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Approximation and Coding             with Orthogonal             Decompositions                 Gabriel Peyréhttp://www.ce...
Overview• Approximation and Compression• Decay of Approximation Error• Fourier for Smooth Functions• Wavelet for Piecewise...
Sparse Approximation in a Basis
Sparse Approximation in a Basis
Sparse Approximation in a Basis
Hard Thresholding
Approximation SpeedApproximation error decay:     (usually polynomial)
Approximation Speed Approximation error decay:       (usually polynomial)                                    fM ||)       ...
Efficiency of Transforms                             fM ||)                            log10 (||f Fourier         DCT     ...
Overview• Approximation and Compression• Decay of Approximation Error• Fourier for Smooth Functions• Wavelet for Piecewise...
Compression by Transform-coding    forwardf    transform                a[m] = ⇥f,   m⇤   R    Image f                  Zo...
Compression by Transform-coding     forward f     transform                 a[m] = ⇥f,   m⇤   R                   q[m]   Z...
Compression by Transform-coding     forward                                                       codi f     transform    ...
Compression by Transform-coding     forward                                                            codi f     transfor...
Compression by Transform-coding       forward                                                              codi f       tr...
Thresholding vs. Quantizing
Non-linear Approximation and Compression
Non-linear Approximation and Compression                2T   T       T   2T    a[m]                              a[m]     ...
Non-linear Approximation and Compression                    2T    T         T     2T         a[m]                         ...
A Naive Support Coding ApproachCoding: relate # bits R to # coe cients M .                                                ...
A Naive Support Coding ApproachCoding: relate # bits R to # coe cients M .                                                ...
A Naive Support Coding ApproachCoding: relate # bits R to # coe cients M .                                                ...
A Naive Support Coding ApproachCoding: relate # bits R to # coe cients M .                                                ...
A Naive Support Coding ApproachCoding: relate # bits R to # coe cients M .                                                ...
Entropic Coders
Entropic Coders
Entropic Coders
JPEG-2000 Overview
JPEG-2000 Overview
JPEG-2000 Overview
JPEG-2000 Overview
JPEG-2000 Overview
Contextual Coding                      3 × 3 context window                    code block width
JPEG-2000 vs. JPEG, 0.2bit/pixel
Overview• Approximation and Compression• Decay of Approximation Error• Fourier for Smooth Functions• Wavelet for Piecewise...
1D Fourier Approximation
1D Fourier Approximation
Sobolev and Fourier
Singularities and Fourier            1           0.8           0.6           0.4           0.2            0            1  ...
Sobolev for Images
Sobolev for Images
Overview• Approximation and Compression• Decay of Approximation Error• Fourier for Smooth Functions• Wavelet for Piecewise...
Magnitude of Wavelet Coefficients                                                     2                                   ...
Magnitude of Wavelet Coefficients                                                                           2             ...
Magnitude of Wavelet Coefficients                                                                                       2 ...
1D Wavelet Coefficient Behavior               1              0.8              0.6              0.4              0.2       ...
1D Wavelet Coefficient Behavior                                                   1                                       ...
1D Wavelet Coefficient Behavior                                                      1                                    ...
Piecewise Regular Functions in 1D   Theorem:       If f is C outside a finite set of discontinuities:                      ...
Examples of 1D Approximations                             −1                            −1.5                             −...
Localizing the Singular Support                    S = {x1 , x2 }                Small coe cient     f (x)                ...
Localizing the Singular Support                              S = {x1 , x2 }                  Small coe cient           f (...
Localizing the Singular Support                                 S = {x1 , x2 }                      Small coe cient       ...
Localizing the Singular Support                                  S = {x1 , x2 }                      Small coe cient      ...
Computing Error and #Coefficients                f (x)                                                          |Cj |     ...
Computing Error and #Coefficients                  f (x)                                                                 |...
Computing Error and #Coefficients                  f (x)                                                                 |...
2D Wavelet ApproximationIf f is C in supp(   j,n ),   then             | f,    j,n ⇥|     2j(   +1)                       ...
Piecewise Regular Functions in 2D     Theorem:        If f is C outside a set of finite length edge curves,                ...
Example of 2D Approximations
Localizing the Singular Support in 2Df (x, y)      Length(S) = L                        j2                        j1      ...
