Compactly Supported Shearlets: Construction, Optimally Sparse Approximation, and Applications

Applied Harmonic Analysis and Imaging Sciences Compactly Supported Shearlet Systems Applications Conclusions

Gitta Kutyniok

(University of Osnabr¨uck, Germany) joint with

Jakob Lemvig & Wang-Q Lim (University of Osnabr¨uck, Germany)

Illinois/Missouri Applied Harmonic Analysis Seminar University of Illinois, October 16, 2010

Gitta Kutyniok, Jakob Lemvig, & Wang-Q Lim Compactly Supported Shearlets Applied Harmonic Analysis and Imaging Sciences Compactly Supported Shearlet Systems Applications Conclusions Outline

Applied Harmonic Analysis and Imaging Sciences Anisotropic Phenomena Goal for Today Compactly Supported Shearlet Systems Review: Discrete Shearlets in 2D Shearlet Systems in 3D Optimal Sparse Approximation Frame Properties Applications Denoising Image Separation Conclusions

Gitta Kutyniok, Jakob Lemvig, & Wang-Q Lim Compactly Supported Shearlets Applied Harmonic Analysis and Imaging Sciences Compactly Supported Shearlet Systems Anisotropic Phenomena Applications Goal for Today Conclusions 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: Layers of earth. ◮ ...implicitely. ◮ Transport Equations: Shock fronts.

Some important tasks: ◮ Denoising. ◮ Inpainting. ◮ Feature Detection/Extraction. ◮ ...

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

Image = Image, ψλ ψλ. λ ◮ Sparse coeﬃcients! ◮ Approximation properties!

Deﬁnition (Donoho; 2001): 2 The set of cartoon-like 2D images E2 (ν) is deﬁned by 2 2 R2 E2 (ν)= {f ∈ L ( ) : f = f0 + f1 χB }, 2 2 2 2 where f0, f1 ∈ C (R ) with supp fi ⊂ [0, 1] and B ⊂ [0, 1] with ∂B a closed C 2-curve whose curvature is bounded by ν.

Theorem (Donoho; 2001): 2 2 Let (ψλ)λ ⊆ L (R ). Allowing only polynomial depth search, the 2 optimal asymptotic approximation error of f ∈E2 (ν) is 2 −2 f − fN 2 ≤ C N , N →∞, where fN = cλψλ. λ∈IN Gitta Kutyniok, Jakob Lemvig, & Wang-Q Lim Compactly Supported Shearlets Applied Harmonic Analysis and Imaging Sciences Compactly Supported Shearlet Systems Anisotropic Phenomena Applications Goal for Today Conclusions What can Wavelets do?

Problem: ◮ 2 2 −1 For E2 (ν), Wavelets only achieve f − fN 2 ≤ C N , N →∞. ◮ Isotropic structure of wavelets:

−j/2 −j 2 {ψj,k = 2 ψ(2 −k) : j ∈ Z, k ∈ Z }

◮ Wavelets cannot sparsely represent curves.

Intuitive explanation:

Design a representation system which... ◮ ...is generated by one ‘mother function’, ◮ ...provides optimally sparse approximation of cartoons, ◮ ...allows for compactly supported analyzing elements, ◮ ...is associated with fast decomposition algorithms, ◮ ...treats the continuum and digital ‘world’ uniformly.

Vision: Introduce a system for 2D data as powerful as wavelets for 1D data!

Non-exhaustive list: ◮ Directional wavelets (Antoine, Murenzi, Vandergheynst; 1999) ◮ Ridgelets (Cand`es and Donoho; 1999) ◮ Curvelets (Cand`es and Donoho; 2002) ◮ Contourlets (Do and Vetterli; 2002) ◮ Bandlets (LePennec and Mallat; 2003) ◮ Wavelets with composite dilations (Guo, Labate, Lim, Weiss, and Wilson; 2004) ◮ Shearlets (K, Labate, Lim, and Weiss; 2005), (Guo, K, and Labate; 2006) ◮ . . .

Question: Why is the 3D situation such crucial?

Question: Why is the 3D situation such crucial? Answer: ◮ Our world is 3-dimensional. ◮ 3D data is essential for Biology, Seismology,...

◮ Transition 2D → 3D is unique. ◮ Anisotropic features occur in 3D for the ﬁrst time in diﬀerent dimensions.

Goal: Introduce shearlets for 3D data which... ◮ ...are generated by one ‘mother function’, ◮ ...provide optimally sparse approximation of cartoons, ◮ ...allow for compactly supported analyzing elements, ◮ ...are associated with fast decomposition algorithms, ◮ ...treat the continuum and digital ‘world’ uniformly.

Remark: Even new in 2D: ◮ Construction of compactly supported shearlets. ◮ Optimal sparse approximation of such shearlets.

