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Shearlet Frames and Optimally Sparse
Approximations

Jakob Lemvig — Technical University of Denmark (DTU)
joint work with Gitta Kutyniok & Wang-Q Lim (TU Berlin)
Outline


1   Applied Harmonic Analysis and Imaging Sciences

2   Fourier and Wavelet Analysis

3   Shearlet Theory

4   Sparse Approximation using 3D Shearlets

5   Conclusion




2/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
Outline

1   Applied Harmonic Analysis and Imaging Sciences
      Anisotropic Phenomena
      Image Model
      Goal for Today

2   Fourier and Wavelet Analysis

3   Shearlet Theory

4   Sparse Approximation using 3D Shearlets

5   Conclusion


3/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
Anisotropic Phenomena in Multivariate Data
Many important multivariate problem classes are governed by
anisotropic features, which require efficient encoding strategies.
The anisotropic structure can be given. . .
• . . . explicitly
      • Image Processing: Edges
      • Seismology: Earth layers
      •
• . . . implicitely
       • PDEs: Shock waves, Boundary layers




4/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
Modern Imaging
Some important tasks:
• Denoising
• Inpainting
• Feature Detection/Extraction
• ...

Imaging Sciences using Applied Harmonic Analysis:
Exploit a carefully designed representation system (ψλ )λ ⊂ L2 (Rd ) :

                              Image =              Image, ψλ ψλ .
                                              λ


• Sparse coefficients
• Approximation properties

5/42 DTU Mathematics, Technical University of Denmark               Shearlet Frames   26.3.2012
Geometric Structures of Multidimensional Images

• What are anisotropic features in a 2D Image? Curves

        Objects
                                                               Cartoon-
          with
                                                               like
      ‘smooth’
                                                               image
    boundaries




6/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
Geometric Structures of Multidimensional Images

• What are anisotropic features in a 2D Image? Curves

        Objects
                                                                Cartoon-
          with
                                                                like
      ‘smooth’
                                                                image
    boundaries

• What are anisotropic features in 3D data? Curves & surfaces
                                                        Video: objects
                                                        carve out
   3D medical
                                                        spatial-
       scans:
                                                        temporal
                                                        tubes


6/42 DTU Mathematics, Technical University of Denmark    Shearlet Frames   26.3.2012
Geometric Structures of Multidimensional Images

• What are anisotropic features in a 2D Image? Curves

        Objects
                                                               Cartoon-
          with
                                                               like
      ‘smooth’
                                                               image
    boundaries

• What are anisotropic features in 3D data? Curves & surfaces
                                                Video: objects
                                                carve out
  3D medical
                                                spatial-
        scans:
                                                temporal
                                                tubes
• Most information is contained in lower dimensional structures!

6/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
Reasonable Model for 2D Images
Definition (Donoho; 2001)
The set of cartoon-like 2D images E2 (R2 ) is defined by
                                   2


E2 (R2 ) = {f ∈ L2 (R2 ) : f = f0 +f1 χB } fi ∈ C 2 (R2 ), supp fi ⊂ [0, 1]2 ,
 2


where B ⊂ [0, 1]2 and the boundary curve ∂B is a closed C 2 -curve
with curvature bounded by ν.




                                     f0=0




7/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
Reasonable Model for 2D Images
Definition (Donoho; 2001)
The set of cartoon-like 2D images E2 (R2 ) is defined by
                                   2


E2 (R2 ) = {f ∈ L2 (R2 ) : f = f0 +f1 χB } fi ∈ C 2 (R2 ), supp fi ⊂ [0, 1]2 ,
 2


where B ⊂ [0, 1]2 and the boundary curve ∂B is a closed C 2 -curve
with curvature bounded by ν.


Theorem (Donoho; 2001)
Let (ψλ )λ ⊂ L2 (R2 ). The optimal asymptotic approximation error
of f ∈ E2 (R2 ) is
         2


                       2
            f − fN     2      N −2 ,      N → ∞,        where fN =          cλ ψλ .
                                                                     λ∈IN

7/42 DTU Mathematics, Technical University of Denmark            Shearlet Frames      26.3.2012
Approaches for 2D
Non-exhaustive list for 2D data:
• Directional wavelets (Antoine, Murenzi, Vandergheynst; 1999)
• Curvelets (Candès & Donoho; 2002)
• Contourlets (Do & Vetterli; 2002)
• Bandlets (LePennec & Mallat; 2003)
• Shearlets (Guo, Kutyniok, Labate, Lim & Weiss; 2006)

Known results (band-limited generators):
Curvelets, Contourlets, and Shearlets provide (almost) optimally
sparse expansions for 2D cartoon-like images f ∈ E2 (R2 ) with
                                                   2

asymptotic error
                               2
                    f − fN     2   = O(N −2 (log N)3 ),   N → ∞.


8/42 DTU Mathematics, Technical University of Denmark      Shearlet Frames   26.3.2012
Approaches for 2D
Known results (band-limited generators):
Curvelets, Contourlets, and Shearlets provide (almost) optimally
sparse expansions for 2D cartoon-like images f ∈ E2 (R2 ) with
                                                   2

asymptotic error
                               2
                    f − fN     2   = O(N −2 (log N)3 ),        N → ∞.
Success of 2D shearlets: Kutyniok and Lim (2011) proved that this
asymptotic rate can be achieved with compactly supported
shearlets.
Intuitive explanation (shearlets vs. wavelets):




                N −2
8/42 DTU Mathematics, Technical University of Denmark
                                                        N −1    Shearlet Frames   26.3.2012
Reasonable Model for 3D Images
Definition
The set of 3D images E2 (R3 ) is defined by
                      2


E2 (R3 ) = {f ∈ L2 (R3 ) : f = f0 +f1 χB } fi ∈ C 2 (R3 ), supp fi ⊂ [0, 1]3 ,
 2


where B ⊂ [0, 1]3 and the boundary surface ∂B is a closed
C 2 -surface for which the principal curvatures are bounded by ν.



                                     f0=0




9/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
Reasonable Model for 3D Images
Definition
The set of 3D images E2 (R3 ) is defined by
                      2


E2 (R3 ) = {f ∈ L2 (R3 ) : f = f0 +f1 χB } fi ∈ C 2 (R3 ), supp fi ⊂ [0, 1]3 ,
 2


where B ⊂ [0, 1]3 and the boundary surface ∂B is a closed
C 2 -surface for which the principal curvatures are bounded by ν.


Theorem (Kutyniok, L, Lim; 2011)
Let (ψλ )λ ⊂ L2 (R3 ). The optimal asymptotic approximation error
of f ∈ E2 (R3 ) is
         2


                       2
            f − fN     2      N −1 ,      N → ∞,        where fN =          cλ ψλ .
                                                                     λ∈IN

9/42 DTU Mathematics, Technical University of Denmark            Shearlet Frames      26.3.2012
Most Approaches are for 2D Data


Question:

                         Why is the 3D situation so crucial?

Answer:
• Our world is 3-dimensional.
    • 3D data is essential for Biology, Seismology,...
• Transition 2D ❀ 3D is special.
    • In 3D anisotropic features occur for the first time in different
      dimensions.




10/42 DTU Mathematics, Technical University of Denmark    Shearlet Frames   26.3.2012
Success of 1D Wavelets


• The wavelet system generated by ψ ∈ L2 (R) is

                         ψj,k = 2j/2 ψ(2j · −m) : j ∈ Z, m ∈ Z .

• Wavelets provide sparse expansions for ‘1D cartoon images’ (C 2
   apart from point discontinuities) with asymptotic error
                                         2
                             f − fN      2   = O(N −4 ),   N → ∞.

• In comparison, Fourier series only gives O(N −1 ).
❀ Wavelets are very good at detecting singularities in 1D!




 11/42 DTU Mathematics, Technical University of Denmark        Shearlet Frames   26.3.2012
Goal for Today

Design a representation system which...
• ...forms a orthonormal basis frame for L2 (R3 ),
• ...captures anisotropic structures, e.g., singularities concentrated
  on lower dimensional manifolds,
• ...provides optimally sparse approximation of cartoon images,
• ...allows for compactly supported analyzing elements,
• ...is associated with fast decomposition algorithms,
• ...treats the continuum and digital ’world’ uniformly.

                Vision: Introduce a system for 3D data that is
                   as powerful as wavelets are for 1D data!



