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Signal Processing Course: Non-linear Approximation and Coding

Signal Processing Course: Non-linear Approximation and Coding

Gabriel Peyré

January 01, 2012
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  1. Approximation and Coding
    with Orthogonal
    Decompositions
    Gabriel Peyré
    http://www.ceremade.dauphine.fr/~peyre/numerical-tour/

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  2. Overview
    • Approximation and Compression
    • Decay of Approximation Error
    • Fourier for Smooth Functions
    • Wavelet for Piecewise Smooth Functions
    • Curvelets and Finite Elements for Cartoons

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  3. Sparse Approximation in a Basis

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  4. Sparse Approximation in a Basis

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  5. Sparse Approximation in a Basis

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  6. Hard Thresholding

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  7. (usually polynomial)
    Approximation Speed
    Approximation error decay:

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  8. (usually polynomial)
    Approximation Speed
    Approximation error decay:
    log
    10
    (||f fM
    ||)
    Log/Log plot: approx. a ne curve
    log
    10
    (M/N)

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  9. Efficiency of Transforms
    Fourier DCT
    Local DCT Wavelets
    log
    10
    (||f fM
    ||)
    log
    10
    (M/N)

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  10. Overview
    • Approximation and Compression
    • Decay of Approximation Error
    • Fourier for Smooth Functions
    • Wavelet for Piecewise Smooth Functions
    • Curvelets and Finite Elements for Cartoons

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

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  12. f
    forward
    Compression by Transform-coding
    a[m] = ⇥f, m
    ⇤ R
    Quantization: q[m] = sign(a[m])
    |a[m]|
    T

    Z
    Image f Zoom on f
    transform
    ˜
    a[m]
    T T 2T
    2T a[m
    Quantized q[m]
    bin T
    q[m] Z

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  13. f
    forward coding
    Compression by Transform-coding
    a[m] = ⇥f, m
    ⇤ R
    Quantization: q[m] = sign(a[m])
    |a[m]|
    T

    Z
    Image f Zoom on f
    transform
    Entropic coding: use statistical redundancy (many 0’s).
    ˜
    a[m]
    T T 2T
    2T a[m
    Quantized q[m]
    bin T
    q[m] Z

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  14. f
    forward coding
    Compression by Transform-coding
    a[m] = ⇥f, m
    ⇤ R
    Quantization: q[m] = sign(a[m])
    |a[m]|
    T

    Z
    Image f Zoom on f
    decoding
    q[m] Z
    ˜
    a[m] dequantization
    transform
    Entropic coding: use statistical redundancy (many 0’s).
    ˜
    a[m]
    T T 2T
    2T a[m
    Quantized q[m]
    bin T
    q[m] Z
    Dequantization:

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  15. f
    forward coding
    Compression by Transform-coding
    a[m] = ⇥f, m
    ⇤ R
    Quantization: q[m] = sign(a[m])
    |a[m]|
    T

    Z
    Image f Zoom on f fR
    , R =0.2 bit/pixel
    decoding
    q[m] Z
    ˜
    a[m] dequantization
    transform
    backward
    fR
    =
    m IT
    ˜
    a[m]
    m
    transform
    Entropic coding: use statistical redundancy (many 0’s).
    ˜
    a[m]
    T T 2T
    2T a[m
    Quantized q[m]
    bin T
    q[m] Z
    Dequantization:

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  16. Thresholding vs. Quantizing

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  17. Non-linear Approximation and Compression

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  18. Non-linear Approximation and Compression
    ˜
    a[m]
    T T 2T
    2T a[m]
    Quantization: q[m] = sign(a[m])
    |a[m]|
    T

    Z
    =⇥ |a[m] ˜
    a[m]|
    T
    2
    Dequantization: ˜
    a[m] = sign(q[m]) |q[m]| +
    1
    2

    T

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  19. Non-linear Approximation and Compression
    ||f fM
    ||2 # = M
    Theorem: ||f fR
    ||2 ||f fM
    ||2 + MT2/4
    where M = # {m \ ˜
    a[m] = 0}.
    ||f fR
    ||2 =

    m
    (a[m] ˜
    a[m])2

    |a[m]||a[m]|2 +

    |a[m]| T
    T
    2
    ⇥2
    ˜
    a[m]
    T T 2T
    2T a[m]
    Quantization: q[m] = sign(a[m])
    |a[m]|
    T

    Z
    =⇥ |a[m] ˜
    a[m]|
    T
    2
    Dequantization: ˜
    a[m] = sign(q[m]) |q[m]| +
    1
    2

    T

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  20. A Naive Support Coding Approach
    Coding: relate # bits R to # coe cients M.
    (H
    1
    ) Ordered coe cients | f, m
    ⇥| decays like m +1
    2
    .
    =⇤ ||f fM
    ||2 ⇥ M .