Localizing the Singular Support in 2D         f (x, y)             Length(S) = L                                          ...
Localizing the Singular Support in 2D          f (x, y)               Length(S) = L                                       ...
Localizing the Singular Support in 2D          f (x, y)                Length(S) = L                                      ...
Computing Error and #Coefficients                f (x, y)                                                        |Cj |    ...
Computing Error and #Coefficients                    f (x, y)                                                             ...
Computing Error and #Coefficients                    f (x, y)                                                             ...
Overview• Approximation and Compression• Decay of Approximation Error• Fourier for Smooth Functions• Wavelet for Piecewise...
Geometrically Regular ImagesGeometric image model: f is C outside a set of C edge curves.     BV image: level sets have fin...
Geometic Construction : Finite ElementsApproximation of f , C2 outside C2 edges.                                          ...
Geometic Construction : Finite ElementsApproximation of f , C2 outside C2 edges.             Regular areas:               ...
Geometic Construction : Finite ElementsApproximation of f , C2 outside C2 edges.             Regular areas:               ...
Geometic Construction : Finite ElementsApproximation of f , C2 outside C2 edges.             Regular areas:               ...
Greedy Triangulation OptimizationBougleux, Peyr´, Cohen, ECCV’08              e
Curvelet Atoms                [Candes, Donoho] [Candes, Demanet, Ying, Donoho]Parabolic dyadic scaling:Rotation:          ...
Curvelet Tight FrameAngular sampling:Spacial sampling:Tight frame of L2 (R2 ):
Curvelet Approximation M -term curvelet approximation:  Theorem:   If f is C2 outside a set of C2 edges, ||f   fM ||2 = O(...
Curvelets DenoisingWorks on elongated edges.Works also on locally parallel textures !
Conclusion
Conclusion
Conclusion              1             0.8             0.6             0.4             0.2              0
Conclusion              1             0.8             0.6             0.4             0.2              0
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Signal Processing Course : Approximation

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Slides for a course on signal and image processing.

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Signal Processing Course : Approximation

  1. 1. Approximation and Coding with Orthogonal Decompositions Gabriel Peyréhttp://www.ceremade.dauphine.fr/~peyre/numerical-tour/
  2. 2. Overview• Approximation and Compression• Decay of Approximation Error• Fourier for Smooth Functions• Wavelet for Piecewise Smooth Functions• Curvelets and Finite Elements for Cartoons
  3. 3. Sparse Approximation in a Basis
  4. 4. Sparse Approximation in a Basis
  5. 5. Sparse Approximation in a Basis
  6. 6. Hard Thresholding
  7. 7. Approximation SpeedApproximation error decay: (usually polynomial)
  8. 8. Approximation Speed Approximation error decay: (usually polynomial) fM ||) log10 (||fLog/Log plot: approx. a ne curve log10 (M/N )
  9. 9. Efficiency of Transforms fM ||) log10 (||f Fourier DCT log10 (M/N )Local DCT Wavelets
  10. 10. Overview• Approximation and Compression• Decay of Approximation Error• Fourier for Smooth Functions• Wavelet for Piecewise Smooth Functions• Curvelets and Finite Elements for Cartoons
