Shearlet theory has become a central tool in analyzing and representing 2D and 3D data with anisotropic features. Shearlet systems are systems of functions generated by one single generator with parabolic scaling, shearing, and translation operators applied to it, in much the same way wavelet systems are dyadic scalings and translations of a single function, but including a directional parameter. The success of shearlets owes to an extensive list of desirable properties: Shearlet systems can be generated by one function, they provide precise resolution of wavefront sets, they allow compactly supported analyzing elements, they are associated with fast decomposition algorithms, and they provide a unified treatment of the continuum and the digital world.
This talk gives an introduction to shearlet theory with focus on separable and compactly supported shearlets in 2D and 3D. We will consider constructions of band-limited and compactly supported shearlet frames in those dimensions. Finally, we will show that compactly supported shearlet frames satisfying weak decay, smoothness, and directional moment conditions provide optimally sparse approximations of a generalized model of cartoon-like images comprising of piecewise C² functions that are smooth apart from piecewise C² discontinuity edges.
This talk is based on joint with G. Kutyniok and W.-Q Lim (TU Berlin).
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.
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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!
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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!
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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
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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)
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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
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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
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34. Action of Anisotropic Scaling and Shearing in 2D
ψj,k,m for j = 0, k = 0, m = (0, 0)
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35. Action of Anisotropic Scaling and Shearing in 2D
ψj,k,m for j = 1, k = 0, m = (0, 0)
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36. Action of Anisotropic Scaling and Shearing in 2D
ψj,k,m for j = 2, k = 0, m = (0, 0)
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37. Action of Anisotropic Scaling and Shearing in 2D
ψj,k,m for j = 1, k = 0, m = (0, 0)
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38. Action of Anisotropic Scaling and Shearing in 2D
ψj,k,m for j = 1, k = 0, m = (1, −1)
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39. Action of Anisotropic Scaling and Shearing in 2D
ψj,k,m for j = 1, k = 0, m = (0, 0)
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40. Action of Anisotropic Scaling and Shearing in 2D
ψj,k,m for j = 1, k = −1, m = (0, 0)
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41. Action of Anisotropic Scaling and Shearing in 2D
ψj,k,m for j = 1, k = −2, m = (0, 0)
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42. Action of Anisotropic Scaling and Shearing in 2D
ψj,k,m for j = 1, k = −3, m = (0, 0)
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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:
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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.
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49. 3D Action of Anisotropic Scaling and Shearing
Spatial Domain:
Frequency Domain:
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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.
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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
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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
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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
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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
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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)
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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.
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71. Thank You!
Preprints: http://www.shearlets.org
http://www2.mat.dtu.dk/people/J.Lemvig/
Software: http://www.shearlab.org
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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.
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