image compression using singular value decomposition

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Image Compression using Singular Value Decomposition

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Page 1: Image compression using singular value decomposition

Image Compression using Singular Value Decomposition

Page 2: Image compression using singular value decomposition

Why Do We Need Compression?

To save• Memory• Bandwidth• Cost

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How Can We Compress?• Coding redundancy

– Neighboring pixels are not independent but correlated

• Interpixel redundancy

• Psychovisual redundancy

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Information vs Data

REDUNDANTDATA

INFORMATION

DATA = INFORMATION + REDUNDANT DATA

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Image Compression

•Lossless Compression

•Lossy Compression

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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

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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.

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Step 1:Calculate AAT and ATA.

• Let then

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Step 2: Eigenvalues and S.

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• Singular Values are

• Therefore

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Step 3: Finding U.

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Step 4: Finding V.

• Similarly

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Step 5:Complete SVD

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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.

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• 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.

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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

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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)

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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)

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1. www.wikipedia.com

2. www.google.com

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

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