  11. 11. Compression by Transform-coding forwardf transform a[m] = ⇥f, m⇤ R Image f Zoom on f
  12. 12. Compression by Transform-coding forward f transform a[m] = ⇥f, m⇤ R q[m] Z bin T ⇥ 2T T T 2T a[m |a[m]|Quantization: q[m] = sign(a[m]) Z T a[m] ˜ Image f Zoom on f Quantized q[m]
  13. 13. Compression by Transform-coding forward codi f transform a[m] = ⇥f, m⇤ R q[m] Z ng bin T ⇥ 2T T T 2T a[m |a[m]|Quantization: q[m] = sign(a[m]) Z T a[m] ˜Entropic coding: use statistical redundancy (many 0’s). Image f Zoom on f Quantized q[m]
  14. 14. Compression by Transform-coding forward codi f transform a[m] = ⇥f, m⇤ R q[m] Z ng bin T ing decod dequantization a[m] ˜ q[m] Z ⇥ 2T T T 2T a[m |a[m]|Quantization: q[m] = sign(a[m]) Z T a[m] ˜Entropic coding: use statistical redundancy (many 0’s).Dequantization: Image f Zoom on f Quantized q[m]
  15. 15. Compression by Transform-coding forward codi f transform a[m] = ⇥f, m⇤ R q[m] Z ng bin T ing decod backward dequantizationfR = a[m] ˜ m a[m] ˜ q[m] Z transform m IT ⇥ 2T T T 2T a[m |a[m]|Quantization: q[m] = sign(a[m]) Z T a[m] ˜Entropic coding: use statistical redundancy (many 0’s).Dequantization: Image f Zoom on f Quantized q[m] fR , R =0.2 bit/pixel
  16. 16. Thresholding vs. Quantizing
  17. 17. Non-linear Approximation and Compression
  18. 18. Non-linear Approximation and Compression 2T T T 2T a[m] a[m] ˜ ⇥ |a[m]|Quantization: q[m] = sign(a[m]) Z T T =⇥ |a[m] a[m]| ˜ ⇥ 1 2Dequantization: a[m] = sign(q[m]) |q[m]| + ˜ T 2
  19. 19. Non-linear Approximation and Compression 2T T T 2T a[m] a[m] ˜ ⇥ |a[m]|Quantization: q[m] = sign(a[m]) Z T T =⇥ |a[m] a[m]| ˜ ⇥ 1 2Dequantization: a[m] = sign(q[m]) |q[m]| + ˜ T 2 ⇤ ⇤ ⇤ ⇥2 T ||f fR || = 2 (a[m] a[m]) ˜ 2 |a[m]| + 2 m 2 |a[m]|<T |a[m]| T ||f fM ||2 #=M Theorem: ||f fR ||2 ||f fM ||2 + M T 2 /4 where M = # {m a[m] = 0}. ˜
  20. 20. A Naive Support Coding ApproachCoding: relate # bits R to # coe cients M . +1 (H1 ) Ordered coe cients | f, m ⇥| decays like m 2 . =⇤ ||f fM ||2 ⇥ M .
  21. 21. A Naive Support Coding ApproachCoding: relate # bits R to # coe cients M . +1 (H1 ) Ordered coe cients | f, m ⇥| decays like m 2 . =⇤ ||f fM ||2 ⇥ M .f RN sampled from f0 , error: ||f f0 ||2 ⇥ N (H2 ) To ensure ||f f0 ||2 ⇥ ||f fM ||2 : M ⇥ N ⇥/ .
  22. 22. A Naive Support Coding ApproachCoding: relate # bits R to # coe cients M . +1 (H1 ) Ordered coe cients | f, m ⇥| decays like m 2 . =⇤ ||f fM ||2 ⇥ M .f RN sampled from f0 , error: ||f f0 ||2 ⇥ N (H2 ) To ensure ||f f0 ||2 ⇥ ||f fM ||2 : M ⇥ N ⇥/ .Simple coding strategy: R = Rval + Rind ⇥ N = Rind log2 = O(M log2 (N/M )) = O(M log2 (M )). M
  23. 23. A Naive Support Coding ApproachCoding: relate # bits R to # coe cients M . +1 (H1 ) Ordered coe cients | f, m ⇥| decays like m 2 . =⇤ ||f fM ||2 ⇥ M .f RN sampled from f0 , error: ||f f0 ||2 ⇥ N (H2 ) To ensure ||f f0 ||2 ⇥ ||f fM ||2 : M ⇥ N ⇥/ .Simple coding strategy: R = Rval + Rind ⇥ N = Rind log2 = O(M log2 (N/M )) = O(M log2 (M )). M = Rval = O(M | log2 (T )|) = O(M log2 (M ))
  24. 24. A Naive Support Coding ApproachCoding: relate # bits R to # coe cients M . +1 (H1 ) Ordered coe cients | f, m ⇥| decays like m 2 . =⇤ ||f fM ||2 ⇥ M .f RN sampled from f0 , error: ||f f0 ||2 ⇥ N (H2 ) To ensure ||f f0 ||2 ⇥ ||f fM ||2 : M ⇥ N ⇥/ .Simple coding strategy: R = Rval + Rind ⇥ N = Rind log2 = O(M log2 (N/M )) = O(M log2 (M )). M = Rval = O(M | log2 (T )|) = O(M log2 (M )) Theorem: Under hypotheses (H1 ) and (H2 ), ||f fR ||2 = O(R log (R)).
