Activity 6: Properties of the 2D Fourier Transform

Objective: To increase familiarity with the various

Tools: Scilab with SIP toolbox.

Procedure:

1. Familiarization of FT of Different 2D Patterns.

Using the same routines as with the previous activity, we take the FTs of the following apertures:
This yields the following FT patterns:

The first two patterns are identical to the ones from the previous activity. The third image is from an annulus and if we compute the far-field diffraction pattern(which is the same as the FT) we see that the result is a difference between two orders of Bessel functions. Hence we see a set of missing orders or fringes. The fourth is essentially the same only now the functions are sinc functions in x and y. The fourth is the double slit pattern at far field. This can be thought of as coherent oscillators interfering with each other. This means that the upper and lower sections cancel each others contribution. This leaves the central strip hence we only see a pattern at the center of the strip. The last image represents two circular apertures, this results in a modulation of the phase from the other hole causing the interfernce fringes that are superposed on the circular diffraction fringes.

2. Anamorphic Property of the Fourier Transform

A. We take a 2-D Sinusoid with frequency f=4:

We then take the FT of this pattern which yields two points in the transform plane. This corresponds to the frequency of the input. We do this for two other frequencies,f=1 and f=15 to get the following images arranged in increasing f:

We see that as the frequency increases, the points move further from the center. This makes sense if we treat the points as the frequncies of the sine waves (which they are). Thus as the frequency is increased the points move away from the center.

Now if we add a constant bias, we essentially add a field with all the possible frequencies present. Hence from Fourier theory, the FT of this is a single point. We see this when we get the mesh() of the biased sine wave. For comparison, we place on the left a mesh plot of the FT of the same wave without a bias:

The spike at the center represents the frequency of the bias or what is known as the zero-order component. Note that this is of a greater intesity than the sine spikes. If in a real image this becomes an issue, it is possible to subtract the constant value from the input image. This would leave the two spikes represnting the sine. The position of the spikes corresponds directly to the frequency hence we can find the spatial frequency of the fringes in areal image. On the other hand, if the bias is not constantand is a low f sine, we can use the FTs and remove the lower frequency component. We can then take the inverse transform to leave us with the filtered sinusoidal wave.

If we rotate the sine waves like the ones below:

We get the following patterns, which are consistent with the previous result:

If we multiply two sines or any function with two frequencies, we get a superposition of the transforms of both. In this case we use a product of two sne functions:

Which has a transform like:

This corresponds with what we expected as we see two sets of spikes which individually look like the pattern for a simple sine function.

For the last part, if wee add this function to the rotated sines above, we get an image which looks like:

Which has a transform like:

All this essentially tells us that the 2D FT obeys the priciple of superposition.

Reference: AP 186 Activity 6 Manual

Evaluation: For the proper results, a grade of 10 is warranted.