MATLAB TUTORIAL FOUR

Image Enhancement in Frequency Domain.

Find the 2D discrete Fourier transform for given input images

         1.    Impulse

         2.    Rectangle

         3.     Sine

1.                   Computing and Visualizing 2D DFT for impulse image

clear all;

close all;

clc;

display('method of taking 2D DFT of impulse image')

f1 = imread('impulse.jpg');

% f2 = im2bw(f1);

f = imresize(f1,[512 512]);

F = fft2(f);

s = abs(F);

Fc = fftshift(s);

s2 = log(1 + abs(Fc));

subplot(2,2,1); imshow(f);

title('Impulse Image');

subplot(2,2,2); imshow(s, []);

title('Fourier Spectrum of given image "can be seen on four corners"');

subplot(2,2,3); imshow(abs(Fc), []);

title('Centered Fourier Spectrum of given image');

subplot(2,2,4); imshow(s2, []);

title('Spectrum Visually enhanced by a log trasformation');

 

Computing 2D DFT from both programming logic and direct inbuilt function

clear all;

close all;

clc;

a = imread('impulse.jpg');

a1 = imresize(a,[512 512])

s = double(a1);

b = size(s)

m = b(1)

n = b(2)

e = 2.718

for u=0:1:m-1

for x=0:1:n-1

        w1 = e.^(-j*2*pi*u*x/m);

        w(u+1,x+1)=w1;

end

end

for i=1:1:b(1)

    f11 = (w)*(s(i,:).');

    f1(i,:)=f11;   

end

for i=1:1:b(2)

    f21 = (w)*(f1(:,i));

    f2(:,i)=f21;

end

i2 = abs(f2);

g = fft2(a1); % computing direct 2D DFT of impulse image

g1 = abs(g);

subplot(2,2,1)

imshow(a); title('Original Impulse image');

subplot(2,2,2)

imshow(a1); title('Binary image of original Impulse image');

subplot(2,2,3);colormap(gray);

imagesc(fftshift(log(1+i2))); title('2D DFT(I2) of binary image by programming');

subplot(2,2,4);colormap(gray);

imagesc(fftshift(log(1+g1)));title('2D DFT(G1) of binary image by inbuilt function');

 

Computing 2D inverse DFT of Previously generated 2D DFT of impulse image.

clear all;

close all;

clc;

display('method of taking 2D inverse DFT of Sine image')

f1 = imread('impulse.jpg');

f = imresize(f1,[512 512]);

F = fft2(f);

s = abs(F);

Fc = fftshift(s);

s2 = log(1 + abs(Fc));

f12 = ifftshift(Fc);

f1 = ifft2(f12);

subplot(2,3,1); imshow(f);

title('Original Impulse Image');

subplot(2,3,2); imshow(s, []);

title('Fourier Spectrum of given image "can be seen on four corners"');

subplot(2,3,3); imshow(abs(Fc), []);

title('Centered Fourier Spectrum of given image');

subplot(2,3,4); imshow(s2, []);

title('Spectrum Visually enhanced by a log trasformation');

subplot(2,3,5); imshow(abs(f12), []);

title('Back to Fourier spectrum');

subplot(2,3,6); imshow(abs(f1), []);

title('inverse fourier transform');

 

2.                   Computing and Visualizing 2D DFT for Sine image

clear all;

close all;

clc;

display('method of taking 2D DFT of impulse image')

f1 = imread('sine2.gif');

f = imresize(f1,[64 64]);

F = fft2(f);

s = abs(F);

Fc = fftshift(s);

s2 = log(1 + abs(Fc));

subplot(2,2,1); imshow(f);

title('Original Sine Image');

subplot(2,2,2); imshow(s, []);

title('Fourier Spectrum of given image "can be seen corners"');

subplot(2,2,3); imshow(abs(Fc), []);

title('Centered Fourier Spectrum of given image');

subplot(2,2,4); imshow(s2, []);

title('Spectrum Visually enhanced by a log trasformation');

 

Computing 2D DFT from both programming logic and direct inbuilt function

clear all;

close all;

clc;

a = imread('sine2.gif');

s1 = im2bw(a);

s12 = imresize(s1,[128 128]);

s = double(s12);

b = size(s)

m = b(1)

n = b(2)

e = 2.718

for u=0:1:m-1

for x=0:1:n-1

        w1 = e.^(-j*2*pi*u*x/m);

        w(u+1,x+1)=w1;

end

end

for i=1:1:b(1)

    f11 = (w)*(s(i,:).');

    f1(i,:)=f11;   

end

for i=1:1:b(2)

    f21 = (w)*(f1(:,i));

    f2(:,i)=f21;

end

g = fft2(s);

g1 = abs(g);

i2 = abs(f2);

subplot(2,2,1)

imshow(a); title('Original Sine image');

subplot(2,2,2)

imshow(s); title('Binary image of original image');

subplot(2,2,3)

colormap(gray);

imagesc(fftshift(log(1+i2))); title('2D DFT of binary image by programming');

subplot(2,2,4)

colormap(gray);

imagesc(fftshift(log(1+g1))); title('2D DFT of binary image by direct')

