Anamorphic property of the Fourier Transform
We created a 2D sinusoid using scilab then we took its Fourier Transform and displayed the FT modulus. The images below are the 2D sinusoid(right) and its FT (left). The frequency of the sinusoid is 4.
We see that the sinusoid is propagating in the horizontal axis. The FT image has two dots in the horizontal axis. We can consider only one dot because we know that the FT gives a mirror image of the frequencies. In the FT of a 1D sinusoid, we see that if you increase the frequency of the sinusoid then the peak would be seen farther in the axis. We checked if this is also the case for a 2D sinusoid. It seems that this was also the case since we saw that the dots are farther away for the origin(center of image) which means that the peak is farther from the axis. The images below are the sinusoids(left) and their FTs(right).
We created a 2D sinusoid using scilab then we took its Fourier Transform and displayed the FT modulus. The images below are the 2D sinusoid(right) and its FT (left). The frequency of the sinusoid is 4.
We see that the sinusoid is propagating in the horizontal axis. The FT image has two dots in the horizontal axis. We can consider only one dot because we know that the FT gives a mirror image of the frequencies. In the FT of a 1D sinusoid, we see that if you increase the frequency of the sinusoid then the peak would be seen farther in the axis. We checked if this is also the case for a 2D sinusoid. It seems that this was also the case since we saw that the dots are farther away for the origin(center of image) which means that the peak is farther from the axis. The images below are the sinusoids(left) and their FTs(right).
Frequency = 8
Frequency = 16
We also checked the effect of rotation of the sinusoid to its FT. We observed that the rotation of the sinusoid also rotates its FT. Below are the images of the rotated sinusoids(left) and their FTs(right). It seems that the rotation in the vertical axis in the spatial domain causes a rotation in the horizontal axis in the frequency domain as seen below.
theta = 30
theta = 45
theta = 60
We also tried a combination of sinusoids. The resulting image and its FT is shown below. For a combination of sinusoids, one in the x axis and one in the y axis, the FT has 4 dots which are in different quadrants. This represents the product of the supposedly 2 dots in the x axis and 2 dots in the y axis.
Fingerprints : Ridge Enhancement
We opened an image of a finger print as a grayscale image in scilab. We get the FT to find where the frequencies of the ridges lie. Using the mkfftfilter command in scilab we make a high-pass filter which allows only the frequencies that we specify. We then get the convolved image of the filter and the finger print. Below are the image of the FT of the finger print as well as of the mask. We can use the log function to show all the frequencies that are present if they are not visible.
The images below are the fingerprint(left) and the enhanced image(right) using the filter. The filter used is a high-pass filter which lets the components with high frequencies pass through and discards the low frequency components. We can see that the fingerprint image is greatly improved.
The finger print is from the site
http://www.dailymail.co.uk/news/article-518628/Foreigners-thrown-Britain-refuse-details-ID-cards.html
Lunar Landing Scanned Pictures : Line removal
In this part of the activity we attempt to remove the line in the lunar landing scanned picture below.
Fingerprints : Ridge Enhancement
We opened an image of a finger print as a grayscale image in scilab. We get the FT to find where the frequencies of the ridges lie. Using the mkfftfilter command in scilab we make a high-pass filter which allows only the frequencies that we specify. We then get the convolved image of the filter and the finger print. Below are the image of the FT of the finger print as well as of the mask. We can use the log function to show all the frequencies that are present if they are not visible.
The images below are the fingerprint(left) and the enhanced image(right) using the filter. The filter used is a high-pass filter which lets the components with high frequencies pass through and discards the low frequency components. We can see that the fingerprint image is greatly improved.
The finger print is from the site
http://www.dailymail.co.uk/news/article-518628/Foreigners-thrown-Britain-refuse-details-ID-cards.html
Lunar Landing Scanned Pictures : Line removal
In this part of the activity we attempt to remove the line in the lunar landing scanned picture below.
In the first part of our activity we noticed that horizontal lines appear to be dots in the vertical axis of our FT. We take the FT of this image and see if we can compare it to what we have seen in the first part of the activity. We expect to see dots or lines in the horizontal axis of our FT since we have vertical lines in our image. Below is the FT of the image.
We see that there are indeed dots and a line in the horizontal axis. We could interpret this as the line that we see in the spatial domain. We prepare a mask that would filter out this dots and line in the frequency domain in the same way retaining the other vital information. The filter looks like this.
The resulting image using this filter is shown below.
We see that our image is now free of the line that we see before but there are some information that were also erased. We could not exactly point out where the frequency of that vertical line lie in the frequency domain. Still, we have enough information to some extent to show the lunar picture quite nicely.
Thanks for Cole Fabros for lots of help. Thanks also for Rafael Jaculbia and Jorge Michael Presto.
I rate myself 10 since I have successfully finished the activity and have done what was required. Also, I have gained quite an understanding of FTs although I believe I still need more practice and learning to do.
No comments:
Post a Comment