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  21. A Naive Support Coding Approach
    Coding: relate # bits R to # coe cients M.
    (H
    1
    ) Ordered coe cients | f, m
    ⇥| decays like m +1
    2
    .
    f RN sampled from f0
    , error:
    (H
    2
    ) To ensure ||f f0
    ||2 ⇥ ||f fM
    ||2: M ⇥ N⇥/ .
    ||f f0
    ||2 ⇥ N
    =⇤ ||f fM
    ||2 ⇥ M .

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  22. A Naive Support Coding Approach
    Simple coding strategy: R = Rval
    + Rind
    = Rind
    log
    2
    N
    M

    = O(M log
    2
    (N/M)) = O(M log
    2
    (M)).
    Coding: relate # bits R to # coe cients M.
    (H
    1
    ) Ordered coe cients | f, m
    ⇥| decays like m +1
    2
    .
    f RN sampled from f0
    , error:
    (H
    2
    ) To ensure ||f f0
    ||2 ⇥ ||f fM
    ||2: M ⇥ N⇥/ .
    ||f f0
    ||2 ⇥ N
    =⇤ ||f fM
    ||2 ⇥ M .

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  23. A Naive Support Coding Approach
    Simple coding strategy: R = Rval
    + Rind
    = Rind
    log
    2
    N
    M

    = O(M log
    2
    (N/M)) = O(M log
    2
    (M)).
    = Rval
    = O(M| log
    2
    (T)|) = O(M log
    2
    (M))
    Coding: relate # bits R to # coe cients M.
    (H
    1
    ) Ordered coe cients | f, m
    ⇥| decays like m +1
    2
    .
    f RN sampled from f0
    , error:
    (H
    2
    ) To ensure ||f f0
    ||2 ⇥ ||f fM
    ||2: M ⇥ N⇥/ .
    ||f f0
    ||2 ⇥ N
    =⇤ ||f fM
    ||2 ⇥ M .

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  24. A Naive Support Coding Approach
    Simple coding strategy: R = Rval
    + Rind
    = Rind
    log
    2
    N
    M

    = O(M log
    2
    (N/M)) = O(M log
    2
    (M)).
    Theorem: Under hypotheses (H
    1
    ) and (H
    2
    ), ||f fR
    ||2 = O(R log (R)).
    = Rval
    = O(M| log
    2
    (T)|) = O(M log
    2
    (M))
    Coding: relate # bits R to # coe cients M.
    (H
    1
    ) Ordered coe cients | f, m
    ⇥| decays like m +1
    2
    .
    f RN sampled from f0
    , error:
    (H
    2
    ) To ensure ||f f0
    ||2 ⇥ ||f fM
    ||2: M ⇥ N⇥/ .
    ||f f0
    ||2 ⇥ N
    =⇤ ||f fM
    ||2 ⇥ M .

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  25. Entropic Coders

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  26. Entropic Coders

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  27. Entropic Coders

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  28. JPEG-2000 Overview

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  29. JPEG-2000 Overview

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  30. JPEG-2000 Overview

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  31. JPEG-2000 Overview

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  32. JPEG-2000 Overview

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  33. Contextual Coding
    code block width
    3 × 3 context window

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

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  35. Overview
    • Approximation and Compression
    • Decay of Approximation Error
    • Fourier for Smooth Functions
    • Wavelet for Piecewise Smooth Functions
    • Curvelets and Finite Elements for Cartoons

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  36. 1D Fourier Approximation

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  37. 1D Fourier Approximation

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  38. Sobolev and Fourier

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  39. Singularities and Fourier
    0
    0.2
    0.4
    0.6
    0.8
    1
    0
    0.2
    0.4
    0.6
    0.8
    1
    0
    0.2
    0.4
    0.6
    0.8
    1
    0
    0.2
    0.4
    0.6
    0.8
    1