 

Computing 2D inverse DFT of Previously generated 2D DFT of sine image.

clear all;

close all;

clc;

display('method of taking 2D inverse DFT of Sine image')

f1 = imread('sine2.gif');

f = imresize(f1,[64 64]);

F = fft2(f);

s = abs(F);

Fc = fftshift(s);

s2 = log(1 + abs(Fc));

f12 = ifftshift(Fc);

f1 = ifft2(f12);

subplot(2,3,1); imshow(f);

title('Sine Image');

subplot(2,3,2); imshow(s, []);

title('Fourier Spectrum of given image "can be seen on four corners"');

subplot(2,3,3); imshow(abs(Fc), []);

title('Centered Fourier Spectrum of given image');

subplot(2,3,4); imshow(s2, []);

title('Spectrum Visually enhanced by a log trasformation');

subplot(2,3,5); imshow(abs(f12), []);

title('Back to Fourier spectrum');

subplot(2,3,6); imshow(abs(f1), []);

title('inverse fourier transform');

 

3.                   Computing and Visualizing 2D DFT for Rectangle image

clear all;

close all;

clc;

display('method of taking 2D DFT of rectangle image')

f1 = imread('rectangle.jpg');

f2 = im2bw(f1);

 f = imresize(f2,[64 64]);

 F = fft2(f);

 s = abs(F);

Fc = fftshift(s);

s2 = log(1 + abs(Fc));

subplot(2,2,1); imshow(f2);

title('Rectangle Original Image');

subplot(2,2,2); imshow(s, []);

title('Fourier Spectrum of given image "can be seen on four corners"');

subplot(2,2,3); imshow(abs(Fc), []);

title('Centered Fourier Spectrum of given image');

subplot(2,2,4); imshow(s2, []);

title('Spectrum Visually enhanced by a log trasformation');

 

Computing 2D DFT from both programming logic and direct inbuilt function

clear all;

close all;

clc;

a = imread('rectangle.jpg');

s1 = im2bw(a);

s12 = imresize(s1,[64 64]);

s = double(s12);

b = size(s)

m = b(1)

n = b(2)

e = 2.718

for u=0:1:m-1

for x=0:1:n-1

        w1 = e.^(-j*2*pi*u*x/m);

        w(u+1,x+1)=w1;

end

end

for i=1:1:b(1)

    f11 = (w)*(s(i,:).');

    f1(i,:)=f11;   

end

for i=1:1:b(2)

    f21 = (w)*(f1(:,i));

    f2(:,i)=f21;

end

g = fft2(s);

g1 = abs(g);

i2 = abs(f2);

subplot(2,2,1)

imshow(a); title('Original image');

subplot(2,2,2)

imshow(s); title('Binary image of original image');

subplot(2,2,3)

colormap(gray);

imagesc(fftshift(log(1+i2))); title('2D DFT of binary image by programming');

subplot(2,2,4)

colormap(gray);

imagesc(fftshift(log(1+g1))); title('2D DFT of binary image by direct')

 

Computing 2D inverse DFT of Previously generated 2D DFT of rectangle image.

clear all;

close all;

clc;

display('method of taking 2D DFT of rectangle image')

f1 = imread('rectangle.jpg');

f2 = im2bw(f1);

 f = imresize(f2,[64 64]);

 F = fft2(f);

 s = abs(F);

Fc = fftshift(s);

s2 = log(1 + abs(Fc));

Fc1= ifftshift(Fc);

F1 = ifft2(Fc1);

subplot(2,3,1); imshow(f2);

title('Rectangle Original Image');

subplot(2,3,2); imshow(s, []);

title('Fourier Spectrum of given image "can be seen on four corners"');

subplot(2,3,3); imshow(abs(Fc), []);

title('Centered Fourier Spectrum of given image');

subplot(2,3,4); imshow(s2, []);

title('Spectrum Visually enhanced by a log trasformation');

subplot(2,3,5); imshow(abs(Fc1), []);

title('Back to Fourier Transform');

subplot(2,3,6); imshow(abs(F1), []);

title('Inverse Fourier Transform');