SVD coding - img=imread("img.png");% Reading image B=im2double(img);% coverting to double precsion image imgBlack=rgb2gray(B); %imshow(imgBlack); [U S V]=svd(imgBlack); [U1 S1 V1]=svd(imgBlack,'econ'); ylabel("Singuler value") xlabel("value") plot(diag(S1),'o k')

1. The Singular Value
Decomposition

2. SVD
• SVD exists for any matrix
• Definition:
• For square matrices A  Cnm
, there exist orthogonal matrices U  Cnn
and a diagonal matrix   Cnm
,
such that all the diagonal values i of  are non-negative and first value is largest .V is also orthogonal
matrix V  C
mm
.
• SVD used to obtain low-rank approximations to matrices and to perform pseudo-inverses of non-square
matrices to find the solution of a system of equations Ax = b .
T
A U V
 
=
A U 
T
V

3. Properties of SVD
• The diagonal values of  (1, …, n) are called the singular values. It is sort like: 1  2 …  n
• The columns of U (u1, …, un) are called the left singular vectors. They are the axes of the ellipsoid.
• The columns of V (v1, …, vn) are called the right singular vectors. They are the pre-images
T
A U V
 
=
A U  T
V

4. Singular values
• If m>=n if m<n
Zeros
Zeros
Full SVD Reduce SVD
Zeros

5. Code
img=imread("img.png");% Reading image
B=im2double(img);% coverting to double precsion
image
imgBlack=rgb2gray(B);
%imshow(imgBlack);
[U1 S1 V1]=svd(imgBlack,'econ');
img1=U1(:,1:100)*S1(1:100,1:100)*V1(:,1:100)';
%imshow(img1)
img1=U1(:,1:400)*S1(1:400,1:400)*V1(:,1:400)';
%imshow(img1)
img1=U1(:,1:700)*S1(1:700,1:700)*V1(:,1:700)';
%imshow(img1)

6. First Reduction (100 pixel)
Original Image After SVD

7. Second Reduction (400 pixel)
Original Image After SVD

8. Third Reduction (700 pixel)
Original Image After SVD

9. Singular values of above images