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  40. Sobolev for Images

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  41. Sobolev for Images

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  42. Overview
    • Approximation and Compression
    • Decay of Approximation Error
    • Fourier for Smooth Functions
    • Wavelet for Piecewise Smooth Functions
    • Curvelets and Finite Elements for Cartoons

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  43. Vanishing moments:
    p = 3
    p = 4
    Magnitude of Wavelet Coefficients
    f(x)
    −1 0 1 2
    −2
    −1
    0
    1
    2
    −2 −1 0 1 2
    −1
    0
    1
    2
    −2 0 2 4
    −1
    −0.5
    0
    0.5
    1
    1.5
    k < p, (x)xkdx = 0
    p = 2

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  44. Vanishing moments:
    p = 3
    p = 4
    Magnitude of Wavelet Coefficients
    f(x)
    −1 0 1 2
    −2
    −1
    0
    1
    2
    −2 −1 0 1 2
    −1
    0
    1
    2
    −2 0 2 4
    −1
    −0.5
    0
    0.5
    1
    1.5
    k < p, (x)xkdx = 0
    p = 2
    | f, j,n
    ⇥| Cf
    || ||1
    2j( +d/2)
    t = x 2jn
    2j
    ⇥f, j,n
    ⇤ =
    1
    2j d
    2

    f(x) x 2jn
    2j

    dx = 2j d
    2

    R(2jt) (t)dt
    If f is C on supp(⇥j,n
    ), p :
    f(x) = P(x 2jn) + R(x 2jn) = P(2jt) + R(2jt)

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  45. Vanishing moments:
    p = 3
    p = 4
    Magnitude of Wavelet Coefficients
    f(x)
    −1 0 1 2
    −2
    −1
    0
    1
    2
    −2 −1 0 1 2
    −1
    0
    1
    2
    −2 0 2 4
    −1
    −0.5
    0
    0.5
    1
    1.5
    k < p, (x)xkdx = 0
    | f, j,n
    ⇥| ||f|| || ||1
    2j d
    2
    p = 2
    | f, j,n
    ⇥| Cf
    || ||1
    2j( +d/2)
    t = x 2jn
    2j
    ⇥f, j,n
    ⇤ =
    1
    2j d
    2

    f(x) x 2jn
    2j

    dx = 2j d
    2

    R(2jt) (t)dt
    If f is C on supp(⇥j,n
    ), p :
    f(x) = P(x 2jn) + R(x 2jn) = P(2jt) + R(2jt)

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  46. 1D Wavelet Coefficient Behavior
    −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
    0
    0.2
    0.4
    0.6
    0.8
    1

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  47. 1D Wavelet Coefficient Behavior
    If f is C in supp(
    j,n
    ), then
    | f, j,n
    ⇥| 2j( +1/2)||f||C
    || ||1
    −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
    0
    0.2
    0.4
    0.6
    0.8
    1

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  48. 1D Wavelet Coefficient Behavior
    If f is C in supp(
    j,n
    ), then
    | f, j,n
    ⇥| 2j( +1/2)||f||C
    || ||1
    | f, j,n
    ⇥| 2j/2||f|| || ||1
    If f is bounded (e.g. around
    a singularity), then
    −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
    0
    0.2
    0.4
    0.6
    0.8
    1

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  49. For Fourier, linear non-linear, sub-optimal.
    For wavelets, linear non-linear, optimal.
    Piecewise Regular Functions in 1D
    Theorem: If f is C outside a finite set of discontinuities:
    n[M] = ||f fn
    M
    ||2 = O(M 1) (Fourier),
    O(M 2 ) (wavelets).

