Engineering Math - Quick Reference                                 Home : www.sharetechnote.com

 

 

 

SVD (Singular Value Decomposition) - Application - Image Compression

 

In this application, I assign the whole pixel data of an black-and-white image to the matrix M

    M = imgBW; % 511 x 513 image, 511 x 513 data

Then I did SVD as follows.

 

Then take out only n colums from U and V matrix and take out (n x n) submatrix from S as shown below.

    Uc = U(:,1:n);

    Sc = S(1:n,1:n);

    Vc = V(:,1:n);

Then reconstruct the image from these submatrix as follows.

    CompressedImage = Uc * Sc * Vc   (Note : '*' indicate the matrix multiplication(Inner Product))

Following result shows you the Original Image and all the diagonal values of S matrix and CompressedImage. This is the case when n = 30; In this case, Uc is (511 x 30) matrix, Sc is (30 x 30) matrix, Vc is (513 x 30) matrix.

 

 

Following result shows you the Original Image and all the diagonal values of S matrix and CompressedImage. This is the case when n = 50; In this case, Uc is (511 x 50) matrix, Sc is (50 x 50) matrix, Vc is (513 x 50) matrix.

 

 

< List 1 >

 

    clear all;

     

    img=imread('Lena.png');

    imgBW = rgb2gray(img);

     

    M = imgBW;

     

    [U,S,V]=svd(double(M));

    Sd = diag(S);

     

    n = 30;

    Uc = U(:,1:n);

    Sc = S(1:n,1:n);

    Vc = V(:,1:n);

    CompressedImag = Uc*Sc*Vc';

     

    subplot(1,3,1);

    imshow(M);

    title('Original');

     

    subplot(1,3,2);

    plot(Sd,'ro','MarkerFaceColor',[1 0 0],'MarkerSize',2);

    hold on;

    plot(Sd,'k-');

    hold on;

    xlim([0 length(Sd)]);

    title('Diag(S)');

     

    subplot(1,3,3);

    imshow(uint8(CompressedImag));

    title('Compressed');