MATLAB: How to rconstruct image using eigenvectors and eigenvalues

Question

I am trying to reconstruct an image by evaluating its eigenvalues and eigenvectors. Some of the eigenvalues are negative and when I reconstruct the image using:

imge_rec = (eig_vec)*(eig_values)*(eig_vec)'

I do not obtain the same image. Following is my code and test image: img_in = double(rgb2gray(imread('input.jpg')));

[eig_vec,eig_val] = eig(img_in);

img_rec = eig_vec * eig_val * eig_vec';

figure;

imshow(img_rec,[]);

input image

output image

Answer 1

I find it easier to work with the SVD-decomposition instead of the eigenvalue-decomposition. Since your matrix is not symmetric it gives complex-valued eigenvalues, which makes it much harder to use the eigenvalue-decomposition. The SVD gives you singular values that are real and >= 0. This makes it easier to implement straight filters and compressions and whatnot. So try:

[U,S,V] = svd(img_in);
imagesc(U*S*V.')

HTH