Signal Processing Course: Signal Processing in Ortho-bases

E34ded36efe4b7abb12510d4e525fee8?s=47 Gabriel Peyré
January 01, 2012

Signal Processing Course: Signal Processing in Ortho-bases

E34ded36efe4b7abb12510d4e525fee8?s=128

Gabriel Peyré

January 01, 2012
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  1. Signal and Image Processing with Orthogonal Decompositions Gabriel Peyré www.numerical-tours.com

  2. Signals, Images and More Continuous signal: f 2 L 2

    ([0 , 1]).
  3. Continuous image: f 2 L 2 ([0 , 1] 2

    ). Signals, Images and More Continuous signal: f 2 L 2 ([0 , 1]).
  4. Continuous image: f 2 L 2 ([0 , 1] 2

    ). General setting: f : [0, 1]d ! Rs Videos: d = 3 , s = 1. Signals, Images and More Continuous signal: f 2 L 2 ([0 , 1]).
  5. Continuous image: f 2 L 2 ([0 , 1] 2

    ). General setting: f : [0, 1]d ! Rs Videos: d = 3 , s = 1. Color image: d = 2 , s = 3. Signals, Images and More Continuous signal: f 2 L 2 ([0 , 1]).
  6. Continuous image: f 2 L 2 ([0 , 1] 2

    ). General setting: f : [0, 1]d ! Rs Videos: d = 3 , s = 1. Color image: d = 2 , s = 3. Multi-spectral: d = 2, s 3. Signals, Images and More Continuous signal: f 2 L 2 ([0 , 1]).
  7. Overview •Orthogonal Representations •Linear and Non-linear Approximations •Compression and Denoising

  8. Orthogonal basis { m }m of L2([0, 1]d) Continuous signal/image

    f L2([0, 1]d). Orthogonal Decompositions
  9. Orthogonal basis { m }m of L2([0, 1]d) f =

    m f, m m ||f|| = |f(x)|2dx = m | f, m ⇥|2 Continuous signal/image f L2([0, 1]d). Orthogonal Decompositions
  10. Orthogonal basis { m }m of L2([0, 1]d) f =

    m f, m m ||f|| = |f(x)|2dx = m | f, m ⇥|2 Continuous signal/image f L2([0, 1]d). Orthogonal Decompositions m
  11. 1-D Wavelet Basis Wavelets: j,n (x) = 1 2j/2 x

    2jn 2j Position n, scale 2j, m = (n, j).
  12. 1-D Wavelet Basis Wavelets: j,n (x) = 1 2j/2 x

    2jn 2j Position n, scale 2j, m = (n, j).
  13. m1,m2 Basis { m1,m2 (x1, x2 )}m1,m2 of L2([0, 1]2)

    m1,m2 (x1, x2 ) = m1 (x1 ) m2 (x2 ) tensor product 2-D Fourier Basis Basis { m (x)}m of L2([0, 1]) m1 m2
  14. m1,m2 Basis { m1,m2 (x1, x2 )}m1,m2 of L2([0, 1]2)

    m1,m2 (x1, x2 ) = m1 (x1 ) m2 (x2 ) tensor product f(x) f, m1,m2 Fourier transform 2-D Fourier Basis Basis { m (x)}m of L2([0, 1]) m1 m2 x m
  15. 3 elementary wavelets { H, V , D}. Orthogonal basis

    of L2([0, 1]2): k j,n (x) = 2 j (2 jx n) k=H,V,D j<0,2j n [0,1]2 2-D Wavelet Basis V (x) H(x) D(x)
  16. 3 elementary wavelets { H, V , D}. Orthogonal basis

    of L2([0, 1]2): k j,n (x) = 2 j (2 jx n) k=H,V,D j<0,2j n [0,1]2 2-D Wavelet Basis V (x) H(x) D(x)
  17. wavelet f, k j,n Example of Wavelet Decomposition f(x) transform

    x (j, n, k)
  18. Discrete Computations Discrete orthogonal basis { m } of CN

    . f = m f, m m
  19. Fast Fourier Transform (FFT), O(N log(N)) operations. Discrete Computations Discrete

    orthogonal basis { m } of CN . m [n] = 1 N e2i N nm f = m f, m m
  20. Fast Fourier Transform (FFT), O(N log(N)) operations. Fast Wavelet Transform,

    O(N) operations. Discrete Wavelet basis: no closed-form expression. Discrete Computations Discrete orthogonal basis { m } of CN . m [n] = 1 N e2i N nm f = m f, m m
  21. Overview •Orthogonal Representations •Linear and Non-linear Approximations •Compression and Denoising