12/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
Outline

1   Applied Harmonic Analysis and Imaging Sciences

2   Fourier and Wavelet Analysis
      1D Analysis: Fourier Series
      1D Analysis: Wavelets
      Problem with Wavelets for 2D and 3D Data

3   Shearlet Theory

4   Sparse Approximation using 3D Shearlets

5   Conclusion


13/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
Fourier Series
• {e2πinx }n∈Z is an ONB for L2 ([0, 1]): f =             n cn   e2πinx
• Modulation: En f (x) := e2πinx f (x), f ∈ L2 (R).
• Translation: Tm f (x) := f (x − m), f ∈ L2 (R).
• {En Tm χ[0,1] }n,m∈Z ONB for L2 (R): f =               n,m cn,m    En Tm χ[0,1]




14/42 DTU Mathematics, Technical University of Denmark           Shearlet Frames   26.3.2012
Fourier Series
• {e2πinx }n∈Z is an ONB for L2 ([0, 1]): f =                        n cn   e2πinx
• Modulation: En f (x) := e2πinx f (x), f ∈ L2 (R).
• Translation: Tm f (x) := f (x − m), f ∈ L2 (R).
• {En Tm χ[0,1] }n,m∈Z ONB for L2 (R): f =                          n,m cn,m    En Tm χ[0,1]
Simple 1D cartoon-like image (0 < x0 < x1 < 1):

                                               1     x ∈ [x0 , x1 ] ,
                                    f (x) =
                                               0     otherwise

Fourier Coefficients:
                                1                              x1                       1
             |cn,0 | =              f (x) e−2πinx dx =              e−2πinx dx ∼
                            0                                 x0                        n


14/42 DTU Mathematics, Technical University of Denmark                      Shearlet Frames   26.3.2012
1D Fourier Series Approximation
• Parseval’s Equality:

                           |cn,m |2 =              | f , En Tm χ0,1 |2 = f    2
                                                                              2
                  n,m∈Z                   n,m∈Z




15/42 DTU Mathematics, Technical University of Denmark              Shearlet Frames   26.3.2012
1D Fourier Series Approximation
• Parseval’s Equality:

                           |cn,m |2 =              | f , En Tm χ0,1 |2 = f    2
                                                                              2
                  n,m∈Z                   n,m∈Z

• Best N-term approximation:

                    fN =                 cn,m En Tm χ[0,1] ,   # |IN | = N
                             (n,m)∈IN




15/42 DTU Mathematics, Technical University of Denmark              Shearlet Frames   26.3.2012
1D Fourier Series Approximation
• Parseval’s Equality:

                           |cn,m |2 =              | f , En Tm χ0,1 |2 = f          2
                                                                                    2
                  n,m∈Z                   n,m∈Z

• Best N-term approximation:

                    fN =                 cn,m En Tm χ[0,1] ,     # |IN | = N
                             (n,m)∈IN

• By Parseval’s Equality:
                                                         ∞            ∞
                       2                                       1             1       1
            f − fN     2   =              |cn,m |2 ∼              ∼             dx ∼
                               (n,m)∈IN
                                    /                    n=N
                                                               n2     N      x2      N




15/42 DTU Mathematics, Technical University of Denmark                    Shearlet Frames   26.3.2012
1D Fourier Series Approximation
• Parseval’s Equality:

                           |cn,m |2 =              | f , En Tm χ0,1 |2 = f              2
                                                                                        2
                  n,m∈Z                   n,m∈Z

• Best N-term approximation:

                    fN =                 cn,m En Tm χ[0,1] ,          # |IN | = N
                             (n,m)∈IN

• By Parseval’s Equality:
                                                              ∞           ∞
                       2                                           1             1       1
            f − fN     2   =              |cn,m |2 ∼                  ∼             dx ∼
                               (n,m)∈IN
                                    /                        n=N
                                                                   n2     N      x2      N

• Conclusion:
                                                         2
                                          f − fN         2   ∼ N −1

15/42 DTU Mathematics, Technical University of Denmark                        Shearlet Frames   26.3.2012
1D Wavelet Approximations

• The wavelet system generated by ψ ∈ L2 (R) is

                ψj,m = Dj Tm ψ = 2j/2 ψ(2j · −m) : j ∈ Z, m ∈ Z .

• ONB e.g., Haar Wavelet:
                                              
                                              1     1
                                           x ∈ [0, 2 ),
                                              
                                              
                                 ψ(x) = −1 x ∈ [ 1 , 1),
                                                 2
                                       
                                       0
                                       
                                           otherwise.

• Haar Scaling function Tm φ, m ∈ Z (replaces ψj,m , j < 0, m ∈ Z):
  φ(x) = χ[0,1] (x)



16/42 DTU Mathematics, Technical University of Denmark     Shearlet Frames   26.3.2012
1D Wavelet Approximations
• At fixed scale j there are exactly two non-zero wavelet coefficients!

                                  |cj,k | = | f , ψj,k | ∼ 2−j/2




17/42 DTU Mathematics, Technical University of Denmark             Shearlet Frames   26.3.2012
1D Wavelet Approximations
• At fixed scale j there are exactly two non-zero wavelet coefficients!

                                  |cj,k | = | f , ψj,k | ∼ 2−j/2

• Best N-term Approximation:

                                                                N
                      f − fN      2
                                  2   =              |cj,k |2         2−j ∼ 2−N
                                          (j,k)∈IN
                                               /                j=1




17/42 DTU Mathematics, Technical University of Denmark                   Shearlet Frames   26.3.2012
1D Wavelet Approximations
• At fixed scale j there are exactly two non-zero wavelet coefficients!

                                  |cj,k | = | f , ψj,k | ∼ 2−j/2

• Best N-term Approximation:

                                                                N
                      f − fN      2
                                  2   =              |cj,k |2         2−j ∼ 2−N
                                          (j,k)∈IN
                                               /                j=1


• For piecewise C 2 functions (1D cartoon-like images):

                                          2
                            f − fN        2   = O(N −4 ),        N → ∞.




17/42 DTU Mathematics, Technical University of Denmark                   Shearlet Frames   26.3.2012
1D Wavelet Approximations
• At fixed scale j there are exactly two non-zero wavelet coefficients!

                                  |cj,k | = | f , ψj,k | ∼ 2−j/2

• Best N-term Approximation:

                                                                N
                      f − fN      2
                                  2   =              |cj,k |2         2−j ∼ 2−N
                                          (j,k)∈IN
                                               /                j=1


• For piecewise C 2 functions (1D cartoon-like images):

                                          2
                            f − fN        2   = O(N −4 ),        N → ∞.

• In comparison, Fourier series only gave O(N −1 ).


17/42 DTU Mathematics, Technical University of Denmark                   Shearlet Frames   26.3.2012
1D Wavelet Approximations
• At fixed scale j there are exactly two non-zero wavelet coefficients!

                                   |cj,k | = | f , ψj,k | ∼ 2−j/2

• Best N-term Approximation:

                                                                 N
                       f − fN      2
                                   2   =              |cj,k |2         2−j ∼ 2−N
                                           (j,k)∈IN
                                                /                j=1


• For piecewise C 2 functions (1D cartoon-like images):

                                           2
                             f − fN        2   = O(N −4 ),        N → ∞.

• In comparison, Fourier series only gave O(N −1 ).
❀ Wavelets are very good at detecting singularities in 1D!

 17/42 DTU Mathematics, Technical University of Denmark                   Shearlet Frames   26.3.2012
Why Not nD Wavelets?
• Isotropic scaling (same scaling in all directions):
                                            
                                                  2j      0    0
                                       Aj =  0           2j   0.
                                                                 
                                                   0      0    2j

• The wavelet system generated by ψ ∈ L2 (R3 ) is

                       ψj,m = 23j/2 ψ(Aj · −m) : j ∈ Z, m ∈ Z3 .

• Wavelets do not provide optimal sparse expansions for 3D cartoon
   images. For f ∈ E2 (R3 ) wavelets only provide an aymptotic error
                     2

   of:
                    f − fN 2 ∼ N −1/2 ,
                            2                N → ∞.
❀ Wavelets are not good at effectively capturing 2D singularities in
  3D: Far from the optimal rate N −1 !
 18/42 DTU Mathematics, Technical University of Denmark               Shearlet Frames   26.3.2012
Outline

1   Applied Harmonic Analysis and Imaging Sciences

2   Fourier and Wavelet Analysis

3   Shearlet Theory
      Discrete Shearlet Systems
      Pyramid-adapted Systems

4   Sparse Approximation using 3D Shearlets

5   Conclusion



19/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
Introducing 2D Shearlets

• Anisotropic scaling Aj :

                                                   2j      0
                                        Aj =                     ,
                                                   0      2j/2

• Shearing Sk (direction parameter ↔ rotations):

                                                         1 k
                                           Sk =
                                                         0 1

• The shearlet system generated by ψ ∈ L2 (R2 ) is

    ψj,k,m = DSk Aj Tm ψ = 23j/4 ψ(Sk A2j · −m) : j ∈ Z, k ∈ Z, m ∈ Z2


20/42 DTU Mathematics, Technical University of Denmark               Shearlet Frames   26.3.2012
Action of Anisotropic Scaling and Shearing in 2D




ψj,k,m for j = 0, k = 0, m = (0, 0)


21/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
Action of Anisotropic Scaling and Shearing in 2D




ψj,k,m for j = 1, k = 0, m = (0, 0)


21/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
Action of Anisotropic Scaling and Shearing in 2D




ψj,k,m for j = 2, k = 0, m = (0, 0)


21/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
Action of Anisotropic Scaling and Shearing in 2D




ψj,k,m for j = 1, k = 0, m = (0, 0)


21/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
Action of Anisotropic Scaling and Shearing in 2D




ψj,k,m for j = 1, k = 0, m = (1, −1)


