Properties and Applications of the 2DFourier Transform

Created patterns for rectangle aperture, wide aperture, two dots and two dots with different spacing and their corresponding FT.

Created 2D sinusoids in the x direction with varying frequencies (2,5,14) with their corresponding FT. We can see that as we increase the sinusoids frequency they get more shifted in the FT space.

Below we add a constant bias of 0.2, and we observe a central dot in the FT of the sinusoids. So for an interferogram of a Young’s double slit you can apply a mask to get the specific frequencies that correspond to the signal being measured. Like a low pass filter to mask out the point in the middle of the sinusoid’s FT.

Next we add a non-constant bias of sin(0.05). Since the bias is non-constant, the noise in the FT space is not only a point. In order to get the signal we are interested with we should also identify first the nature of the noise and create specific masks to cut them out.

Below are the patterns of sine frequency 2, sine with frequency 5 and rotated at 90 degrees and their corresponding sum. We see here that vertical lines in real space corresponds to horizontal lines on the fourier space and vice versa. We can predict the resulting FT of signals from this.

Below are also combinations of rotated and added sine functions with their corresponding labels. From what we have learned, it is evident even before showing the FT of the signals that signals with diagonals will produce both horizontal and vertical lines in the FT space.

These are the corresponding FT of dots and dots with radii.

Below are the FT’s of squares with some width and with gaussians.

Below is a 200 by 200 array with 10 ones, a 9 by 9 pattern of the letter C and their corresponding convolution and FT.

Now a 200 by 200 array is created with evenly spaced ones along the x and y axis.

For the fingerprint ridge enhancement, a low pass and a high pass filter was first applied separately to see their masking effects on the image. Then a combined band pass filter was made in order to have the best result for the image. The results for these processes are found in the attached python notebook. Below is the original image and the image that went through the band pass filter.

The image for the moon landing had horizontal lines and from observing its FT needs to be masked along the y axis by some line and some dots in order to effectively remove the horizontal lines in the image.

For the brickwall, it was hard to remove the pattern just by adding a vertical and horizontal mask along the x and y axis. I believe some spaced frequencies should be used to effectively remove the brick wall pattern. This can be done better by applying the mask found on this site. https://goo.gl/images/cx8Q3V