Two dimensional Fourier transform using Matlab

2-D Fourier transform using 1-D Fourier transform:

The Fourier Transform ( in this case, the 2D Fourier Transform ) is the series expansion of an image function ( over the 2D space domain ) in terms of "cosine" image (orthonormal) basis functions.
The definitons of the transform (to expansion coefficients) and the inverse transform are given below:

 F(u,v) = SUM{ f(x,y)*exp(-j*2*pi*(u*x+v*y)/N) }
    and
 f(x,y) = SUM{ F(u,v)*exp(+j*2*pi*(u*x+v*y)/N) }

    where u = 0,1,2,...,N-1 and v = 0,1,2,...,N-1
   x = 0,1,2,...,N-1 and y = 0,1,2,...,N-1
   j = SQRT( -1 )
   and SUM means double summation  over proper
   x,y or u,v ranges

Program: 

aa=imread('D:\Flower.jpg');          % USE SQAURE IMAGE

aa=rgb2gray(aa)

a1=abs(fft2(aa));                          %Using the inbuilt FFT function for comparison

s=size(a);

figure(1)

imshow(aa)

title('original image')

%To find the Twiddle factor

for k=0:1:s(1)-1                            %Image is square

for n=0:1:s(2)-1

                 p=exp(-i*2*pi*k*n/s(2));

 x(k+1,n+1)=p;     %x is the DFT matrix

end

end

%To find the 1-D Fourier transform again along the rows

for n=1:1:s(1)

    d=x*a(n,:).';

z(:,n)=d;                                               %Arranged in rows

end

%To find the 1-D Fourier transform again along the columns

for n1=1:1:s(2)

    d1=x*z(n1,:).';

z1(:,n1)=d1;

end

%plotting

z=abs(z1);

figure(2);

colormap(gray);

imagesc(fftshift(log(1+z)));

title('Fourier image');

figure(3;)

colormap(gray);

imagesc(fftshift(log(1+a1)));

title('Fourier transform image');

OUTPUT:  

 

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