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  50. Examples of 1D Approximations
    0
    0.2
    0.4
    0.6
    0.8
    1
    0
    0.2
    0.4
    0.6
    0.8
    1
    0
    0.2
    0.4
    0.6
    0.8
    1
    0
    0.2
    0.4
    0.6
    0.8
    1
    0
    0.2
    0.4
    0.6
    0.8
    1
    0
    0.2
    0.4
    0.6
    0.8
    1
    0
    0.2
    0.4
    0.6
    0.8
    1
    −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8
    −6
    −5.5
    −5
    −4.5
    −4
    −3.5
    −3
    −2.5
    −2
    −1.5
    −1

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  51. Large coe cient
    | f, jn
    ⇥| < T
    x1 x2
    S = {x1, x2
    }
    Localizing the Singular Support
    f(x)
    j1
    j2
    Small coe cient
    Singular support:
    |Cj
    | K|S| = constant

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  52. Large coe cient
    | f, jn
    ⇥| < T
    x1 x2
    S = {x1, x2
    }
    Localizing the Singular Support
    f(x)
    j1
    j2
    Small coe cient
    Singular support:
    Coe cient behavior:
    Regular, n Cc
    j
    : |⇥f, j,n
    ⇤| C2j( +1/2)
    Singular, n Cj
    : |⇥f, j,n
    ⇤| C2j/2
    |Cj
    | K|S| = constant

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  53. Large coe cient
    | f, jn
    ⇥| < T
    x1 x2
    S = {x1, x2
    }
    Localizing the Singular Support
    f(x)
    j1
    j2
    Small coe cient
    Singular support:
    Coe cient behavior:
    Regular, n Cc
    j
    : |⇥f, j,n
    ⇤| C2j( +1/2)
    Singular, n Cj
    : |⇥f, j,n
    ⇤| C2j/2
    Cut-o scales (depends on T):
    Singular: 2j1 = (T/C) 1
    +1/2
    Regular: 2j2 = (T/C)2
    |Cj
    | K|S| = constant

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  54. Large coe cient
    | f, jn
    ⇥| < T
    Hand-made approximate: ˜
    fM
    =
    j j2 n Cj
    f, j,n
    ⇥ j,n
    +
    j j1 n Cc
    j
    f, j,n
    ⇥ j,n
    x1 x2
    S = {x1, x2
    }
    Localizing the Singular Support
    f(x)
    j1
    j2
    Small coe cient
    Singular support:
    Coe cient behavior:
    Regular, n Cc
    j
    : |⇥f, j,n
    ⇤| C2j( +1/2)
    Singular, n Cj
    : |⇥f, j,n
    ⇤| C2j/2
    Cut-o scales (depends on T):
    Singular: 2j1 = (T/C) 1
    +1/2
    Regular: 2j2 = (T/C)2
    |Cj
    | K|S| = constant

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  55. ||f fM
    ||2 ||f ˜
    fM
    ||2
    j|⇥f, j,n
    ⇤|2 +
    jj
    |⇥f, j,n
    ⇤|2
    Computing Error and #Coefficients
    f(x)
    j1
    j2
    n Cc
    j
    : |⇥f, j,n
    ⇤| C2j( +1/2)
    n Cj
    : |⇥f, j,n
    ⇤| C2j/2
    2j1 = (T/C) 1
    +1/2
    2j2 = (T/C)2
    j(K|S|) C22j +
    j2 j C22j(2 +1)
    = O(2j2 + 22 j1 ) = O(T2 + T
    2
    +1/2 ) = O(T
    2
    +1/2 )
    |Cj
    | K|S| = constant
    Singulatities
    regular part

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  56. ||f fM
    ||2 ||f ˜
    fM
    ||2
    j|⇥f, j,n
    ⇤|2 +
    jj
    |⇥f, j,n
    ⇤|2
    Computing Error and #Coefficients
    f(x)
    j1
    j2
    n Cc
    j
    : |⇥f, j,n
    ⇤| C2j( +1/2)
    n Cj
    : |⇥f, j,n
    ⇤| C2j/2
    2j1 = (T/C) 1
    +1/2
    2j2 = (T/C)2
    j(K|S|) C22j +
    j2 j C22j(2 +1)
    M
    j j2
    |Cj
    | +
    j j1
    |Cc
    j
    |
    j j2
    K|S| +
    j j1
    2 j
    = O(| log(T)| + T
    1
    +1/2 ) = O(T
    1
    +1/2 )
    = O(2j2 + 22 j1 ) = O(T2 + T
    2
    +1/2 ) = O(T
    2
    +1/2 )
    |Cj
    | K|S| = constant
    Singulatities
    regular part