21/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
Action of Anisotropic Scaling and Shearing in 2D




ψj,k,m for j = 1, k = 0, m = (0, 0)


21/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
Action of Anisotropic Scaling and Shearing in 2D




ψj,k,m for j = 1, k = −1, m = (0, 0)


21/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
Action of Anisotropic Scaling and Shearing in 2D




ψj,k,m for j = 1, k = −2, m = (0, 0)


21/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
Action of Anisotropic Scaling and Shearing in 2D




ψj,k,m for j = 1, k = −3, m = (0, 0)


21/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
Introducing 3D Shearlets
• Anisotropic scaling Aj :
                                                              
                                          2j             0   0
                                   Aj =  0          2j/2    0 ,
                                                              
                                          0           0 2j/2

• Shearing Sk , k = (k1 , k2 ) (direction parameter ↔ rotations):
                                                            
                                             1 k1 k2
                                       Sk = 0 1 0 
                                                    
                                             0 0 1

• The shearlet system generated by ψ ∈ L2 (R3 ) is

               ψj,k,m = 2j ψ(Sk A2j · −m) : j ∈ Z, k ∈ Z2 , m ∈ Z3

22/42 DTU Mathematics, Technical University of Denmark              Shearlet Frames   26.3.2012
Introducing 3D Shearlets
• Anisotropic scaling Aj (α ∈ (1, 2]):
                                                            
                                        2jα/2  0   0
                                 Aj =  0     2j/2 0 ,
                                                     
                                          0    0 2j/2

• Shearing Sk , k = (k1 , k2 ) (direction parameter ↔ rotations):
                                                        
                                             1 k1 k2
                                       Sk = 0 1 0 
                                                    
                                             0 0 1

• The shearlet system generated by ψ ∈ L2 (R3 ) is
                                  1
           ψj,k,m = 2j( 4 + 2 ) ψ(Sk A2j · −m) : j ∈ Z, k ∈ Z2 , m ∈ Z3
                             α




22/42 DTU Mathematics, Technical University of Denmark           Shearlet Frames   26.3.2012
Compactly Supported Generators
Idea: We want generators of the form
                            ψ(x1 , x2 , x3 ) = η(x1 )φ(x2 )φ(x3 )
for the shearlets system associated with the pyramids P1 , where η is
a 1D wavelet and φ is a scaling/bump function. The wavelet η will
then point in the ‘short’ direction of the plate-like elements.

Class of generators: We consider generators with weak directional
vanishing moments and essential support (in Frequency domain) as:




23/42 DTU Mathematics, Technical University of Denmark              Shearlet Frames   26.3.2012
Example: 3D Action of Anisotropic scaling and
Shearing

Example: Suppose supp ψ = [0, 1]3 . Let the translation parameter
m = (0, 0, 0) be fixed.

        Scaling For k = (0, 0):
                supp ψj,0,0 = [0, 2−j ] × [0, 2−j/2 ] × [0, 2−j/2 ] → the
                shearlet becomes a small, plate-like element as j → ∞.




24/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
Example: 3D Action of Anisotropic scaling and
Shearing

Example: Suppose supp ψ = [0, 1]3 . Let the translation parameter
m = (0, 0, 0) be fixed.

        Scaling For k = (0, 0):
                supp ψj,0,0 = [0, 2−j ] × [0, 2−j/2 ] × [0, 2−j/2 ] → the
                shearlet becomes a small, plate-like element as j → ∞.
      Shearing For j > 0: ψ(Sk A2j x) =
               ψ(2j x1 + k1 2j/2 x2 + k2 2j/2 x3 , 2j/2 x2 , 2j/2 x3 )




24/42 DTU Mathematics, Technical University of Denmark       Shearlet Frames   26.3.2012
Example: 3D Action of Anisotropic scaling and
Shearing

Example: Suppose supp ψ = [0, 1]3 . Let the translation parameter
m = (0, 0, 0) be fixed.

        Scaling For k = (0, 0):
                supp ψj,0,0 = [0, 2−j ] × [0, 2−j/2 ] × [0, 2−j/2 ] → the
                shearlet becomes a small, plate-like element as j → ∞.
      Shearing For j > 0: ψ(Sk A2j x) =
               ψ(2j x1 + k1 2j/2 x2 + k2 2j/2 x3 , 2j/2 x2 , 2j/2 x3 )
      Problem To get horizontal plate-elements require very large
              shear parameters |ki | = ∞ ❀ non-uniform treatment
              of directions.



24/42 DTU Mathematics, Technical University of Denmark       Shearlet Frames   26.3.2012
3D Action of Anisotropic Scaling and Shearing
Spatial Domain:




Frequency Domain:




25/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
Pyramid-adapted Shearlet Systems
Idea: We partition the frequency space R3 into 3 pairs of pyramids:
    
     {(ξ1 , ξ2 , ξ3 ) ∈ R3 : |ξ1 | ≥ 1, |ξ2 /ξ1 | ≤ 1, |ξ3 /ξ1 | ≤ 1} : ι = 1,
Pι = {(ξ1 , ξ2 , ξ3 ) ∈ R3 : |ξ2 | ≥ 1, |ξ1 /ξ2 | ≤ 1, |ξ3 /ξ2 | ≤ 1} : ι = 2,
      {(ξ1 , ξ2 , ξ3 ) ∈ R3 : |ξ3 | ≥ 1, |ξ1 /ξ3 | ≤ 1, |ξ2 /ξ3 | ≤ 1} : ι = 3,
    

and a centered cube

                  C ={(ξ1 , ξ2 , ξ3 ) ∈ R3 : (ξ1 , ξ2 , ξ3 )   ∞   < 1}.


                                                                                     P3
              C                                 P1
015
                                                         P2
                        016


        017


❀ For each pair of pyramids we construct a shearlet frame.
26/42 DTU Mathematics, Technical University of Denmark             Shearlet Frames    26.3.2012
Pyramid-adapted Shearlet Systems


                                               ˜ ˘
The (pyramid-adapted) shearlet system SH(φ, ψ, ψ, ψ) generated
         ˜ ψ ∈ L2 (R3 ) is the union of
by φ, ψ, ψ, ˘

            {φm = φ(· − m) : m ∈ Z3 },
        {ψj,k,m = 2j ψ(Sk A2j · −m) : j ≥ 0, |k| ≤ ⌈2j/2 ⌉, m ∈ Z3 },
         ˜           ˜ ˜ ˜
        {ψj,k,m = 2j ψ(Sk A2j · −m) : j ≥ 0, |k| ≤ ⌈2j/2 ⌉, m ∈ Z3 },
         ˘           ˘ ˘ ˘
        {ψj,k,m = 2j ψ(Sk A2j · −m) : j ≥ 0, |k| ≤ ⌈2j/2 ⌉, m ∈ Z3 },

                                                              ˜
where j ∈ N0 and k ∈ Z2 . We call φ a scaling function and ψ, ψ,
    ˘ shearlets.
and ψ




27/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
Pyramid-adapted Shearlet Systems

                                               ˜ ˘
The (pyramid-adapted) shearlet system SH(φ, ψ, ψ, ψ; α) generated
         ˜ ψ ∈ L2 (R3 ) is the union of
by φ, ψ, ψ, ˘

    {φm = φ(· − m) : m ∈ Z3 },
                    α    1
{ψj,k,m = 2j( 4 + 2 ) ψ(Sk A2j · −m) : j ≥ 0, |k| ≤ ⌈2(α−1)j/2 ⌉, m ∈ Z3 },
                    α    1
{ψj,k,m = 2j( 4 + 2 ) ψ(Sk A2j · −m) : j ≥ 0, |k| ≤ ⌈2(α−1)j/2 ⌉, m ∈ Z3 },
 ˜                    ˜ ˜ ˜
                    α    1
 ˘                    ˘ ˘ ˘
{ψj,k,m = 2j( 4 + 2 ) ψ(Sk A2j · −m) : j ≥ 0, |k| ≤ ⌈2(α−1)j/2 ⌉, m ∈ Z3 },

                                                              ˜
where j ∈ N0 and k ∈ Z2 . We call φ a scaling function and ψ, ψ,
    ˘ shearlets.
and ψ



27/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
Outline

1   Applied Harmonic Analysis and Imaging Sciences

2   Fourier and Wavelet Analysis

3   Shearlet Theory

4   Sparse Approximation using 3D Shearlets
      Optimally Theorem
      Sketch of proof

5   Conclusion



28/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
Revisiting 3D Cartoon-like Images
Definition
                                     2
Let 1 < α ≤ 2. The set of 3D images Eα (R3 ) is defined by

Eα (R3 ) = {f ∈ L2 (R3 ) : f = f0 +f1 χB } fi ∈ C 2 (R3 ), supp fi ⊂ [0, 1]3 ,
 2


where B ⊂ [0, 1]3 and the boundary surface ∂B is a closed
C α -surface with ‘curvature’ bounded by ν.

