Your SlideShare is downloading.
×

×
Saving this for later?
Get the SlideShare app to save on your phone or tablet. Read anywhere, anytime – even offline.

Text the download link to your phone

Standard text messaging rates apply

Like this presentation? Why not share!

- Magnetic levitation trai ns by PRADEEP Cheekatla 1657 views
- Image Compression using SVD by Georgios-Marios P... 176 views
- SVD and the Netflix Dataset by Ben Mabey 17404 views
- Electons meet animals by PRADEEP Cheekatla 1779 views
- Image enhancement techniques by Abhi Abhinay 13435 views
- Image color correction and contrast... by Yu Huang 16077 views
- Crusoe processor by PRADEEP Cheekatla 2670 views
- Smart quill by PRADEEP Cheekatla 4780 views
- Redtacton by PRADEEP Cheekatla 505 views
- Struts(mrsurwar) ppt by mrsurwar 370 views
- Haptics touch the virtual by PRADEEP Cheekatla 958 views
- E newspaper by PRADEEP Cheekatla 32154 views

Like this? Share it with your network
Share

1,838

views

views

Published on

Published in:
Education

No Downloads

Total Views

1,838

On Slideshare

0

From Embeds

0

Number of Embeds

0

Shares

0

Downloads

90

Comments

0

Likes

3

No embeds

No notes for slide

- 1. Image Compression usingSingular Value Decomposition
- 2. Why Do We Need Compression?To save• Memory• Bandwidth• Cost
- 3. How Can We Compress?• Coding redundancy – Neighboring pixels are not independent but correlated• Interpixel redundancy• Psychovisual redundancy
- 4. Information vs Data REDUNDANTDAT A INFORMATIONDATA = INFORMATION + REDUNDANT DATA
- 5. Image Compression•Lossless Compression•Lossy Compression
- 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 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.
- 8. Step 1:Calculate AA and A A. T T• Let then
- 9. Step 2: Eigenvalues and S.
- 10. • Singular Values are• Therefore
- 11. Step 3: Finding U.
- 12. Step 4: Finding V.• Similarly
- 13. Step 5:Complete SVD
- 14. 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.
- 15. • 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.
- 16. If we are going to increase the rank then we can improve the quality of the imageand also the memory used is also high
- 17. 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)
- 18. 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)
- 19. 1. www.wikipedia.com2. www.google.com3. www.imagesco.com4. www.idocjax.com5. www.howstuffworks.com6. www.mysvd.com

Be the first to comment