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  57. ||f fM
    ||2 ||f ˜
    fM
    ||2
    j|⇥f, j,n
    ⇤|2 +
    jj
    |⇥f, j,n
    ⇤|2
    Computing Error and #Coefficients
    f(x)
    j1
    j2
    n Cc
    j
    : |⇥f, j,n
    ⇤| C2j( +1/2)
    n Cj
    : |⇥f, j,n
    ⇤| C2j/2
    2j1 = (T/C) 1
    +1/2
    2j2 = (T/C)2
    j(K|S|) C22j +
    j2 j C22j(2 +1)
    M
    j j2
    |Cj
    | +
    j j1
    |Cc
    j
    |
    j j2
    K|S| +
    j j1
    2 j
    = O(| log(T)| + T
    1
    +1/2 ) = O(T
    1
    +1/2 ) ||f fM
    ||2 = O(M 2 )
    = O(2j2 + 22 j1 ) = O(T2 + T
    2
    +1/2 ) = O(T
    2
    +1/2 )
    |Cj
    | K|S| = constant
    Singulatities
    regular part

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  58. 2D Wavelet Approximation
    If f is C in supp(
    j,n
    ), then
    | f, j,n
    ⇥| 2j( +1)||f||C
    || ||1
    | f, j,n
    ⇥| 2j||f|| || ||1
    If f is bounded (e.g. around a singularity), then

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  59. Fourier Wavelet, both sub-optimal.
    Wavelets: same result for BV functions (optimal).
    Piecewise Regular Functions in 2D
    n[M] = ||f fn
    M
    ||2 = O(M 1/2) (Fourier),
    O(M 1) (wavelets).
    Theorem: If f is C outside a set of finite length edge curves,

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  60. Example of 2D Approximations

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  61. Length(S) = L
    Localizing the Singular Support in 2D
    j1
    j2
    f(x, y)
    |Cj
    | LK2 j = constant

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  62. Length(S) = L
    Localizing the Singular Support in 2D
    j1
    j2
    f(x, y)
    Coe cient behavior:
    Regular, n Cc
    j
    : |⇥f, j,n
    ⇤| C2j( +1)
    Singular, n Cj
    : |⇥f, j,n
    ⇤| C2j
    |Cj
    | LK2 j = constant

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  63. Length(S) = L
    Localizing the Singular Support in 2D
    j1
    j2
    f(x, y)
    Coe cient behavior:
    Regular, n Cc
    j
    : |⇥f, j,n
    ⇤| C2j( +1)
    Singular, n Cj
    : |⇥f, j,n
    ⇤| C2j
    Cut-o scales (depends on T):
    Singular: 2j1 = (T/C) 1
    +1
    Regular: 2j2 = T/C
    |Cj
    | LK2 j = constant

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  64. Hand-made approximate: ˜
    fM
    =
    j j2 n Cj
    f, j,n
    ⇥ j,n
    +
    j j1 n Cc
    j
    f, j,n
    ⇥ j,n
    Length(S) = L
    Localizing the Singular Support in 2D
    j1
    j2
    f(x, y)
    Coe cient behavior:
    Regular, n Cc
    j
    : |⇥f, j,n
    ⇤| C2j( +1)
    Singular, n Cj
    : |⇥f, j,n
    ⇤| C2j
    Cut-o scales (depends on T):
    Singular: 2j1 = (T/C) 1
    +1
    Regular: 2j2 = T/C
    |Cj
    | LK2 j = constant

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  65. ||f fM
    ||2 ||f ˜
    fM
    ||2
    j|⇥f, j,n
    ⇤|2 +
    jj
    |⇥f, j,n
    ⇤|2
    Singulatities
    regular part
    Computing Error and #Coefficients
    j1
    j2
    f(x, y)
    n Cc
    j
    : |⇥f, j,n
    ⇤| C2j( +1)
    n Cj
    : |⇥f, j,n
    ⇤| C2j
    2j2 = T/C
    2j1 = (T/C) 1
    +1
    jLK2 j C222j +
    j2 2j C222j( +1)
    |Cj
    | LK2 j = constant
    = O(2j2 + 22 j1 ) = O(T + T 2
    +1 ) = O(T)

