Image compression using singular value decomposition
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Image Compression using Singular Value Decomposition
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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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