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Avanessia Mitchell , final research presentation
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- 3. Image Processing isa process used to convert an image signal into a physical image.
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- 6. The process offinding a corresponding diagonal matrix D. D= λ1 ⋯ 0 ⋮ ⋱ ⋮ 0 ⋯ λ 𝑛 ∋ A=PDP−1 𝑃 = 𝐯1 ⋯ 𝐯 𝑛 𝐯 𝑇 𝑖 𝐯𝑗 = 0 , i ≠ 𝑗
- 7. An nxn matrixA is a 𝐑𝐞𝐚𝐥 𝐒𝐲𝐦𝐦𝐞𝐭𝐫𝐢𝐜 𝐦𝐚𝐭𝐫𝐢𝐱 if A = AT. Real Symmetric Matrices are always diagonalizable!
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- 9. Now P =𝐮1 𝐮2 …… 𝐮n where D = λ1 ⋯ 0 ⋮ ⋱ ⋮ 0 ⋯ λn ∴ 𝐴 = 𝑃𝐷𝑃−1 and P is an orthogonal matrix where 𝑃−1 = 𝑃 𝑇 𝐮𝑖 𝑇 𝐮𝑗 = 0 , 𝑖 ≠ 𝑗 1 , 𝑖 = 𝑗
- 10. The Spectral decompositionof A is the representation of A that breaks up A by the spectrum of A. 𝐴 = 𝑃𝐷𝑃 𝑇= 𝐮1 ⋯ 𝐮 𝑛 λ1 ⋯ 0 ⋮ ⋱ ⋮ 0 ⋯ λ 𝑛 𝐮1 ⋮ 𝐮 𝑛 𝐴 = λ1 𝐮1 𝐮1 𝑇 +λ2 𝐮2 𝐮2 𝑇 + ⋯ + λ 𝑛 𝐮 𝑛 𝐮 𝑛 𝑇 𝐴 = λ𝑖 𝐮𝑖 𝐮𝑖 𝑇 𝑛 𝑖=1
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- 12. A quadratic formon Rn is Q 𝐱 on Rn whose value at a vector 𝐱 can be expressed as Q 𝐱 = 𝐱T A𝐱
- 13. A Quadratic formQ 𝐱 = 𝐱 𝑇 A x Q(x)=𝐱 𝑇 𝐼𝐱 =∥ 𝐱 ∥2 𝑎𝑖𝑗 = 𝑎𝑗𝑖 non diagonal 𝑎𝑖𝑖 diagonal 𝐱 𝑇 Ax= 𝐱 𝑖 𝐱𝑗 𝑎𝑖𝑖 𝑎𝑖𝑗 𝑎𝑖𝑗 𝑎𝑗𝑗 𝐱 𝑖 𝐱𝑗 = 𝑎𝑖𝑖 𝐱 𝑖 2 + 2𝑎𝑖𝑗 𝐱 𝑖 𝐱𝑗+𝑎𝑗𝑗 𝐱𝑗 𝟐
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- 15. m = min*𝐱TA𝐱: ∥ 𝐱 ∥ + = 1 M = max {𝐱TA𝐱: ∥ x ∥ } = 1
- 16. Q 𝐱 =a𝐱1 2 + b𝐱2 2 + c𝐱3 2 , a ≧ b ≧ c a𝐱1 2 + b𝐱2 2 + c𝐱3 2 ≦ a𝐱1 2 + a𝐱2 2 + a𝐱3 2 Q(x) ≦ a(𝐱1 2 + 𝐱2 2 + 𝐱3 2 ) Q(x) ≦ a Q(x) ≦ 𝑐 where 𝐱 𝑇 𝐱 = 1
- 17. 𝑚 ≤ λ𝑖≤ 𝑀 a ≤ 𝑄 𝑥 ≤ 𝑐
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- 19. ∥ 𝐴𝐱 𝑖∥2= 𝐱 𝑖(𝐴 𝑇 𝐴)𝐱 𝑖 = λ𝑖 λ𝑖 ≥ λ𝑖+1≥ ⋯ ≥ λ 𝑛 ≥ 0
- 20. ∑= D 0 0 0 wherethe D= σ1 ⋯ 0 ⋮ ⋱ ⋮ 0 ⋯ σr σi are the singular values of A. 𝜎𝑖 = λ𝑖
- 21. If A isan m × n matrix with rank r. ∃ an m × n matrix ∑ for which the diagonal entries in D are the first singular values of A, σ1 ≧ σ2 ≧ ⋯ ≧ σr > 0 ∃ an m × m orthogonal matrix U and nxn orthogonal matrix V ∋ A = U∑VT
- 22. 𝐬𝑖 = 1 𝜎𝑖 𝐴𝐯𝑖 ,1 ≤ 𝑖 ≤ 𝑟 U = 𝐬1 ⋯ 𝐬 𝑟 V = 𝐮1 … 𝐮 𝑛 unit eigenvectors 𝐴 𝑇 𝐴
- 23. Applications to Image Processing& Statistics
- 24. In image processingan 𝐢𝐦𝐚𝐠𝐞 is a function Q 𝒙 Q 𝐱1, 𝐱2 = 𝐱 𝑇 A𝐱 A pixel in an image is a square matrix within the mxn neighborhood of an image.
- 25. Lay, C. David.3ed.LinearAlgebra and its applications. (chapters 5,6,7) http://www.wisegeek.org/what-is-image-processing.htm www.wisegeek.com Lipschaum’s ,Seymour, PhD. Lipson, Marc ,PhD.5ed.Linear Algebra(Schuam’s outlines) Vuthy, Teav. Pineda, R. Dr. Angel. Using the singular value decomposition (SVD) image compression. Royal University of Phnom Penh.
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