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Image compression

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Niteesh Shanbog

This document discusses singular value decomposition (SVD) and its applications in image processing. SVD decomposes a matrix A into three matrices - left singular vectors U, singular values Σ, and right singular vectors V. This reveals the dominant information encoded in the matrix and allows the matrix to be approximated by its first few terms. SVD can be used for image compression, representing an image with far fewer values by keeping only the first few terms. Other applications of SVD include computer tomography, magnetic resonance imaging, seismology, and image deblurring.

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Singular Value Decomposition - Applications in Image Processing 1 By- Chithaj Mallikarjun Niteesh S Shanbog PES1201702412 PES1201702421 MTech, EEE MTech, EEE
1. Singular value decomposition Consider a (real) matrix A ∈ Rn×m, r = rank (A) ≤ min {n, m} . A has m. columns of length n , n. rows of length m , r is the maximal number of linearly independent columns (rows)of A . 2
3 There exists an S V D decomposition of A in the form A = U Σ V T , where U = [u1, . . . , un] ∈ Rn×n, V = [v1, . . . , vm] ∈ Rm×m are orthogo- nal matrices, and
Singular value decomposition – the matrices: A U  VT {ui} i = 1,...,n {vi} i = 1,...,m {σi } i=1,...,r 4 are left singular vectors (columns of U ), are right singular vectors (columns of V ), are singular values of A.
5 T h e S V D gives us: span (u1, . . . , ur) ≡ range (A ) ⊂ Rn, span (vr+1, . . . , vm) ≡ ker ( A ) ⊂ Rm, span (v1, . . . , vr) ≡ range ( A T ) ⊂ Rm, span (ur+1, . . . , un) ≡ ker ( A T ) ⊂ Rn,
Singular value decomposition – the subspaces: v1 v2 ... vr u1 u2 ... ur vr+1 ... vm ur+1 ... un 1 2 r } }0 0 AT A range(A) ker(AT ) ker(A) range(A )T Rn Rm 6
7 T h e outer product (dyadic) form: We can rewrite A as a sum of rank-one matricesin the dyadic form
Matrix A as a sum of rank-one matrices: A A1 + A2 + ... + Ar -1 + Ar 8 r A  Ai i = 1 SVD reveals the dominating information encoded in a matrix. The first terms are the “ m o s t ” important.
Image compression
Application - image compression Grayscale image =matrix; each entry represents a pixel brightness. 10
Grayscale image: scale0 , . . . , 255 from black to white Colored image: 3 matrices for Red, Green and Blue brightness values 11
12 Memory required to store: An uncompressed image of size (m × n):mn values SVD approximation:k(m + n + 1) values
Other applications: • Computer Tomography ( C T ) ; • Magnetic Resonance; • Seismology; • Crystallography; • Material Sciences; • Image De blurring 13
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Image compression

  • 1. Singular Value Decomposition - Applications in Image Processing 1 By- Chithaj Mallikarjun Niteesh S Shanbog PES1201702412 PES1201702421 MTech, EEE MTech, EEE
  • 2. 1. Singular value decomposition Consider a (real) matrix A ∈ Rn×m, r = rank (A) ≤ min {n, m} . A has m. columns of length n , n. rows of length m , r is the maximal number of linearly independent columns (rows)of A . 2
  • 3. 3 There exists an S V D decomposition of A in the form A = U Σ V T , where U = [u1, . . . , un] ∈ Rn×n, V = [v1, . . . , vm] ∈ Rm×m are orthogo- nal matrices, and
  • 4. Singular value decomposition – the matrices: A U  VT {ui} i = 1,...,n {vi} i = 1,...,m {σi } i=1,...,r 4 are left singular vectors (columns of U ), are right singular vectors (columns of V ), are singular values of A.
  • 5. 5 T h e S V D gives us: span (u1, . . . , ur) ≡ range (A ) ⊂ Rn, span (vr+1, . . . , vm) ≡ ker ( A ) ⊂ Rm, span (v1, . . . , vr) ≡ range ( A T ) ⊂ Rm, span (ur+1, . . . , un) ≡ ker ( A T ) ⊂ Rn,
  • 6. Singular value decomposition – the subspaces: v1 v2 ... vr u1 u2 ... ur vr+1 ... vm ur+1 ... un 1 2 r } }0 0 AT A range(A) ker(AT ) ker(A) range(A )T Rn Rm 6
  • 7. 7 T h e outer product (dyadic) form: We can rewrite A as a sum of rank-one matricesin the dyadic form
  • 8. Matrix A as a sum of rank-one matrices: A A1 + A2 + ... + Ar -1 + Ar 8 r A  Ai i = 1 SVD reveals the dominating information encoded in a matrix. The first terms are the “ m o s t ” important.
  • 10. Application - image compression Grayscale image =matrix; each entry represents a pixel brightness. 10
  • 11. Grayscale image: scale0 , . . . , 255 from black to white Colored image: 3 matrices for Red, Green and Blue brightness values 11
  • 12. 12 Memory required to store: An uncompressed image of size (m × n):mn values SVD approximation:k(m + n + 1) values
  • 13. Other applications: • Computer Tomography ( C T ) ; • Magnetic Resonance; • Seismology; • Crystallography; • Material Sciences; • Image De blurring 13