Shearlet Frames and Optimally Sparse Approximations

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

Anisotropic Phenomena
Image Model
Goal for Today


3 Shearlet Theory


5 Conclusion


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


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


Geometric Structures of Multidimensional Images

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

Objects
Cartoon-
with
like
‘smooth’
image
boundaries




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




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!


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


Deﬁnition (Donoho; 2001)
The set of cartoon-like 2D images E2 (R2 ) is deﬁned by
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


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 → ∞.


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

Deﬁnition
The set of 3D images E2 (R3 ) is deﬁned by
2

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


Deﬁnition
The set of 3D images E2 (R3 ) is deﬁned by
2

2

C 2 -surface for which the principal curvatures are bounded by ν.

Theorem (Kutyniok, L, Lim; 2011)
of f ∈ E2 (R3 ) is
2

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


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.


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!


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!


Outline


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

3 Shearlet Theory


5 Conclusion


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]


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


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



|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



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


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



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


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


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)


• At ﬁxed scale j there are exactly two non-zero wavelet coeﬃcients!

|cj,k | = | f , ψj,k | ∼ 2−j/2



|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



|cj,k | = | f , ψj,k | ∼ 2−j/2


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 → ∞.



|cj,k | = | f , ψj,k | ∼ 2−j/2


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


2
f − fN 2 = O(N −4 ), N → ∞.

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



|cj,k | = | f , ψj,k | ∼ 2−j/2


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


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!


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 !

Outline



3 Shearlet Theory
Discrete Shearlet Systems
Pyramid-adapted Systems


5 Conclusion


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


Action of Anisotropic Scaling and Shearing in 2D

ψj,k,m for j = 0, k = 0, m = (0, 0)



ψj,k,m for j = 1, k = 0, m = (0, 0)



ψj,k,m for j = 2, k = 0, m = (0, 0)



ψj,k,m for j = 1, k = 0, m = (1, −1)



ψj,k,m for j = 1, k = −1, m = (0, 0)



ψj,k,m for j = 1, k = −2, m = (0, 0)



ψj,k,m for j = 1, k = −3, m = (0, 0)


• 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


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


• 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

1
ψj,k,m = 2j( 4 + 2 ) ψ(Sk A2j · −m) : j ∈ Z, k ∈ Z2 , m ∈ Z3
α


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:


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

m = (0, 0, 0) be ﬁxed.

supp ψj,0,0 = [0, 2−j ] × [0, 2−j/2 ] × [0, 2−j/2 ] → the
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 )


Shearing

m = (0, 0, 0) be ﬁxed.

supp ψj,0,0 = [0, 2−j ] × [0, 2−j/2 ] × [0, 2−j/2 ] → the
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.


3D Action of Anisotropic Scaling and Shearing
Spatial Domain:

Frequency Domain:


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.


˜ ˘
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 },
˜ ˜ ˜ ˜
˘ ˘ ˘ ˘

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



˜ ˘
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
˜ ˜ ˜ ˜
α 1
˘ ˘ ˘ ˘

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


Outline



3 Shearlet Theory

Optimally Theorem
Sketch of proof

5 Conclusion


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

C α -surface with ‘curvature’ bounded by ν.

❇➯

✶

✺✼ ✳✵

✵✺ ✳✵

✺✷✳✵

✶

✺✼✳✵
✵

✵ ✵✺ ✳✵

✺✷ ✳✵
✵✺✳✵ ✺✷✳✵

✺✼ ✳✵
✵
✶


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

C α -surface with ‘curvature’ bounded by ν.

of f ∈ Eα (R3 ) is
2

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


Main result: Optimal Sparsity of 3D shearlet
˜ ˘
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)
,


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

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


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.

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


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)



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 .



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

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

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!


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



The sparsity result extends to a class of cartoon-like 3D images with
only piecewise smooth C α boundary ∂B.



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.


Outline



3 Shearlet Theory


5 Conclusion


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.


Thank You!

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

Software: http://www.shearlab.org


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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