                                                                            ❇➯




                                        ✶




                                     ✺✼ ✳✵




                                     ✵✺ ✳✵




                                     ✺✷✳✵


                                                                                                  ✶


                                                                                           ✺✼✳✵
                                        ✵

                                        ✵                                          ✵✺ ✳✵

                                             ✺✷ ✳✵
                                                     ✵✺✳✵                   ✺✷✳✵

                                                            ✺✼ ✳✵
                                                                        ✵
                                                                    ✶




29/42 DTU Mathematics, Technical University of Denmark                                                Shearlet Frames   26.3.2012
Revisiting 3D Cartoon-like Images
Definition
Let 1 < α ≤ 2. The set of 3D images Eα (R3 ) is defined by
                                     2


Eα (R3 ) = {f ∈ L2 (R3 ) : f = f0 +f1 χB } fi ∈ C 2 (R3 ), supp fi ⊂ [0, 1]3 ,
 2


where B ⊂ [0, 1]3 and the boundary surface ∂B is a closed
C α -surface with ‘curvature’ bounded by ν.


Theorem (Kutyniok, L, Lim; 2011)
Let (ψλ )λ ⊂ L2 (R3 ). The optimal asymptotic approximation error
of f ∈ Eα (R3 ) is
         2


                      2
          f − fN      2      N −α/2 ,       N → ∞,       where fN =          cλ ψλ .
                                                                      λ∈IN

29/42 DTU Mathematics, Technical University of Denmark           Shearlet Frames       26.3.2012
Main result: Optimal Sparsity of 3D shearlet
Theorem (Kutyniok, L, Lim; 2011)
                          ˜ ˘
Fix α ∈ (1, 2]. Let φ, ψ, ψ, ψ ∈ L2 (R3 ) be compactly supported.
Assume:
       ˆ
  (i) |ψ(ξ)| min(1, |ξ1 |δ ) · 3 min(1, |ξi |−γ )
                                              i=1
         ∂ ˆ                                    |ξ2 | −γ        |ξ3 | −γ
 (ii)   ∂ξi ψ(ξ)       ≤ |h(ξ1 )| · 1 +         |ξ1 |      1+   |ξ1 |      ,      i = 2, 3,
                                                ˜     ˘
where δ > 8, γ ≥ 4, h ∈ L1 (R), and similar for ψ and ψ. Further,
                       ˜ ψ; α) forms a frame for L2 (R3 ). For
suppose that SH(φ, ψ, ψ, ˘
f ∈ Eα (R3 ),
     2



                   2               O(N −α/2+µ ),           if α < 2,
        f − fN     2   =             −1             2
                                                                               as N → ∞,
                             O(N          (log N) ),       if α = 2,
 where                                           3(2−α)(α−1)(α+2)
                             µ = µ(α) =            2(9α2 +17α−10)
                                                                  ,

30/42 DTU Mathematics, Technical University of Denmark                 Shearlet Frames   26.3.2012
Optimality Gap of Main result
If α < 2, we have for f ∈ Eα (R3 ),
                           2

                                2
                    f − fN      2   = O(N −α/2+µ ),            as N → ∞,

where
• 0 < µ(α) < 0.037 for α (1, 2),
• µ(α) → 0 for α → 1+ or α → 2− .

            1
          0.9
          0.8
          0.7
          0.6
          0.5
                1            1.2            1.4          1.6      1.8                2
                                                     α

❀ Shearlets provide nearly optimally sparse approximations of
Eα (R3 ).
 2
31/42 DTU Mathematics, Technical University of Denmark             Shearlet Frames       26.3.2012
Sketch of proof: Frames and Non-linear
approximations
Definition
A sequence (ψλ )λ ⊂ L2 (Rd ) is a frame for L2 (Rd ) if
                             2
∃A, B > 0 : A f                  ≤       | f , ψλ |2 ≤ B f   2
                                                                  for all f ∈ L2 (Rd ).
                                     λ

                                                             ˜
Fact from frame theory: There exists a canonical dual frame {ψλ }λ
such that
              f =            ˜
                      f , ψλ ψλ for all f ∈ L2 (Rd ).
                                 λ




32/42 DTU Mathematics, Technical University of Denmark           Shearlet Frames   26.3.2012
Sketch of proof: Frames and Non-linear
approximations
Definition
A sequence (ψλ )λ ⊂ L2 (Rd ) is a frame for L2 (Rd ) if
                             2
∃A, B > 0 : A f                  ≤         | f , ψλ |2 ≤ B f       2
                                                                        for all f ∈ L2 (Rd ).
                                       λ

                                                             ˜
Fact from frame theory: There exists a canonical dual frame {ψλ }λ
such that
              f =            ˜
                      f , ψλ ψλ for all f ∈ L2 (Rd ).
                                 λ
As N-term approximation (not necessarily ‘best’) we take:

                                       fN =                 ˜
                                                     f , ψλ ψλ ,
                                              λ∈IN

where (| f , ψ |)λ      λ∈IN         are the N largest coefficients.
32/42 DTU Mathematics, Technical University of Denmark                 Shearlet Frames   26.3.2012
Sketch of proof: Outline

                                                        ˜
• Estimates of f , ψj,k,m for f ∈ Eα (R3 ). Similar for ψj,k,m and
                                   2
  ˘
  ψj,k,m .
• WLOG assume j ≥ 0 is sufficiently large.
• Let |θ(f )|n denote the nth largest shearlet coefficient | f , ψj,k,m |:

                                              2       1
                                  f − fN      2   ≤         |θ(f )|2 .
                                                                   n
                                                      A n>N

• Hence, it suffices to prove:

                              |θ(f )|2 ≤ C N −α/2
                                     n                         as N → ∞.
                       n>N




33/42 DTU Mathematics, Technical University of Denmark                   Shearlet Frames   26.3.2012
Sketch of proof: Heuristic argument of N −α/2
Recall: The shearlet is of the form ψ(x) = η(x1 )ϕ(x2 )ϕ(x3 ), where
η is a wavelet and ϕ a bump (or a scaling) function.
We consider three cases of coefficients f , ψj,k,m :
(a) Shearlets whose support do not overlap with the boundary ∂B.
(b) Shearlets whose support overlap with ∂B and are nearly tangent.
(c) Shearlets whose support overlap with ∂B, but tangentially.




        (a)                               (b)            (c)


34/42 DTU Mathematics, Technical University of Denmark    Shearlet Frames   26.3.2012
Sketch of proof: Heuristic argument of N −α/2

                     Case (b):




• At scale j > 0: At most O(2j ) coefficients since the plate-like
  element is of size 2−j/2 × 2−j/2 (and “thickness” 2−jα/2 ).
  Moreover,
                                                                       α 1                    α    1
                                                                   −j( 4 + 2 )
| f , ψj,k,m | ≤ f      ∞       ψj,k,m     1   ≤ f       ∞   ψ   12             ≤C       2−j( 4 + 2 )
                                                                              1
                                                                         −j( 2 + α )
  ❀ we have          O(2j )    coefficients bounded by C · 2                       4   .




35/42 DTU Mathematics, Technical University of Denmark                     Shearlet Frames     26.3.2012
Sketch of proof: Heuristic argument of N −α/2

                     Case (b):




• At scale j > 0: At most O(2j ) coefficients since the plate-like
  element is of size 2−j/2 × 2−j/2 (and “thickness” 2−jα/2 ).
  Moreover,
                                                                       α 1                       α   1
                                                                   −j( 4 + 2 )
| f , ψj,k,m | ≤ f      ∞       ψj,k,m     1   ≤ f       ∞   ψ   12             ≤C       2−j( 4 + 2 )
                                                                              1
                                                                         −j( 2 + α )
  ❀ we have          O(2j )    coefficients bounded by C · 2                       4   .
• Assuming, case (a) and (c) coefficients are negligible, the nth
                                                                                         1
  largest coefficient |θ(f )|n is bounded by |θ(f )|n ≤ C n−( 2 + 4 ) :
                                                                                             α



                                |θ(f )|2
                                       n                 n−1−α/2      N −α/2 .
                         n>N                   n>N
35/42 DTU Mathematics, Technical University of Denmark                     Shearlet Frames       26.3.2012
Heuristic argument of N −1/2 for Wavelets

                     Wavelets:




• At scale j > 0: At most O(2jα ) coefficients since the cube-like
  element is of size 2−jα/2 × 2−jα/2 (and “thickness” 2−jα/2 ).
  Moreover,
                                                                     3α                  3
                                                                   −j 4
  | f , ψj,k,m | ≤ f           ∞    ψj,k,m     1   ≤ f   ∞   ψ   12      ≤C       2jα(− 4 )
                                                                      jα(− 3 )
  ❀ we have O(2jα ) coefficients bounded by C · 2          .                 4

• Assuming, case (a) and (c) coefficients are negligible, the nth
                                                           3
  largest coefficient |θ(f )|n is bounded by |θ(f )|n ≤ C n− 4 :
                               |θ(f )|2 ≤
                                      n              n−3/2 ≤ C · N −1/2 .
                        n>N                   n>N
36/42 DTU Mathematics, Technical University of Denmark                 Shearlet Frames   26.3.2012
Generalization: Optimal Sparsity of 3D shearlets
Question:
• The above result shows that the ‘plate-like’ shearlet system is
  optimal for describing 2D singularities. But not all singularities in
  3D images are located on C α -smooth surfaces!




