2-D Discrete Cosine Transform

DCT is a technique for converting a signal into elementary frequency components.

It is widely used in image compression. Check Inverse discrete cosine transform for the reverse process.

MATLAB CODE:

%CONSIDER A MATRIX

A=[4 2 9 60; 7 10 5 77;88 66 44 3];



%PREALLOCATE THE MATRICES

B=zeros(size(A));

Temp=zeros(size(A));

display(A);

%FIND THE SIZE OF THE
MATRIX

[M N]=size(A);



for i=1:M

    for j=1:N

        

        if(i==1)

          AlphaP=sqrt(1/M);

        else

          AlphaP=sqrt(2/M);

        end

        

        if(j==1)

          AlphaQ=sqrt(1/N);

        else

          AlphaQ=sqrt(2/N);

        end

For A(1,1),  m=1 and n=1. If m= 1 then AlphaP=sqrt(1/M)=0.5774

If n=1 then AlphaQ=sqrt(1/N)=0.5000

For A(2,3),m=1 and n=3. AlphaP=sqrt(2/M)=0.8165
 AlphaQ =  sqrt(2/N) = 0.7071

Temp=0;

       for x=1:M

            for y=1:N

                    cs1=cos((pi*(2*x-1)*(i-1))/(2*M));

                    cs2=cos((pi*(2*y-1)*(j-1))/(2*N));

                    Temp=Temp+(A(x,y)*cs1*cs2);

             end

        end

        B(i,j)=AlphaP*AlphaQ*Temp;

    end

end

%2-D discrete Cosine Transform



display(B);

EXAMPLE 2

%2-D DISCRETE COSINE TRANSFORM FOR AN IMAGE

%READ THE IMAGE

I=imread('coins.png');

I=I(90:150,140:210);

A=double(I);

%PREALLOCATE THE MATRIX

B=zeros(size(A));

Temp=zeros(size(A));

[M N]=size(A);

%

x=1:M;

x=repmat(x',1,N);

y=repmat(1:N,M,1);

for i=1:M

    for j = 1: N

   

        if(i==1)

          AlphaP=sqrt(1/M);

        else

          AlphaP=sqrt(2/M);

        end

       

        if(j==1)

          AlphaQ=sqrt(1/N);

        else

          AlphaQ=sqrt(2/N);

        end

       

             

               

        cs1=cos((pi*(2*x-1)*(i-1))/(2*M));

        cs2=cos((pi*(2*y-1)*(j-1))/(2*N));

        Temp=A.*cs1.*cs2;

              

      

        B(i,j)=AlphaP*AlphaQ*sum(sum(Temp));

    end

end

%OUTPUT

figure,

subplot(2,1,1);imshow(I);title('Original Image');

subplot(2,1,2),imshow(log(abs(B)),[]);
colormap(jet);title('After DCT');