Activity 5: Fourier Transform Model of Image Formation

For this activity, Fourier Transform was implemented on images. The first part of the activity is to get the FFT of a circle and the letter 'A'. Shown below are the original image, the shifted FFT of that image, and the FFT of its FFT. Notice that the third image is just a reconstruction of the original image, only this time it is inverted. This is a property of the inverse FFT - that the inverse FFT is just the same as the forward FFT with the image inverted. [1] This is not readily seen in the circle since it is symmetric in all directions.


The second part of the activity deals with convolution of two images. The system resembles an imaging system with the circle serving as an aperture and the text 'VIP' as the object. The radius of the circle is varied and the following results were obtained.

r=5

r=10
r=15
r = 48.5

As can be seen, the larger the radius of the aperture, the better the reconstruction. This is because the number of frequencies that can pass through is dependent on the size of the aperture. For larger apertures, more frequencies can pass through. Hence a better reconstruction is obtained as opposed to that of apertures with smaller radii.

The third part of the activity is to find the correlation of the letter 'A' in the sentence "THE RAIN IN SPAIN STAYS MAINLY ON THE PLANE." To do this, the FFT of 'A' is multiplied element per element with the FFT of the text. The result shows that the letter A's in the text are highlighted because of high correlation.
The figure above shows the text (1st), the letter 'A' with the same font style and font size, and the resulting image when their FFTs are multiplied element per element.

The last part of the activity shows edge detection using the convolution integral. Four patterns were used (horizontal, vertical, diagonal, and spot). The following results were obtained.
The pattern allows only certain components to be seen. For example, the reconstruction for the horizontal pattern allows only almost horizontal components. Most vertical components are missing in the horizontal pattern, most horizontal components are missing in the vertical pattern, other vertical and horizontal patterns are missing in the diagonal pattern, and most components are present in the spot pattern.



The code (taken from Reference 1):

//first part
I=imread("C:\Documents and Settings\2004-28298\Desktop\activity 5\circle.bmp");
Igray=im2gray(I);
FIgray= fft2(Igray);
fc=abs(fftshift(FIgray))/max(abs(fftshift(FIgray)));
ffc=abs(fft2(FIgray))/max(abs(fft2(FIgray)));
imwrite(fc,"C:\Documents and Settings\mimie\Desktop\activity 5\fc.bmp");
imwrite(ffc,"C:\Documents and Settings\mimie\Desktop\activity 5\ffc.bmp");


//2nd part
o=imread("C:\Documents and Settings\mimie\Desktop\activity 5\vip.bmp");
I=imread("C:\Documents and Settings\mimie\Desktop\activity 5\30r.bmp");
lgray=im2gray(I);
ogray=im2gray(o);
fl=fftshift(lgray);
fo=fft2(ogray);
col=fo.*(fl);
icol=fft2(col);
fimage=abs(icol);
fimagenorm=fimage/max(fimage);
imwrite(fimagenorm,""C:\Documents and Settings\mimie\Desktop\activity 5\vip30r.bmp");


//3rd part
t=imread("C:\Documents and Settings\mimie\Desktop\activity 5\text.bmp");
a2=imread("C:\Documents and Settings\mimie\Desktop\activity 5\A2.bmp");
tgray=im2gray(t);
a2gray=im2gray(a2);
ft=fft2(tgray);
fa2=fft2(a2gray);
cft=conj(ft);
cta2=fa2.*(cft);
fcta2=fft(cta2);
ta2=fftshift(abs(fcta2))/max(fftshift(abs(fcta2)));
imwrite(ta2,"C:\Documents and Settings\mimie\Desktop\activity 5\ta2.bmp");


//last part
i=imread("C:\Documents and Settings\mimie\Desktop\activity 5\vip.bmp");
igray=im2gray(i);
pattern1=[-1,-1,-1; 2,2,2; -1,-1,-1];
pattern2=[-1,2,-1; -1,2,-1;-1,2,-1];
pattern3=[-1,-1,-1; -1,8,-1;-1,2,-1];
pattern4=[2,-1,-1;-1,2,-1;-1,-1,2];

c1=imcorrcoef(igray,pattern1);
c2=imcorrcoef(igray,pattern2);
c3=imcorrcoef(igray,pattern3);
c4=imcorrcoef(igray,pattern4);

c1norm=c1/max(c1);
c2norm=c2/max(c2);
c3norm=c3/max(c3);
c4norm=c4/max(c4);

imwrite(c1norm,"C:\Documents and Settings\mimie\Desktop\activity 5\c1norm.bmp");
imwrite(c2norm,"C:\Documents and Settings\mimie\Desktop\activity 5\c2norm.bmp");
imwrite(c3norm,"C:\Documents and Settings\mimie\Desktop\activity 5\c3norm.bmp");
imwrite(c4norm,"C:\Documents and Settings\mimie\Desktop\activity 5\c4norm.bmp");

**Remember to comment the part of the code you don't need.

Many thanks to Ma'am Jing for providing the code which guided me through this activity. Thank you also to Kaye and Miguel for the discussions we had, and to Raffy for helping me. I give myself a 9/10 for this activity because I did what was expected from the activity.

[1] Dr. Maricor Soriano. Activity 5 Manual.