Question

I am brand new to MATLAB but am trying to do some image compression code for grayscale images.

Questions

How can I use SVD to trim off low-valued eigenvalues to reconstruct a compressed image?

Work/Attempts so far

My code so far is:

B=imread('images1.jpeg');   
B=rgb2gray(B);  
doubleB=double(B);  
%read the image and store it as matrix B, convert the image to a grayscale
photo and convert the matrix to a class 'double' for values 0-255  
[U,S,V]=svd(doubleB);

This allows me to successfully decompose the image matrix with eigenvalues stored in variable S.

How do I truncate S (which is 167×301, class double)? Let’s say of the 167 eigenvalues I want to take only the top 100 (or any n really), how do I do that and reconstruct the compressed image?

Updated code/thoughts

Instead of putting a bunch of code in the comments section, this is the current draft I have. I have been able to successfully create the compressed image by manually changing N, but I would like to do 2 additional things:

1- Show a pannel of images for various compressions (i/e, run a loop for N = 5,10,25, etc.)

2- Somehow calculate the difference (error) between each image and the original and graph it.

I am horrible with understanding loops and output, but this is what I have tried:

B=imread('images1.jpeg');  
B=rgb2gray(B);  
doubleB=im2double(B);%  
%read the image and store it as matrix B, convert the image to a grayscale  
%photo and convert the image to a class 'double'  
[U,S,V]=svd(doubleB);   
C=S;  
for N=[5,10,25,50,100]  
C(N+1:end,:)=0;  
C(:,N+1:end)=0;  
D=U*C*V';  
%Use singular value decomposition on the image doubleB, create a new matrix  
%C (for Compression diagonal) and zero out all entries above N, (which in  
%this case is 100). Then construct a new image, D, by using the new  
%diagonal matrix C.  
imshow(D);  
error=C-D;  
end

Obviously there are some errors because I don’t get multiple pictures or know how to “graph” the error matrix

Answer 1

Just to start, I assume you’re aware that the SVD is really not the best tool to decorrelate the pixels in a single image. But it is good practice.

OK, so we know that B = U*S*V'. And we know S is diagonal, and sorted by magnitude. So by using only the top few values of S, you’ll get an approximation of your image. Let’s say C=U*S2*V', where S2 is your modified S. The sizes of U and V haven’t changed, so the easiest thing to do for now is to zero the elements of S that you don’t want to use, and run the reconstruction. (Easiest way to do this: S2=S; S2(N+1:end, :) = 0; S2(:, N+1:end) = 0;).

Now for the compression part. U is full, and so is V, so no matter what happens to S2, your data volume doesn’t change. But look at what happens to U*S2. (Plot the image). If you kept N singular values in S2, then only the first N rows of S2 are nonzero. Compression! Except you still have to deal with V. You can’t use the same trick after you’ve already done (U*S2), since more of U*S2 is nonzero than S2 was by itself. How can we use S2 on both sides? Well, it’s diagonal, so use D=sqrt(S2), and now C=U*D*D*V'. So now U*D has only N nonzero rows, and D*V' has only N nonzero columns. Transmit only those quantities, and you can reconstruct C, which is approximately like B.