37/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
Generalization: Optimal Sparsity of 3D shearlets
Question:
• The above result shows that the ‘plate-like’ shearlet system is
  optimal for describing 2D singularities. But not all singularities in
  3D images are located on C α -smooth surfaces!
• ❀ We would also like to consider point, curve and surface
  singularities:


                                                 B
                                f0=0
                                                         f1=1




37/42 DTU Mathematics, Technical University of Denmark          Shearlet Frames   26.3.2012
Generalization: Optimal Sparsity of 3D shearlets


Theorem (Kutyniok, L, Lim; 2011)
The sparsity result extends to a class of cartoon-like 3D images with
only piecewise smooth C α boundary ∂B.




38/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
Generalization: Optimal Sparsity of 3D shearlets


Theorem (Kutyniok, L, Lim; 2011)
The sparsity result extends to a class of cartoon-like 3D images with
only piecewise smooth C α boundary ∂B.

Observations:
1   The optimal rate is still the same when introducing “corner
    points” and “curves”
2   Pyramid-adapted ‘plate-like’ shearlets still achieve the optimal
    rate.




38/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
Outline


1   Applied Harmonic Analysis and Imaging Sciences

2   Fourier and Wavelet Analysis

3   Shearlet Theory

4   Sparse Approximation using 3D Shearlets

5   Conclusion




39/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
What To Take Home?


• Anisotropic features in multivariate data require special efficient
  encoding strategies
• The shearlet theory is perfectly suited to this problem
• One main advantage of shearlets is that they provide a unified
  treatment of the continuum and digital setting.
• The Shearlet Theory for 2D and 3D data provides:
    • Compactly supported shearlet frames
    • Explicit estimates for frame bounds
    • Optimal sparse approximation of cartoon-like images.




40/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
Thank You!


                 Preprints: http://www.shearlets.org
              http://www2.mat.dtu.dk/people/J.Lemvig/




                     Software: http://www.shearlab.org




41/42 DTU Mathematics, Technical University of Denmark   Shearlet Frames   26.3.2012
References
     D. L. Donoho, Sparse components of images and optimal atomic
     decomposition, Constr. Approx. 17 (2001), 353–382.
     E. J. Candés and D. L. Donoho, New tight frames of curvelets and optimal
     representations of objects with piecewise C 2 singularities, Comm. Pure and
     Appl. Math. 56 (2004), 216–266.
     K. Guo and D. Labate, Optimally sparse multidimensional representation
     using shearlets, SIAM J. Math Anal. 39 (2007), 298–318.
     M. N. Do and M. Vetterli, The contourlet transform: an efficient directional
     multiresolution image representation, IEEE Trans. Image Process. 14
     (2005), 2091–2106.
     G. Kutyniok and W.-Q Lim, Compactly supported shearlets are optimally
     sparse, J. Approx. Theory 163 (2011), 1564–1589.
     S. Dahlke, G. Steidl, and G. Teschke, The continuous shearlet transform in
     arbitrary space dimensions, J. Fourier Anal. Appl. 16 (2010), 340–364.
     G. Kutyniok, J. Lemvig, and W.-Q Lim, Compactly supported shearlets, in
     Approximation Theory XIII (San Antonio, TX, 2011), Springer.
42/42 DTU Mathematics, Technical University of Denmark     Shearlet Frames   26.3.2012

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Shearlet Frames and Optimally Sparse Approximations