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  66. ||f fM
    ||2 ||f ˜
    fM
    ||2
    j|⇥f, j,n
    ⇤|2 +
    jj
    |⇥f, j,n
    ⇤|2
    Singulatities
    regular part
    Computing Error and #Coefficients
    j1
    j2
    f(x, y)
    n Cc
    j
    : |⇥f, j,n
    ⇤| C2j( +1)
    n Cj
    : |⇥f, j,n
    ⇤| C2j
    2j2 = T/C
    2j1 = (T/C) 1
    +1
    jLK2 j C222j +
    j2 2j C222j( +1)
    |Cj
    | LK2 j = constant
    = O(2j2 + 22 j1 ) = O(T + T 2
    +1 ) = O(T)
    M
    j j2
    |Cj
    | +
    j j1
    |Cc
    j
    |
    j j2
    LK2 j +
    j j1
    2 2j
    = O(T 1 + T
    1
    +1 ) = O(T 1)

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  67. ||f fM
    ||2 = O(M 1)
    ||f fM
    ||2 ||f ˜
    fM
    ||2
    j|⇥f, j,n
    ⇤|2 +
    jj
    |⇥f, j,n
    ⇤|2
    Singulatities
    regular part
    Computing Error and #Coefficients
    j1
    j2
    f(x, y)
    n Cc
    j
    : |⇥f, j,n
    ⇤| C2j( +1)
    n Cj
    : |⇥f, j,n
    ⇤| C2j
    2j2 = T/C
    2j1 = (T/C) 1
    +1
    jLK2 j C222j +
    j2 2j C222j( +1)
    |Cj
    | LK2 j = constant
    = O(2j2 + 22 j1 ) = O(T + T 2
    +1 ) = O(T)
    M
    j j2
    |Cj
    | +
    j j1
    |Cc
    j
    |
    j j2
    LK2 j +
    j j1
    2 2j
    = O(T 1 + T
    1
    +1 ) = O(T 1)

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  68. Overview
    • Approximation and Compression
    • Decay of Approximation Error
    • Fourier for Smooth Functions
    • Wavelet for Piecewise Smooth Functions
    • Curvelets and Finite Elements for Cartoons

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  69. Geometric 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.
    Geometry = cartoon image Sharp edges Smoothed edges
    Geometrically Regular Images
    | f|

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  70. Approximation of f, C2 outside C2 edges.
    Piecewise linear approximation on M triangles: ˜
    fM
    .
    Geometic Construction : Finite Elements

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  71. Approximation of f, C2 outside C2 edges.
    Piecewise linear approximation on M triangles: ˜
    fM
    .
    Geometic Construction : Finite Elements
    Regular areas:
    M/2 equilateral triangles.
    M 1/2
    M 1/2

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  72. Approximation of f, C2 outside C2 edges.
    Piecewise linear approximation on M triangles: ˜
    fM
    .
    Geometic Construction : Finite Elements
    Regular areas:
    M/2 equilateral triangles.
    M 1/2
    M 1/2
    M/2 anisotropic triangles.
    Singular areas:

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  73. Approximation of f, C2 outside C2 edges.
    Piecewise linear approximation on M triangles: ˜
    fM
    .
    Di culties to build e cient approximations.
    No optimal strategies (greedy solutions).
    Theorem: If f is C2 outside a set of C2 contours, then one has for an adapted
    triangulation ||f ˜
    fM
    ||2 = O(M 2).
    Geometic Construction : Finite Elements
    Regular areas:
    M/2 equilateral triangles.
    M 1/2
    M 1/2
    M/2 anisotropic triangles.
    Singular areas:

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  74. Greedy Triangulation Optimization
    Bougleux, Peyr´
    e, Cohen, ECCV’08

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  75. Curvelet Atoms
    Parabolic dyadic scaling:
    Rotation:
    [Candes, Donoho] [Candes, Demanet, Ying, Donoho]
    “width length2”

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  76. Curvelet Tight Frame
    Spacial sampling:
    Tight frame of L2(R2):
    Angular sampling:

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  77. Discrete curvelets: O(N log(N)) algorithm.
    Redundancy 5 =⇥ not e cient for compression.
    M-term curvelet approximation:
    Curvelet Approximation
    Theorem: If f is C2 outside a set of C2 edges, ||f fM
    ||2 = O(M 2(log M)3).

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  78. Works on elongated edges.
    Works also on locally parallel textures !
    Curvelets Denoising

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  79. Conclusion
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  80. Conclusion
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