  25. 25. Entropic Coders
  26. 26. Entropic Coders
  27. 27. Entropic Coders
  28. 28. JPEG-2000 Overview
  29. 29. JPEG-2000 Overview
  30. 30. JPEG-2000 Overview
  31. 31. JPEG-2000 Overview
  32. 32. JPEG-2000 Overview
  33. 33. Contextual Coding 3 × 3 context window code block width
  34. 34. JPEG-2000 vs. JPEG, 0.2bit/pixel
  35. 35. Overview• Approximation and Compression• Decay of Approximation Error• Fourier for Smooth Functions• Wavelet for Piecewise Smooth Functions• Curvelets and Finite Elements for Cartoons
  36. 36. 1D Fourier Approximation
  37. 37. 1D Fourier Approximation
  38. 38. Sobolev and Fourier
  39. 39. Singularities and Fourier 1 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 0.2 0
  40. 40. Sobolev for Images
  41. 41. Sobolev for Images
  42. 42. Overview• Approximation and Compression• Decay of Approximation Error• Fourier for Smooth Functions• Wavelet for Piecewise Smooth Functions• Curvelets and Finite Elements for Cartoons
  43. 43. Magnitude of Wavelet Coefficients 2 p=2Vanishing moments: k < p, (x)x dx = 0 k 1 0 −1 −2 −1 0 1 2 2 p=3 1 0 −1 −2 −1 0 1 2 1.5 p=4 1 0.5 f (x) 0 −0.5 −1 −2 0 2 4
  44. 44. Magnitude of Wavelet Coefficients 2 p=2Vanishing moments: k < p, (x)x dx = 0 k 1 x 2j n If f is C on supp(⇥j,n ), p : t= 0 2j ⇤ ⇥ ⇤ 1 x 2 n j d −1⇥f, j,n ⇤ = d f (x) dx = 2j 2 R(2j t) (t)dt 2j 2 2j −2 −1 0 1 2 2 f (x) = P (x 2 n) + R(x j 2 n) = P (2 t) + R(2 t) j j j p=3 1 0| f, j,n ⇥| Cf || ||1 2j( +d/2) −1 −2 −1 0 1 2 1.5 p=4 1 0.5 f (x) 0 −0.5 −1 −2 0 2 4
  45. 45. Magnitude of Wavelet Coefficients 2 p=2Vanishing moments: k < p, (x)x dx = 0 k 1 x 2j n If f is C on supp(⇥j,n ), p : t= 0 2j ⇤ ⇥ ⇤ 1 x 2 n j d −1⇥f, j,n ⇤ = d f (x) dx = 2j 2 R(2j t) (t)dt 2j 2 2j −2 −1 0 1 2 2 f (x) = P (x 2 n) + R(x j 2 n) = P (2 t) + R(2 t) j j j p=3 1 0 d| f, j,n ⇥| Cf || ||1 2j( +d/2) | f, j,n ⇥| ||f || || ||1 2j 2 −1 −2 −1 0 1 2 1.5 p=4 1 0.5 f (x) 0 −0.5 −1 −2 0 2 4
  46. 46. 1D Wavelet Coefficient Behavior 1 0.8 0.6 0.4 0.2 0 0.2 0.1 0 −0.1 −0.2 0.2 0.1 0 −0.1 −0.2 0.5 0 −0.5 0.5 0 −0.5
  47. 47. 1D Wavelet Coefficient Behavior 1 0.8 0.6 0.4 0.2 0 0.2 0.1 0 −0.1 −0.2 0.2If f is C in supp( j,n ), then 0.1 0 2 j( +1/2) −0.1 | f, j,n ⇥| ||f ||C || ||1 −0.2 0.5 0 −0.5 0.5 0 −0.5
  48. 48. 1D Wavelet Coefficient Behavior 1 0.8 0.6 0.4 0.2 0 0.2 0.1 0 −0.1 −0.2 0.2If f is C in supp( j,n ), then 0.1 0 2 j( +1/2) −0.1 | f, j,n ⇥| ||f ||C || ||1 −0.2 0.5If f is bounded (e.g. around 0 a singularity), then −0.5 0.5 | f, j,n ⇥| 2 j/2 ||f || || ||1 0 −0.5
  49. 49. Piecewise Regular Functions in 1D Theorem: If f is C outside a finite set of discontinuities: O(M 1 ) (Fourier), n [M ] = ||f fM ||2 n = O(M 2 ) (wavelets).For Fourier, linear non-linear, sub-optimal.For wavelets, linear non-linear, optimal.
