Image compression using singular value decomposition

Image Compression using Singular Value Decomposition

Why Do We Need Compression?

To save• Memory• Bandwidth• Cost

How Can We Compress?• Coding redundancy

– Neighboring pixels are not independent but correlated

• Interpixel redundancy

• Psychovisual redundancy

Information vs Data

REDUNDANTDATA

INFORMATION

DATA = INFORMATION + REDUNDANT DATA

Image Compression

•Lossless Compression

•Lossy Compression

Overview of SVD

• The purpose of (SVD) is to factor matrix A into

USVT.• 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 σ1

2· · · > σn2 are the

eigenvalues of AAT and ATA.

A= USVT

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.

Step 1:Calculate AAT and ATA.

• Let then

Step 2: Eigenvalues and S.

• Singular Values are

• Therefore

Step 3: Finding U.

Step 4: Finding V.

• Similarly

Step 5:Complete SVD

SVD Compression

How 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 image and also the memory used is also high

SVD vs Memory• Non-compressed image, I, requires

With 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 + kAM = k(m + n + 1)

Limitations • There are important limits on k for

which SVD actually saves memory.AM ≤IM

k(m + n + 1) < mnk <mn/(m+n+1)

• The same rule for k applies to color images.

• In the case of color IM =3mn. WhileAM =3k(m+n+1)

AM ≤IM→ 3k(m+n+1) < 3mn

Thus, k <mn/(m+n+1)

1. www.wikipedia.com

2. www.google.com

3. www.imagesco.com4. www.idocjax.com5. www.howstuffworks.com6. www.mysvd.com