Fourier Transform Model of Image Formation

Fourier Transform (FT) is the transformation of any input signal found in spatial domain into its frequency domain (1). For images, FT is used to decompose the each point into its corresponding sine or cosine components . However, images do not fully show all the frequencies composing the images. Thus, this activity is limited to the discrete Fourier Transform which shows limited number of frequencies but are enough to depict the spatial domain of the image.

In this activity, the Fourier Transform of different images are taken. Moreover, the properties and extension of Fourier Transform such as convolution and correlation are implemented.

The figures above describe the transformation of the original image (first image) as it undergoes FFT2 and FFTSHIFT. The third picture represents the Fourier transformed image which is a central bright spot. This agrees with the ideal Fourier transformation of circle. Applying another fourier transformation to the previously transformed image inverts the image with respect to the horizontal axis.

Applying the same procedure to letter "A", I arrived at the figures above. The inversion of "A" is still experience upon the application of another Fourier transform as shown in the last figure above. The star-like spread of light in the image just above the last image shows the fftshift of "A".

The figure above shows the effect of the size of the aperture to the quality of the image. Notice that as the aperture gets larger, the image's resolution becomes better.

For the images above, it can be seen that the sentence "The rain in Spain stays mainly in the plain" correlates with the letter "A" by producing more intense light in the part of the sentence which contains "A".

The Last three figures shows the corresponding image produced when an image pattern with a sum of zero is convolved with another image. It can be observed that some parts of the word "VIP" are lost depending on the location of the zeroed-sum in the matrix. The figures also show that the spot pattern gives out the best image quality among the three matrix patterns.

I would like to thank Master and Jaya for reminding me of the deadline of this activity as well as Gary and Rara for giving suggestions on improving my blog.

For this activity, I am giving myself a grade of 9. I was not able to attend the class and I finished the activity in a later time.

References:
1. http://homepages.inf.ed.ac.uk/rbf/HIPR2/fourier.htm