  • 1. Shearlet Frames and Optimally Sparse Approximations Jakob Lemvig — Technical University of Denmark (DTU) joint work with Gitta Kutyniok & Wang-Q Lim (TU Berlin)
  • 2. Outline 1 Applied Harmonic Analysis and Imaging Sciences 2 Fourier and Wavelet Analysis 3 Shearlet Theory 4 Sparse Approximation using 3D Shearlets 5 Conclusion 2/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 3. Outline 1 Applied Harmonic Analysis and Imaging Sciences Anisotropic Phenomena Image Model Goal for Today 2 Fourier and Wavelet Analysis 3 Shearlet Theory 4 Sparse Approximation using 3D Shearlets 5 Conclusion 3/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 4. Anisotropic Phenomena in Multivariate Data Many important multivariate problem classes are governed by anisotropic features, which require efficient encoding strategies. The anisotropic structure can be given. . . • . . . explicitly • Image Processing: Edges • Seismology: Earth layers • • . . . implicitely • PDEs: Shock waves, Boundary layers 4/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 5. Modern Imaging Some important tasks: • Denoising • Inpainting • Feature Detection/Extraction • ... Imaging Sciences using Applied Harmonic Analysis: Exploit a carefully designed representation system (ψλ )λ ⊂ L2 (Rd ) : Image = Image, ψλ ψλ . λ • Sparse coefficients • Approximation properties 5/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 6. Geometric Structures of Multidimensional Images • What are anisotropic features in a 2D Image? Curves Objects Cartoon- with like ‘smooth’ image boundaries 6/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 7. Geometric Structures of Multidimensional Images • What are anisotropic features in a 2D Image? Curves Objects Cartoon- with like ‘smooth’ image boundaries • What are anisotropic features in 3D data? Curves & surfaces Video: objects carve out 3D medical spatial- scans: temporal tubes 6/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 8. Geometric Structures of Multidimensional Images • What are anisotropic features in a 2D Image? Curves Objects Cartoon- with like ‘smooth’ image boundaries • What are anisotropic features in 3D data? Curves & surfaces Video: objects carve out 3D medical spatial- scans: temporal tubes • Most information is contained in lower dimensional structures! 6/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 9. Reasonable Model for 2D Images Definition (Donoho; 2001) The set of cartoon-like 2D images E2 (R2 ) is defined by 2 E2 (R2 ) = {f ∈ L2 (R2 ) : f = f0 +f1 χB } fi ∈ C 2 (R2 ), supp fi ⊂ [0, 1]2 , 2 where B ⊂ [0, 1]2 and the boundary curve ∂B is a closed C 2 -curve with curvature bounded by ν. f0=0 7/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 10. Reasonable Model for 2D Images Definition (Donoho; 2001) The set of cartoon-like 2D images E2 (R2 ) is defined by 2 E2 (R2 ) = {f ∈ L2 (R2 ) : f = f0 +f1 χB } fi ∈ C 2 (R2 ), supp fi ⊂ [0, 1]2 , 2 where B ⊂ [0, 1]2 and the boundary curve ∂B is a closed C 2 -curve with curvature bounded by ν. Theorem (Donoho; 2001) Let (ψλ )λ ⊂ L2 (R2 ). The optimal asymptotic approximation error of f ∈ E2 (R2 ) is 2 2 f − fN 2 N −2 , N → ∞, where fN = cλ ψλ . λ∈IN 7/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 11. Approaches for 2D Non-exhaustive list for 2D data: • Directional wavelets (Antoine, Murenzi, Vandergheynst; 1999) • Curvelets (Candès & Donoho; 2002) • Contourlets (Do & Vetterli; 2002) • Bandlets (LePennec & Mallat; 2003) • Shearlets (Guo, Kutyniok, Labate, Lim & Weiss; 2006) Known results (band-limited generators): Curvelets, Contourlets, and Shearlets provide (almost) optimally sparse expansions for 2D cartoon-like images f ∈ E2 (R2 ) with 2 asymptotic error 2 f − fN 2 = O(N −2 (log N)3 ), N → ∞. 8/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 12. Approaches for 2D Known results (band-limited generators): Curvelets, Contourlets, and Shearlets provide (almost) optimally sparse expansions for 2D cartoon-like images f ∈ E2 (R2 ) with 2 asymptotic error 2 f − fN 2 = O(N −2 (log N)3 ), N → ∞. Success of 2D shearlets: Kutyniok and Lim (2011) proved that this asymptotic rate can be achieved with compactly supported shearlets. Intuitive explanation (shearlets vs. wavelets): N −2 8/42 DTU Mathematics, Technical University of Denmark N −1 Shearlet Frames 26.3.2012
  • 13. Reasonable Model for 3D Images Definition The set of 3D images E2 (R3 ) is defined by 2 E2 (R3 ) = {f ∈ L2 (R3 ) : f = f0 +f1 χB } fi ∈ C 2 (R3 ), supp fi ⊂ [0, 1]3 , 2 where B ⊂ [0, 1]3 and the boundary surface ∂B is a closed C 2 -surface for which the principal curvatures are bounded by ν. f0=0 9/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 14. Reasonable Model for 3D Images Definition The set of 3D images E2 (R3 ) is defined by 2 E2 (R3 ) = {f ∈ L2 (R3 ) : f = f0 +f1 χB } fi ∈ C 2 (R3 ), supp fi ⊂ [0, 1]3 , 2 where B ⊂ [0, 1]3 and the boundary surface ∂B is a closed C 2 -surface for which the principal curvatures are bounded by ν. Theorem (Kutyniok, L, Lim; 2011) Let (ψλ )λ ⊂ L2 (R3 ). The optimal asymptotic approximation error of f ∈ E2 (R3 ) is 2 2 f − fN 2 N −1 , N → ∞, where fN = cλ ψλ . λ∈IN 9/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 15. Most Approaches are for 2D Data Question: Why is the 3D situation so crucial? Answer: • Our world is 3-dimensional. • 3D data is essential for Biology, Seismology,... • Transition 2D ❀ 3D is special. • In 3D anisotropic features occur for the first time in different dimensions. 10/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 16. Success of 1D Wavelets • The wavelet system generated by ψ ∈ L2 (R) is ψj,k = 2j/2 ψ(2j · −m) : j ∈ Z, m ∈ Z . • Wavelets provide sparse expansions for ‘1D cartoon images’ (C 2 apart from point discontinuities) with asymptotic error 2 f − fN 2 = O(N −4 ), N → ∞. • In comparison, Fourier series only gives O(N −1 ). ❀ Wavelets are very good at detecting singularities in 1D! 11/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 17. Goal for Today Design a representation system which... • ...forms a orthonormal basis frame for L2 (R3 ), • ...captures anisotropic structures, e.g., singularities concentrated on lower dimensional manifolds, • ...provides optimally sparse approximation of cartoon images, • ...allows for compactly supported analyzing elements, • ...is associated with fast decomposition algorithms, • ...treats the continuum and digital ’world’ uniformly. Vision: Introduce a system for 3D data that is as powerful as wavelets are for 1D data! 12/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 18. Outline 1 Applied Harmonic Analysis and Imaging Sciences 2 Fourier and Wavelet Analysis 1D Analysis: Fourier Series 1D Analysis: Wavelets Problem with Wavelets for 2D and 3D Data 3 Shearlet Theory 4 Sparse Approximation using 3D Shearlets 5 Conclusion 13/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 19. Fourier Series • {e2πinx }n∈Z is an ONB for L2 ([0, 1]): f = n cn e2πinx • Modulation: En f (x) := e2πinx f (x), f ∈ L2 (R). • Translation: Tm f (x) := f (x − m), f ∈ L2 (R). • {En Tm χ[0,1] }n,m∈Z ONB for L2 (R): f = n,m cn,m En Tm χ[0,1] 14/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 20. Fourier Series • {e2πinx }n∈Z is an ONB for L2 ([0, 1]): f = n cn e2πinx • Modulation: En f (x) := e2πinx f (x), f ∈ L2 (R). • Translation: Tm f (x) := f (x − m), f ∈ L2 (R). • {En Tm χ[0,1] }n,m∈Z ONB for L2 (R): f = n,m cn,m En Tm χ[0,1] Simple 1D cartoon-like image (0 < x0 < x1 < 1): 1 x ∈ [x0 , x1 ] , f (x) = 0 otherwise Fourier Coefficients: 1 x1 1 |cn,0 | = f (x) e−2πinx dx = e−2πinx dx ∼ 0 x0 n 14/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 21. 1D Fourier Series Approximation • Parseval’s Equality: |cn,m |2 = | f , En Tm χ0,1 |2 = f 2 2 n,m∈Z n,m∈Z 15/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 22. 1D Fourier Series Approximation • Parseval’s Equality: |cn,m |2 = | f , En Tm χ0,1 |2 = f 2 2 n,m∈Z n,m∈Z • Best N-term approximation: fN = cn,m En Tm χ[0,1] , # |IN | = N (n,m)∈IN 15/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 23. 1D Fourier Series Approximation • Parseval’s Equality: |cn,m |2 = | f , En Tm χ0,1 |2 = f 2 2 n,m∈Z n,m∈Z • Best N-term approximation: fN = cn,m En Tm χ[0,1] , # |IN | = N (n,m)∈IN • By Parseval’s Equality: ∞ ∞ 2 1 1 1 f − fN 2 = |cn,m |2 ∼ ∼ dx ∼ (n,m)∈IN / n=N n2 N x2 N 15/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 24. 1D Fourier Series Approximation • Parseval’s Equality: |cn,m |2 = | f , En Tm χ0,1 |2 = f 2 2 n,m∈Z n,m∈Z • Best N-term approximation: fN = cn,m En Tm χ[0,1] , # |IN | = N (n,m)∈IN • By Parseval’s Equality: ∞ ∞ 2 1 1 1 f − fN 2 = |cn,m |2 ∼ ∼ dx ∼ (n,m)∈IN / n=N n2 N x2 N • Conclusion: 2 f − fN 2 ∼ N −1 15/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 25. 