A6 – Fourier Transform Model of Image

In this activity we familiarized ourselves with discrete FFts. We made a 128*128 image of a circle that is centered. The images below are the original image, the unshifted FFT image, the shifted FFT image and the image with FFT done twice.

The FFT image observed is consistent with the analytical fourier transform of a circle which are airy discs. We see that performing the FFT twice on the image reverted it back to the original image. Theoretically it would invert the image upside down but since the circle is symmetric we cannot see if this really happened.

The same procedure is done on a 128*128 image of letter A that is centered. The images below are the original image, the unshifted FFT image, the shifted FFT image and the image with FFT done twice.

We see that the FFT of the image has a high intensity at its center. Also the image with FFT done twice is the original image inverted upside down which was expected.

In the second part of the activity, we have done a simulation of an imaging device. We made an aperture represented by a white circle with a black background. Our image is the text VIP centered. We get the resulting image of the convolution of the aperture and the FFT of our image. The images below are the image, aperture(small, medium and large) and the resulting image.

A larger aperture seems to have a better image but still not a perfect one.

The next part of the activity is template matching using correlation. We get the FFT of a text “THE RAIN IN SPAIN STAYS MAINLY IN THE PLAIN.” then the FFT of a text "A" then we multiply the latter to the conjugate of the former. After that, we get the inverse FFT of the result. Below are the two text and the resulting image.

We see that the resulting image looks like a inverted blurred image of the phrase. We can also see that the places where there a letter 'A' is brighter than the rest. This means that letter 'A' was emphasized and matched.

The last part of the activity is template matching of edge pattern. We created a 3*3 matrix edge pattern such that the total sum is zero. I used a horizontal, a vertical, a diagonal and a spot edge pattern. The image used is the text 'VIP'. The image and the results are shown below.(horizontal, vertical, diagonal and spot)

Notice that the areas that are bright in the resulting image seems to correspond to the pattern that was used. If a horizontal pattern was used then the bright areas are the horizontal edges. This is the same for the rest.

Collaborators: Rafael Jaculbia, Cole Fabros and Marge Maallo

I rate myself 9 out of 10 because I have finished the task and also learned some things but I think it is not yet enough.