  50. 50. Examples of 1D Approximations −1 −1.5 −2 −2.5 −3 −3.5 10.8 −40.6 −4.50.4 −50.2 0 −5.5 −6 −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 1 10.8 0.80.6 0.60.4 0.40.2 0.2 0 0 1 10.8 0.80.6 0.60.4 0.40.2 0.2 0 0 1 10.8 0.80.6 0.60.4 0.40.2 0.2 0 0
  51. 51. Localizing the Singular Support S = {x1 , x2 } Small coe cient f (x) | f, jn ⇥| < Tx1 x2 Large coe cient j2 j1 Singular support: |Cj | K|S| = constant
  52. 52. Localizing the Singular Support S = {x1 , x2 } Small coe cient f (x) | f, jn ⇥| < T x1 x2 Large coe cient j2 j1 Singular support: |Cj | K|S| = constant Regular, n Cj : c |⇥f, j,n ⇤| C2j( +1/2)Coe cient behavior: Singular, n Cj : |⇥f, j,n ⇤| C2j/2
  53. 53. Localizing the Singular Support S = {x1 , x2 } Small coe cient f (x) | f, jn ⇥| < T x1 x2 Large coe cient j2 j1 Singular support: |Cj | K|S| = constant Regular, n Cj : c |⇥f, j,n ⇤| C2j( +1/2)Coe cient behavior: Singular, n Cj : |⇥f, j,n ⇤| C2j/2 1 Singular: 2j1 = (T /C) +1/2Cut-o scales (depends on T ): Regular: 2j2 = (T /C)2
  54. 54. Localizing the Singular Support S = {x1 , x2 } Small coe cient f (x) | f, jn ⇥| < T x1 x2 Large coe cient j2 j1 Singular support: |Cj | K|S| = constant Regular, n Cj : c |⇥f, j,n ⇤| C2j( +1/2)Coe cient behavior: Singular, n Cj : |⇥f, j,n ⇤| C2j/2 1 Singular: 2j1 = (T /C) +1/2Cut-o scales (depends on T ): Regular: 2j2 = (T /C)2Hand-made approximate: ˜ fM = f, j,n ⇥ j,n + f, j,n ⇥ j,n j j2 n Cj c j j1 n Cj
  55. 55. Computing Error and #Coefficients f (x) |Cj | K|S| = constant n Cj : |⇥f, c j,n ⇤| C2j( +1/2) j2 j1 n Cj : |⇥f, j,n ⇤| C2j/2 1 2 j1 = (T /C) +1/2 2j2 = (T /C)2||f fM ||2 ||f ˜ fM ||2 |⇥f, j,n ⇤| 2 + |⇥f, j,n ⇤| 2 j<j2 ,n Cj c j<j1 ,n Cj (K|S|) C 2 2j + 2 j C 2 2j(2 +1) Singulatities j<j2 j<j1 = O(2 + 2j2 2 j1 ) = O(T + T 2 2 +1/2 ) = O(T 2 +1/2 ) regular part
  56. 56. Computing Error and #Coefficients f (x) |Cj | K|S| = constant n Cj : |⇥f, c j,n ⇤| C2j( +1/2) j2 j1 n Cj : |⇥f, j,n ⇤| C2j/2 1 2 j1 = (T /C) +1/2 2j2 = (T /C)2||f fM ||2 ||f ˜ fM ||2 |⇥f, j,n ⇤| 2 + |⇥f, j,n ⇤| 2 j<j2 ,n Cj c j<j1 ,n Cj (K|S|) C 2 2j + 2 j C 2 2j(2 +1) Singulatities j<j2 j<j1 = O(2 + 2 j2 2 j1 ) = O(T + T2 2 +1/2 ) = O(T 2 +1/2 ) regular partM |Cj | + c |Cj | K|S| + 2 j j j2 j j1 j j2 j j1 1 1 = O(| log(T )| + T +1/2 ) = O(T +1/2 )
  57. 57. Computing Error and #Coefficients f (x) |Cj | K|S| = constant n Cj : |⇥f, c j,n ⇤| C2j( +1/2) j2 j1 n Cj : |⇥f, j,n ⇤| C2j/2 1 2 j1 = (T /C) +1/2 2j2 = (T /C)2||f fM ||2 ||f ˜ fM ||2 |⇥f, j,n ⇤| 2 + |⇥f, j,n ⇤| 2 j<j2 ,n Cj c j<j1 ,n Cj (K|S|) C 2 2j + 2 j C 2 2j(2 +1) Singulatities j<j2 j<j1 = O(2 + 2 j2 2 j1 ) = O(T + T2 2 +1/2 ) = O(T 2 +1/2 ) regular partM |Cj | + c |Cj | K|S| + 2 j j j2 j j1 j j2 j j1 1 1 = O(| log(T )| + T +1/2 ) = O(T +1/2 ) ||f fM ||2 = O(M 2 )