1D Wavelet Approximations • The wavelet system generated by ψ ∈ L2 (R) is ψj,m = Dj Tm ψ = 2j/2 ψ(2j · −m) : j ∈ Z, m ∈ Z . • ONB e.g., Haar Wavelet:  1 1 x ∈ [0, 2 ),   ψ(x) = −1 x ∈ [ 1 , 1), 2  0  otherwise. • Haar Scaling function Tm φ, m ∈ Z (replaces ψj,m , j < 0, m ∈ Z): φ(x) = χ[0,1] (x) 16/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 26. 1D Wavelet Approximations • At fixed scale j there are exactly two non-zero wavelet coefficients! |cj,k | = | f , ψj,k | ∼ 2−j/2 17/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 27. 1D Wavelet Approximations • At fixed scale j there are exactly two non-zero wavelet coefficients! |cj,k | = | f , ψj,k | ∼ 2−j/2 • Best N-term Approximation: N f − fN 2 2 = |cj,k |2 2−j ∼ 2−N (j,k)∈IN / j=1 17/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 28. 1D Wavelet Approximations • At fixed scale j there are exactly two non-zero wavelet coefficients! |cj,k | = | f , ψj,k | ∼ 2−j/2 • Best N-term Approximation: N f − fN 2 2 = |cj,k |2 2−j ∼ 2−N (j,k)∈IN / j=1 • For piecewise C 2 functions (1D cartoon-like images): 2 f − fN 2 = O(N −4 ), N → ∞. 17/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 29. 1D Wavelet Approximations • At fixed scale j there are exactly two non-zero wavelet coefficients! |cj,k | = | f , ψj,k | ∼ 2−j/2 • Best N-term Approximation: N f − fN 2 2 = |cj,k |2 2−j ∼ 2−N (j,k)∈IN / j=1 • For piecewise C 2 functions (1D cartoon-like images): 2 f − fN 2 = O(N −4 ), N → ∞. • In comparison, Fourier series only gave O(N −1 ). 17/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 30. 1D Wavelet Approximations • At fixed scale j there are exactly two non-zero wavelet coefficients! |cj,k | = | f , ψj,k | ∼ 2−j/2 • Best N-term Approximation: N f − fN 2 2 = |cj,k |2 2−j ∼ 2−N (j,k)∈IN / j=1 • For piecewise C 2 functions (1D cartoon-like images): 2 f − fN 2 = O(N −4 ), N → ∞. • In comparison, Fourier series only gave O(N −1 ). ❀ Wavelets are very good at detecting singularities in 1D! 17/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 31. Why Not nD Wavelets? • Isotropic scaling (same scaling in all directions):   2j 0 0 Aj =  0 2j 0.   0 0 2j • The wavelet system generated by ψ ∈ L2 (R3 ) is ψj,m = 23j/2 ψ(Aj · −m) : j ∈ Z, m ∈ Z3 . • Wavelets do not provide optimal sparse expansions for 3D cartoon images. For f ∈ E2 (R3 ) wavelets only provide an aymptotic error 2 of: f − fN 2 ∼ N −1/2 , 2 N → ∞. ❀ Wavelets are not good at effectively capturing 2D singularities in 3D: Far from the optimal rate N −1 ! 18/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 32. Outline 1 Applied Harmonic Analysis and Imaging Sciences 2 Fourier and Wavelet Analysis 3 Shearlet Theory Discrete Shearlet Systems Pyramid-adapted Systems 4 Sparse Approximation using 3D Shearlets 5 Conclusion 19/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 33. Introducing 2D Shearlets • Anisotropic scaling Aj : 2j 0 Aj = , 0 2j/2 • Shearing Sk (direction parameter ↔ rotations): 1 k Sk = 0 1 • The shearlet system generated by ψ ∈ L2 (R2 ) is ψj,k,m = DSk Aj Tm ψ = 23j/4 ψ(Sk A2j · −m) : j ∈ Z, k ∈ Z, m ∈ Z2 20/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 34. Action of Anisotropic Scaling and Shearing in 2D ψj,k,m for j = 0, k = 0, m = (0, 0) 21/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 35. Action of Anisotropic Scaling and Shearing in 2D ψj,k,m for j = 1, k = 0, m = (0, 0) 21/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 36. Action of Anisotropic Scaling and Shearing in 2D ψj,k,m for j = 2, k = 0, m = (0, 0) 21/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 37. Action of Anisotropic Scaling and Shearing in 2D ψj,k,m for j = 1, k = 0, m = (0, 0) 21/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 38. Action of Anisotropic Scaling and Shearing in 2D ψj,k,m for j = 1, k = 0, m = (1, −1) 21/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 39. Action of Anisotropic Scaling and Shearing in 2D ψj,k,m for j = 1, k = 0, m = (0, 0) 21/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 40. Action of Anisotropic Scaling and Shearing in 2D ψj,k,m for j = 1, k = −1, m = (0, 0) 21/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 41. Action of Anisotropic Scaling and Shearing in 2D ψj,k,m for j = 1, k = −2, m = (0, 0) 21/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 42. Action of Anisotropic Scaling and Shearing in 2D ψj,k,m for j = 1, k = −3, m = (0, 0) 21/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 43. Introducing 3D Shearlets • Anisotropic scaling Aj :   2j 0 0 Aj =  0 2j/2 0 ,   0 0 2j/2 • Shearing Sk , k = (k1 , k2 ) (direction parameter ↔ rotations):   1 k1 k2 Sk = 0 1 0    0 0 1 • The shearlet system generated by ψ ∈ L2 (R3 ) is ψj,k,m = 2j ψ(Sk A2j · −m) : j ∈ Z, k ∈ Z2 , m ∈ Z3 22/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 44. Introducing 3D Shearlets • Anisotropic scaling Aj (α ∈ (1, 2]):   2jα/2 0 0 Aj =  0 2j/2 0 ,   0 0 2j/2 • Shearing Sk , k = (k1 , k2 ) (direction parameter ↔ rotations):   1 k1 k2 Sk = 0 1 0    0 0 1 • The shearlet system generated by ψ ∈ L2 (R3 ) is 1 ψj,k,m = 2j( 4 + 2 ) ψ(Sk A2j · −m) : j ∈ Z, k ∈ Z2 , m ∈ Z3 α 22/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 45. Compactly Supported Generators Idea: We want generators of the form ψ(x1 , x2 , x3 ) = η(x1 )φ(x2 )φ(x3 ) for the shearlets system associated with the pyramids P1 , where η is a 1D wavelet and φ is a scaling/bump function. The wavelet η will then point in the ‘short’ direction of the plate-like elements. Class of generators: We consider generators with weak directional vanishing moments and essential support (in Frequency domain) as: 23/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 46. Example: 3D Action of Anisotropic scaling and Shearing Example: Suppose supp ψ = [0, 1]3 . Let the translation parameter m = (0, 0, 0) be fixed. Scaling For k = (0, 0): supp ψj,0,0 = [0, 2−j ] × [0, 2−j/2 ] × [0, 2−j/2 ] → the shearlet becomes a small, plate-like element as j → ∞. 24/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 47. Example: 3D Action of Anisotropic scaling and Shearing Example: Suppose supp ψ = [0, 1]3 . Let the translation parameter m = (0, 0, 0) be fixed. Scaling For k = (0, 0): supp ψj,0,0 = [0, 2−j ] × [0, 2−j/2 ] × [0, 2−j/2 ] → the shearlet becomes a small, plate-like element as j → ∞. Shearing For j > 0: ψ(Sk A2j x) = ψ(2j x1 + k1 2j/2 x2 + k2 2j/2 x3 , 2j/2 x2 , 2j/2 x3 ) 24/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 48. Example: 3D Action of Anisotropic scaling and Shearing Example: Suppose supp ψ = [0, 1]3 . Let the translation parameter m = (0, 0, 0) be fixed. Scaling For k = (0, 0): supp ψj,0,0 = [0, 2−j ] × [0, 2−j/2 ] × [0, 2−j/2 ] → the shearlet becomes a small, plate-like element as j → ∞. Shearing For j > 0: ψ(Sk A2j x) = ψ(2j x1 + k1 2j/2 x2 + k2 2j/2 x3 , 2j/2 x2 , 2j/2 x3 ) Problem To get horizontal plate-elements require very large shear parameters |ki | = ∞ ❀ non-uniform treatment of directions. 24/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 49. 3D Action of Anisotropic Scaling and Shearing Spatial Domain: Frequency Domain: 25/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 50. Pyramid-adapted Shearlet Systems Idea: We partition the frequency space R3 into 3 pairs of pyramids:   {(ξ1 , ξ2 , ξ3 ) ∈ R3 : |ξ1 | ≥ 1, |ξ2 /ξ1 | ≤ 1, |ξ3 /ξ1 | ≤ 1} : ι = 1, Pι = {(ξ1 , ξ2 , ξ3 ) ∈ R3 : |ξ2 | ≥ 1, |ξ1 /ξ2 | ≤ 1, |ξ3 /ξ2 | ≤ 1} : ι = 2, {(ξ1 , ξ2 , ξ3 ) ∈ R3 : |ξ3 | ≥ 1, |ξ1 /ξ3 | ≤ 1, |ξ2 /ξ3 | ≤ 1} : ι = 3,  and a centered cube C ={(ξ1 , ξ2 , ξ3 ) ∈ R3 : (ξ1 , ξ2 , ξ3 ) ∞ < 1}. P3 C P1 015 P2 016 017 ❀ For each pair of pyramids we construct a shearlet frame. 