think is 4 and -4).

The Fourier transform of a sinusoid with a constant bias was also observed.bias = 0.1, frequency = 4

bias = 0.5, frequency = 4

bias = 2, frequency = 4

Adding a constant bias in the sinusoids results to a peak in the origin in the frequency space of the Fourier transformed image. As can be seen from the images above, the Fourier transform of the images for a sinusoid with frequency of 4 and bias equal to 0.1,0.5 and 2 have peaks at the origin. It can be noticed that at values much less than 1, the peak found at the origin has a lesser intensity compared to the peaks found at the value of the frequencies. And as the value of the bias increases (greater than 1), the intensity at the origin becomes greater than the value of the frequencies. This is because the range of values of sine is from -1 to 1 only. Hence, biases at this range does not exceed the peak of the frequency values. However, biases out of this range has a peak higher than the frequency values.

When the generated sinsusoids were rotated by a desired angle, the resulting Fourier transfrom were the following:

frequency = 4, angle = 0 degreesfrequency = 4, angle = 45 degrees

frequency = 9, angle = 90 degrees

When the sinusoid was rotated by an angle, the resulting Fourier transformed was also rotated by the same angle. The images abve shows sinusoids of frequency 4 and 9 with angle of rotation equal to 0, 45 and 90 degrees.

When the combination of two sinusoids was Fourier transformed, the resulting image is the following:

frequency = 4

The Fourier transform of the new sinusoid which is the product of two sinusoids (corrugated roof) running in the x and y direction shows the frequencies present in the product.

When several sinusoids are added to the previous sinusoid shown above the resulting image and its Fourier transfrom is found to be..

frequency = 4 at angles 30, 90, 120 and 330 and previously generated sinusoid

The Fourier transform of the image is the same as my predicted result. One property of Fourier transform is linearity. It means that given a function h which is expressed by f and g and its corresponding Fourier transform H, F and G, the Fourier of h is given by..

h = f + g
H = F + G

Because of this property of Fourier transform the image of the generated sinusoid above will just be sum of the Fourier of the previously generated sinusoid which is the corrugated roof, and the sinusoids with frequency equal to 4 rotated at angles 30, 90, 120 and 330 degrees.

I give myself a grade of 10 since Iwas able to do all the required tasks and I was able to observe and understand the properties of Fourier transform.