  58. 58. 2D Wavelet ApproximationIf f is C in supp( j,n ), then | f, j,n ⇥| 2j( +1) ||f ||C || ||1If f is bounded (e.g. around a singularity), then | f, j,n ⇥| 2j ||f || || ||1
  59. 59. Piecewise Regular Functions in 2D Theorem: If f is C outside a set of finite length edge curves, O(M 1/2 ) (Fourier), n [M ] = ||f fM ||2 n = O(M 1 ) (wavelets).Fourier Wavelet, both sub-optimal.Wavelets: same result for BV functions (optimal).
  60. 60. Example of 2D Approximations
  61. 61. Localizing the Singular Support in 2Df (x, y) Length(S) = L j2 j1 |Cj | LK2 j = constant
  62. 62. Localizing the Singular Support in 2D f (x, y) Length(S) = L j2 j1 |Cj | LK2 j = constant Regular, n Cj : c |⇥f, j,n ⇤| C2j( +1)Coe cient behavior: Singular, n Cj : |⇥f, j,n ⇤| C2j
  63. 63. Localizing the Singular Support in 2D f (x, y) Length(S) = L j2 j1 |Cj | LK2 j = constant Regular, n Cj : c |⇥f, j,n ⇤| C2j( +1)Coe cient behavior: Singular, n Cj : |⇥f, j,n ⇤| C2j 1 Singular: 2j1 = (T /C) +1Cut-o scales (depends on T ): Regular: 2j2 = T /C
  64. 64. Localizing the Singular Support in 2D f (x, y) Length(S) = L j2 j1 |Cj | LK2 j = constant Regular, n Cj : c |⇥f, j,n ⇤| C2j( +1)Coe cient behavior: Singular, n Cj : |⇥f, j,n ⇤| C2j 1 Singular: 2 j1 = (T /C) +1Cut-o scales (depends on T ): Regular: 2j2 = T /CHand-made approximate: ˜ fM = f, j,n ⇥ j,n + f, j,n ⇥ j,n j j2 n Cj c j j1 n Cj
  65. 65. Computing Error and #Coefficients f (x, y) |Cj | LK2 j = constant n Cj : |⇥f, c j,n ⇤| C2j( +1) j2 n Cj : |⇥f, j,n ⇤| C2j j1 1 2 j1 = (T /C) +1 2j2 = T /C||f fM ||2 ||f ˜ fM ||2 |⇥f, j,n ⇤| 2 + |⇥f, j,n ⇤| 2 j<j2 ,n Cj c j<j1 ,n Cj LK2 j C 2 22j + 2 2j C 2 22j( +1) Singulatities j<j2 j<j1 = O(2 + 2 j2 2 j1 ) = O(T + T 2 +1 ) = O(T ) regular part
  66. 66. Computing Error and #Coefficients f (x, y) |Cj | LK2 j = constant n Cj : |⇥f, c j,n ⇤| C2j( +1) j2 n Cj : |⇥f, j,n ⇤| C2j j1 1 2 j1 = (T /C) +1 2j2 = T /C||f fM ||2 ||f ˜ fM ||2 |⇥f, j,n ⇤| 2 + |⇥f, j,n ⇤| 2 j<j2 ,n Cj c j<j1 ,n Cj LK2 j C 2 22j + 2 2j C 2 22j( +1) Singulatities j<j2 j<j1 = O(2 + 2 j2 2 j1 ) = O(T + T 2 +1 ) = O(T ) regular partM |Cj | + c |Cj | LK2 j + 2 2j j j2 j j1 j j2 j j1 1 = O(T 1 +T +1 ) = O(T 1 )