  22. Linear Approximation

  23. Linear Approximation

  24. Linear Approximation

  25. Non-Linear Approximation

  26. Coe cients Non-Linear Approximation

  27. Coe cients Non-Linear Approximation

  28. Best basis Fastest error decay ||f fM ||2 log(||f fM

    ||) log(M) Efficiency of Transforms Fourier DCT Local DCT Wavelets
  29. Efficient Approximation

  30. Efficient Approximation

  31. Efficient Approximation

  32. Efficient Approximation

  33. Efficient Approximation

  34. Efficient Approximation

  35. Efficient Approximation

  36. Overview •Orthogonal Representations •Linear and Non-linear Approximations •Compression and Denoising

  37. JPEG-2000 vs. JPEG, 0.2bit/pixel

  38. Compression by Transform-coding Image f Zoom on f f forward

    a[m] = ⇥f, m ⇤ R transform
  39. Compression by Transform-coding Image f Zoom on f f forward

    a[m] = ⇥f, m ⇤ R transform Quantization: q[m] = sign(a[m]) |a[m]| T ⇥ Z ˜ a[m] T T 2T 2T a[m] Quantized q[m] bin T q[m] Z
  40. Compression by Transform-coding Image f Zoom on f f forward

    a[m] = ⇥f, m ⇤ R coding transform Entropic coding: use statistical redundancy (many 0’s). Quantization: q[m] = sign(a[m]) |a[m]| T ⇥ Z ˜ a[m] T T 2T 2T a[m] Quantized q[m] bin T q[m] Z
  41. Compression by Transform-coding Image f Zoom on f f forward

    a[m] = ⇥f, m ⇤ R coding decoding q[m] Z transform Entropic coding: use statistical redundancy (many 0’s). Quantization: q[m] = sign(a[m]) |a[m]| T ⇥ Z ˜ a[m] T T 2T 2T a[m] Quantized q[m] bin T q[m] Z
  42. Compression by Transform-coding Image f Zoom on f f forward

    Dequantization: ˜ a[m] = sign(q[m]) |q[m] + 1 2 ⇥ T a[m] = ⇥f, m ⇤ R coding decoding q[m] Z ˜ a[m] dequantization transform Entropic coding: use statistical redundancy (many 0’s). Quantization: q[m] = sign(a[m]) |a[m]| T ⇥ Z ˜ a[m] T T 2T 2T a[m] Quantized q[m] bin T q[m] Z
  43. Compression by Transform-coding Image f Zoom on f f ,

    R =0.2 bit/pixel f forward Dequantization: ˜ a[m] = sign(q[m]) |q[m] + 1 2 ⇥ T a[m] = ⇥f, m ⇤ R coding decoding q[m] Z ˜ a[m] dequantization transform backward fR = m IT ˜ a[m] m transform Entropic coding: use statistical redundancy (many 0’s). Quantization: q[m] = sign(a[m]) |a[m]| T ⇥ Z ˜ a[m] T T 2T 2T a[m] Quantized q[m] bin T q[m] Z
  44. Compression by Transform-coding Image f Zoom on f f ,

    R =0.2 bit/pixel f forward Dequantization: ˜ a[m] = sign(q[m]) |q[m] + 1 2 ⇥ T a[m] = ⇥f, m ⇤ R coding decoding q[m] Z ˜ a[m] dequantization transform backward fR = m IT ˜ a[m] m transform Entropic coding: use statistical redundancy (many 0’s). Quantization: q[m] = sign(a[m]) |a[m]| T ⇥ Z ˜ a[m] T T 2T 2T a[m] Quantized q[m] bin T q[m] Z ||f fM ||2 = O(M ) =⇥ ||f fR ||2 = O(log (R)R ) Theorem:
  45. Noise in Images

  46. Denoising

  47. Denoising thresh. f = N 1 m=0 f, m ⇥

    m ˜ f = | f, m ⇥|>T f, m ⇥ m
  48. Denoising thresh. f = N 1 m=0 f, m ⇥

    m ˜ f = | f, m ⇥|>T f, m ⇥ m In practice: T 3 for T = 2 log(N) Theorem: if ||f0 f0,M ||2 = O(M ), E(|| ˜ f f0 ||2) = O( 2 +1 )
  49. Inverse Problems

  50. Inverse Problems

  51. Inverse Problems

  52. Inverse Problems

  53. Inverse Problems

  54. Restoration with Sparsity

  55. Restoration with Sparsity

  56. Restoration with Sparsity

  57. Conclusion

  58. Conclusion

  59. Conclusion