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Image compression using singular value decomposition
Singular value decomposition (SVD) can be used to compress images by decomposing images into orthogonal matrices and a diagonal matrix. This decomposition allows images to be approximated using only the first few terms in the decomposition series, reducing memory usage. However, there are limits to the compression rate that can be achieved while still saving memory. The rank of the approximation must be less than mn/(m+n+1) for memory savings, where m and n are the image dimensions. Higher ranks improve quality but also increase memory usage.
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Image compression using singular value decomposition
- 1. Image Compression using SingularValue Decomposition
- 2. Why Do WeNeed Compression? To save • Memory • Bandwidth • Cost
- 3. How Can WeCompress? • Coding redundancy – Neighboring pixels are not independent but correlated • Interpixel redundancy • Psychovisual redundancy
- 4. Information vs Data REDUNDANTDAT A INFORMATION DATA = INFORMATION + REDUNDANT DATA
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- 6. 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
- 7. Procedure to findSVD • Step 1:Calculate AAT and ATA. • Step 2: Eigenvalues and S. • Step 3: Finding U. • Step 4: Finding V. • Step 5: The complete SVD.
- 8. Step 1:Calculate AAand A A. T T • Let then
- 9. Step 2: Eigenvaluesand S.
- 10. • Singular Valuesare • Therefore
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- 13. Step 4: FindingV. • Similarly
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- 15. SVD Compression How SVDcan 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.
- 17. • 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.
- 18. If we aregoing to increase the rank then we can improve the quality of the image and also the memory used is also high
- 19. 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 + k AM = k(m + n + 1)
- 20. Limitations • There areimportant 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)
- 21. 1. www.wikipedia.com 2. www.google.com 3.www.imagesco.com 4. www.idocjax.com 5. www.howstuffworks.com 6. www.mysvd.com
Editor's Notes
- #22 6.www.mysvd.com www.imagesco.com www.mysvd.com