  67. 67. Computing Error and #Coefficients f (x, y) |Cj | LK2 j = constant n Cj : |⇥f, c j,n ⇤| C2j( +1) j2 n Cj : |⇥f, j,n ⇤| C2j j1 1 2 j1 = (T /C) +1 2j2 = T /C||f fM ||2 ||f ˜ fM ||2 |⇥f, j,n ⇤| 2 + |⇥f, j,n ⇤| 2 j<j2 ,n Cj c j<j1 ,n Cj LK2 j C 2 22j + 2 2j C 2 22j( +1) Singulatities j<j2 j<j1 = O(2 + 2 j2 2 j1 ) = O(T + T 2 +1 ) = O(T ) regular partM |Cj | + c |Cj | LK2 j + 2 2j j j2 j j1 j j2 j j1 1 = O(T 1 +T +1 ) = O(T 1 ) ||f fM ||2 = O(M 1 )
  68. 68. Overview• Approximation and Compression• Decay of Approximation Error• Fourier for Smooth Functions• Wavelet for Piecewise Smooth Functions• Curvelets and Finite Elements for Cartoons
  69. 69. Geometrically Regular ImagesGeometric image model: f is C outside a set of C edge curves. BV image: level sets have finite lengths. Geometric image: level sets are regular. | f| Geometry = cartoon image Sharp edges Smoothed edges
  70. 70. Geometic Construction : Finite ElementsApproximation of f , C2 outside C2 edges. ˜Piecewise linear approximation on M triangles: fM .
  71. 71. Geometic Construction : Finite ElementsApproximation of f , C2 outside C2 edges. Regular areas: M/2 equilateral triangles. ˜Piecewise linear approximation on M triangles: fM . 1/2 M 1/2 M
  72. 72. Geometic Construction : Finite ElementsApproximation of f , C2 outside C2 edges. Regular areas: M/2 equilateral triangles. ˜Piecewise linear approximation on M triangles: fM . 1/2 M 1/2 M Singular areas: M/2 anisotropic triangles.
  73. 73. Geometic Construction : Finite ElementsApproximation of f , C2 outside C2 edges. Regular areas: M/2 equilateral triangles. ˜Piecewise linear approximation on M triangles: fM . 1/2 M 1/2 M Singular areas: M/2 anisotropic triangles. Theorem: If f is C2 outside a set of C2 contours, then one has for an adapted ˜ triangulation ||f fM ||2 = O(M 2 ). Di culties to build e cient approximations. No optimal strategies (greedy solutions).
  74. 74. Greedy Triangulation OptimizationBougleux, Peyr´, Cohen, ECCV’08 e
  75. 75. Curvelet Atoms [Candes, Donoho] [Candes, Demanet, Ying, Donoho]Parabolic dyadic scaling:Rotation: “width length2 ”
  76. 76. Curvelet Tight FrameAngular sampling:Spacial sampling:Tight frame of L2 (R2 ):
  77. 77. Curvelet Approximation M -term curvelet approximation: Theorem: If f is C2 outside a set of C2 edges, ||f fM ||2 = O(M 2 (log M )3 ).Discrete curvelets: O(N log(N )) algorithm.Redundancy 5 =⇥ not e cient for compression.
  78. 78. Curvelets DenoisingWorks on elongated edges.Works also on locally parallel textures !
  79. 79. Conclusion
  80. 80. Conclusion
  81. 81. Conclusion 1 0.8 0.6 0.4 0.2 0
  82. 82. Conclusion 1 0.8 0.6 0.4 0.2 0

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