Overview of SVD• The purpose of (SVD) is to factor matrix A into T USV .• U and V are orthonormal matrices.• S is a diagonal matrix• . The singular values σ1 > · · · > σn > 0 appear in descending order along the main diagonal of S. The numbers σ12· · · > σn2 are the eigenvalues of T T AA and A A. T A= USV
Procedure to find SVD• Step 1:Calculate AAT and ATA.• Step 2: Eigenvalues and S.• Step 3: Finding U.• Step 4: Finding V.• Step 5: The complete SVD.
SVD CompressionHow SVD can compress any form of data.• SVD takes a matrix, square or non- square, and divides it into two orthogonal matrices and a diagonal matrix.• This allows us to rewrite our original matrix as a sum of much simpler rank one matrices.
• Since σ1 > · · · > σn > 0 , the first term of this series will have the largest impact on the total sum, followed by the second term, then the third term, etc.• This means we can approximate the matrix A by adding only the first few terms of the series!• As k increases, the image quality increases, but so too does the amount of memory needed to store the image. This means smaller ranked SVD approximations are preferable.
If we are going to increase the rank then we can improve the quality of the imageand also the memory used is also high
SVD vs Memory• Non-compressed image, I, requiresWith rank k approximation of I,• Originally U is an m×m matrix, but we only want the first k columns. Then UM = mk.• similarly VM = nk. AM = UM+ VM+∑ M AM = mk + nk + k AM = k(m + n + 1)
Limitations• There are important limits on k for which SVD actually saves memory. AM ≤IM k(m + n + 1) < mn k <mn/(m+n+1)• The same rule for k applies to color images.• In the case of color IM =3mn. While AM =3k(m+n+1) AM ≤IM → 3k(m+n+1) < 3mn Thus, k <mn/(m+n+1)