26/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 51. Pyramid-adapted Shearlet Systems ˜ ˘ The (pyramid-adapted) shearlet system SH(φ, ψ, ψ, ψ) generated ˜ ψ ∈ L2 (R3 ) is the union of by φ, ψ, ψ, ˘ {φm = φ(· − m) : m ∈ Z3 }, {ψj,k,m = 2j ψ(Sk A2j · −m) : j ≥ 0, |k| ≤ ⌈2j/2 ⌉, m ∈ Z3 }, ˜ ˜ ˜ ˜ {ψj,k,m = 2j ψ(Sk A2j · −m) : j ≥ 0, |k| ≤ ⌈2j/2 ⌉, m ∈ Z3 }, ˘ ˘ ˘ ˘ {ψj,k,m = 2j ψ(Sk A2j · −m) : j ≥ 0, |k| ≤ ⌈2j/2 ⌉, m ∈ Z3 }, ˜ where j ∈ N0 and k ∈ Z2 . We call φ a scaling function and ψ, ψ, ˘ shearlets. and ψ 27/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 52. Pyramid-adapted Shearlet Systems ˜ ˘ The (pyramid-adapted) shearlet system SH(φ, ψ, ψ, ψ; α) generated ˜ ψ ∈ L2 (R3 ) is the union of by φ, ψ, ψ, ˘ {φm = φ(· − m) : m ∈ Z3 }, α 1 {ψj,k,m = 2j( 4 + 2 ) ψ(Sk A2j · −m) : j ≥ 0, |k| ≤ ⌈2(α−1)j/2 ⌉, m ∈ Z3 }, α 1 {ψj,k,m = 2j( 4 + 2 ) ψ(Sk A2j · −m) : j ≥ 0, |k| ≤ ⌈2(α−1)j/2 ⌉, m ∈ Z3 }, ˜ ˜ ˜ ˜ α 1 ˘ ˘ ˘ ˘ {ψj,k,m = 2j( 4 + 2 ) ψ(Sk A2j · −m) : j ≥ 0, |k| ≤ ⌈2(α−1)j/2 ⌉, m ∈ Z3 }, ˜ where j ∈ N0 and k ∈ Z2 . We call φ a scaling function and ψ, ψ, ˘ shearlets. and ψ 27/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 53. Outline 1 Applied Harmonic Analysis and Imaging Sciences 2 Fourier and Wavelet Analysis 3 Shearlet Theory 4 Sparse Approximation using 3D Shearlets Optimally Theorem Sketch of proof 5 Conclusion 28/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 54. Revisiting 3D Cartoon-like Images Definition 2 Let 1 < α ≤ 2. The set of 3D images Eα (R3 ) is defined by Eα (R3 ) = {f ∈ L2 (R3 ) : f = f0 +f1 χB } fi ∈ C 2 (R3 ), supp fi ⊂ [0, 1]3 , 2 where B ⊂ [0, 1]3 and the boundary surface ∂B is a closed C α -surface with ‘curvature’ bounded by ν. ❇➯ ✶ ✺✼ ✳✵ ✵✺ ✳✵ ✺✷✳✵ ✶ ✺✼✳✵ ✵ ✵ ✵✺ ✳✵ ✺✷ ✳✵ ✵✺✳✵ ✺✷✳✵ ✺✼ ✳✵ ✵ ✶ 29/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 55. Revisiting 3D Cartoon-like Images Definition Let 1 < α ≤ 2. The set of 3D images Eα (R3 ) is defined by 2 Eα (R3 ) = {f ∈ L2 (R3 ) : f = f0 +f1 χB } fi ∈ C 2 (R3 ), supp fi ⊂ [0, 1]3 , 2 where B ⊂ [0, 1]3 and the boundary surface ∂B is a closed C α -surface with ‘curvature’ bounded by ν. Theorem (Kutyniok, L, Lim; 2011) Let (ψλ )λ ⊂ L2 (R3 ). The optimal asymptotic approximation error of f ∈ Eα (R3 ) is 2 2 f − fN 2 N −α/2 , N → ∞, where fN = cλ ψλ . λ∈IN 29/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 56. Main result: Optimal Sparsity of 3D shearlet Theorem (Kutyniok, L, Lim; 2011) ˜ ˘ Fix α ∈ (1, 2]. Let φ, ψ, ψ, ψ ∈ L2 (R3 ) be compactly supported. Assume: ˆ (i) |ψ(ξ)| min(1, |ξ1 |δ ) · 3 min(1, |ξi |−γ ) i=1 ∂ ˆ |ξ2 | −γ |ξ3 | −γ (ii) ∂ξi ψ(ξ) ≤ |h(ξ1 )| · 1 + |ξ1 | 1+ |ξ1 | , i = 2, 3, ˜ ˘ where δ > 8, γ ≥ 4, h ∈ L1 (R), and similar for ψ and ψ. Further, ˜ ψ; α) forms a frame for L2 (R3 ). For suppose that SH(φ, ψ, ψ, ˘ f ∈ Eα (R3 ), 2 2 O(N −α/2+µ ), if α < 2, f − fN 2 = −1 2 as N → ∞, O(N (log N) ), if α = 2, where 3(2−α)(α−1)(α+2) µ = µ(α) = 2(9α2 +17α−10) , 30/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 57. Optimality Gap of Main result If α < 2, we have for f ∈ Eα (R3 ), 2 2 f − fN 2 = O(N −α/2+µ ), as N → ∞, where • 0 < µ(α) < 0.037 for α (1, 2), • µ(α) → 0 for α → 1+ or α → 2− . 1 0.9 0.8 0.7 0.6 0.5 1 1.2 1.4 1.6 1.8 2 α ❀ Shearlets provide nearly optimally sparse approximations of Eα (R3 ). 2 31/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 58. Sketch of proof: Frames and Non-linear approximations Definition A sequence (ψλ )λ ⊂ L2 (Rd ) is a frame for L2 (Rd ) if 2 ∃A, B > 0 : A f ≤ | f , ψλ |2 ≤ B f 2 for all f ∈ L2 (Rd ). λ ˜ Fact from frame theory: There exists a canonical dual frame {ψλ }λ such that f = ˜ f , ψλ ψλ for all f ∈ L2 (Rd ). λ 32/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 59. Sketch of proof: Frames and Non-linear approximations Definition A sequence (ψλ )λ ⊂ L2 (Rd ) is a frame for L2 (Rd ) if 2 ∃A, B > 0 : A f ≤ | f , ψλ |2 ≤ B f 2 for all f ∈ L2 (Rd ). λ ˜ Fact from frame theory: There exists a canonical dual frame {ψλ }λ such that f = ˜ f , ψλ ψλ for all f ∈ L2 (Rd ). λ As N-term approximation (not necessarily ‘best’) we take: fN = ˜ f , ψλ ψλ , λ∈IN where (| f , ψ |)λ λ∈IN are the N largest coefficients. 32/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 60. Sketch of proof: Outline ˜ • Estimates of f , ψj,k,m for f ∈ Eα (R3 ). Similar for ψj,k,m and 2 ˘ ψj,k,m . • WLOG assume j ≥ 0 is sufficiently large. • Let |θ(f )|n denote the nth largest shearlet coefficient | f , ψj,k,m |: 2 1 f − fN 2 ≤ |θ(f )|2 . n A n>N • Hence, it suffices to prove: |θ(f )|2 ≤ C N −α/2 n as N → ∞. n>N 33/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 61. Sketch of proof: Heuristic argument of N −α/2 Recall: The shearlet is of the form ψ(x) = η(x1 )ϕ(x2 )ϕ(x3 ), where η is a wavelet and ϕ a bump (or a scaling) function. We consider three cases of coefficients f , ψj,k,m : (a) Shearlets whose support do not overlap with the boundary ∂B. (b) Shearlets whose support overlap with ∂B and are nearly tangent. (c) Shearlets whose support overlap with ∂B, but tangentially. (a) (b) (c) 34/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 62. Sketch of proof: Heuristic argument of N −α/2 Case (b): • At scale j > 0: At most O(2j ) coefficients since the plate-like element is of size 2−j/2 × 2−j/2 (and “thickness” 2−jα/2 ). Moreover, α 1 α 1 −j( 4 + 2 ) | f , ψj,k,m | ≤ f ∞ ψj,k,m 1 ≤ f ∞ ψ 12 ≤C 2−j( 4 + 2 ) 1 −j( 2 + α ) ❀ we have O(2j ) coefficients bounded by C · 2 4 . 35/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 63. Sketch of proof: Heuristic argument of N −α/2 Case (b): • At scale j > 0: At most O(2j ) coefficients since the plate-like element is of size 2−j/2 × 2−j/2 (and “thickness” 2−jα/2 ). Moreover, α 1 α 1 −j( 4 + 2 ) | f , ψj,k,m | ≤ f ∞ ψj,k,m 1 ≤ f ∞ ψ 12 ≤C 2−j( 4 + 2 ) 1 −j( 2 + α ) ❀ we have O(2j ) coefficients bounded by C · 2 4 . • Assuming, case (a) and (c) coefficients are negligible, the nth 1 largest coefficient |θ(f )|n is bounded by |θ(f )|n ≤ C n−( 2 + 4 ) : α |θ(f )|2 n n−1−α/2 N −α/2 . n>N n>N 35/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 64. Heuristic argument of N −1/2 for Wavelets Wavelets: • At scale j > 0: At most O(2jα ) coefficients since the cube-like element is of size 2−jα/2 × 2−jα/2 (and “thickness” 2−jα/2 ). Moreover, 3α 3 −j 4 | f , ψj,k,m | ≤ f ∞ ψj,k,m 1 ≤ f ∞ ψ 12 ≤C 2jα(− 4 ) jα(− 3 ) ❀ we have O(2jα ) coefficients bounded by C · 2 . 4 • Assuming, case (a) and (c) coefficients are negligible, the nth 3 largest coefficient |θ(f )|n is bounded by |θ(f )|n ≤ C n− 4 : |θ(f )|2 ≤ n n−3/2 ≤ C · N −1/2 . n>N n>N 36/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 65. Generalization: Optimal Sparsity of 3D shearlets Question: • The above result shows that the ‘plate-like’ shearlet system is optimal for describing 2D singularities. But not all singularities in 3D images are located on C α -smooth surfaces! 37/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 66. Generalization: Optimal Sparsity of 3D shearlets Question: • The above result shows that the ‘plate-like’ shearlet system is optimal for describing 2D singularities. But not all singularities in 3D images are located on C α -smooth surfaces! • ❀ We would also like to consider point, curve and surface singularities: B f0=0 f1=1 37/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 67. Generalization: Optimal Sparsity of 3D shearlets Theorem (Kutyniok, L, Lim; 2011) The sparsity result extends to a class of cartoon-like 3D images with only piecewise smooth C α boundary ∂B. 38/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 68. Generalization: Optimal Sparsity of 3D shearlets Theorem (Kutyniok, L, Lim; 2011) The sparsity result extends to a class of cartoon-like 3D images with only piecewise smooth C α boundary ∂B. Observations: 1 The optimal rate is still the same when introducing “corner points” and “curves” 2 Pyramid-adapted ‘plate-like’ shearlets still achieve the optimal rate. 38/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 69. Outline 1 Applied Harmonic Analysis and Imaging Sciences 2 Fourier and Wavelet Analysis 3 Shearlet Theory 4 Sparse Approximation using 3D Shearlets 5 Conclusion 39/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 70. What To Take Home? • Anisotropic features in multivariate data require special efficient encoding strategies • The shearlet theory is perfectly suited to this problem • One main advantage of shearlets is that they provide a unified treatment of the continuum and digital setting. • The Shearlet Theory for 2D and 3D data provides: • Compactly supported shearlet frames • Explicit estimates for frame bounds • Optimal sparse approximation of cartoon-like images. 40/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 71. Thank You! Preprints: http://www.shearlets.org http://www2.mat.dtu.dk/people/J.Lemvig/ Software: http://www.shearlab.org 41/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012
  • 72. References D. L. Donoho, Sparse components of images and optimal atomic decomposition, Constr. Approx. 17 (2001), 353–382. E. J. Candés and D. L. Donoho, New tight frames of curvelets and optimal representations of objects with piecewise C 2 singularities, Comm. Pure and Appl. Math. 56 (2004), 216–266. K. Guo and D. Labate, Optimally sparse multidimensional representation using shearlets, SIAM J. Math Anal. 39 (2007), 298–318. M. N. Do and M. Vetterli, The contourlet transform: an efficient directional multiresolution image representation, IEEE Trans. Image Process. 14 (2005), 2091–2106. G. Kutyniok and W.-Q Lim, Compactly supported shearlets are optimally sparse, J. Approx. Theory 163 (2011), 1564–1589. S. Dahlke, G. Steidl, and G. Teschke, The continuous shearlet transform in arbitrary space dimensions, J. Fourier Anal. Appl. 16 (2010), 340–364. G. Kutyniok, J. Lemvig, and W.-Q Lim, Compactly supported shearlets, in Approximation Theory XIII (San Antonio, TX, 2011), Springer. 42/42 DTU Mathematics, Technical University of Denmark Shearlet Frames 26.3.2012