Gitta Kutyniok, Jakob Lemvig, & Wang-Q Lim Compactly Supported Shearlets Applied Harmonic Analysis and Imaging Sciences Review: Discrete Shearlets in 2D Compactly Supported Shearlet Systems Shearlet Systems in 3D Applications Optimal Sparse Approximation Conclusions Frame Properties Which is the Correct Scaling...?

In harmonic analysis there have been a number of important applications of decompositions based on parabolic scaling

1 fa(x1, x2)= f (a 2 x1, ax2),

2 which leaves invariant the parabola x2 = x1 .

History: ◮ 1970’s: Feﬀerman and later Seeger, Sogge, and Stein studied boundedness of certain operators. ◮ 1998: Hart Smith deﬁned special molecular decompositions of Fourier Integral Operators.

Angle versus slope: ◮ It seems that the appropriate measure for directions is the angle (↔ rotation).

◮ However, in today’s digital world slope (↔ shearing) is much more appropriate. ◮ Example: Parametrization by slope is more suitable for measurements in seismology.

◮ Supports the treating of the digital setting:

1 k Z2 = Z2 for all k ∈ Z. 0 1

Idea: ◮ Consider (aﬃne) systems:

1/2 2 {| det M| ψ(Mx − m) : M ∈ G ⊆ GLn, m ∈ Z }.

◮ Recall: For wavelets, G = {diag(2j , 2j ) : j ∈ Z}. ◮ Construct G to be a special 2-parameter dilation group by:

2j 0 1 k Aj = and Sk = , j, k ∈ Z. 0 2j/2 0 1

Aj : Parabolic scaling matrix, Sk : Shear matrix. ◮ Dilation matrices: G = {Sk Aj : j, k ∈ Z}.

Deﬁnition: Let ψ ∈ L2(R2). Then the associated discrete shearlet system is deﬁned by

3j 2 {2 4 ψ(Sk Aj −m) : j, k ∈ Z, m ∈ Z }.

Remarks: ◮ Advantage: Associated shearlet group S = (R+ × R) ⋉R2. ◮ Disadvantage: Biased treatment of directions.

Induced Tiling of the Frequency Domain:

Deﬁnition: The cone-adapted shearlet system SH(φ, ψ, ψ˜; c) generated by φ ∈ L2(R2) and ψ, ψ˜ ∈ L2(R2) is

{φ(− cm) : m ∈ Z2}

3j ∪{2 4 ψ(Sk Aj −cm) : (j, k, m) ∈ Λcone } 3j ˜ T ˜ ∪{2 4 ψ(Sk Aj −cm) : (j, k, m) ∈ Λcone }, where

j/2 2 Λcone = {(j, k, m) : j ≥ 0, |k| ≤ ⌈2 ⌉, m ∈ Z }, c > 0.

Classical (Band-Limited) Shearlet: Let ψ ∈ L2(R2) be deﬁned by

ψˆ(ξ)= ψˆ(ξ ,ξ )= ψˆ (ξ ) ψˆ ( ξ2 ), 1 2 1 1 2 ξ1 where ◮ ˆ 1 1 ˆ ∞ R ψ1 wavelet, supp(ψ1) ⊆ [−2, − 2 ] ∪ [ 2 , 2], and ψ1 ∈ C ( ). ◮ ∞ ψ2 ‘bump’ function, supp(ψˆ2) ⊆ [−1, 1], and ψˆ2 ∈ C (R). Illustration:

Hypothesis: ◮ Let φ be a scaling function. ◮ Let ψ be a classical shearlet, and ψ˜ deﬁned analogously.

Theorem (K, Labate, Lim, Weiss; 2006): SH(φ, ψ, ψ˜;1) is a tight frame for L2(R2).

Theorem (Guo, Labate; 2007): SH(φ, ψ, ψ˜; 1) provides (almost) optimally sparse approximations 2 of functions f ∈E2 (ν) with asymptotic error

2 −2 3 f − fN 2 ≤ C N (log N) , N →∞.

Deﬁnition (Donoho; 2001): 2 The set of 2D images E2 (ν) is deﬁned by 2 2 R2 E2 (ν)= {f ∈ L ( ) : f = f0 + f1 χB }, 2 2 2 2 where fi ∈ C (R ) with supp fi ⊂ [0, 1] and B ⊂ [0, 1] with ∂B a closed C 2-curve whose curvature is bounded by ν.

Theorem (Donoho; 2001): 2 2 Let (ψλ)λ ⊆ L (R ). Allowing only polynomial depth search, the 2 optimal asymptotic approximation error of f ∈E2 (ν) is 2 −2 f − fN 2 ≤ C N , N →∞.

Deﬁnition: 2 The set of 3D images E3 (ν) is deﬁned by 2 2 R3 E3 = {f ∈ L ( ) : f = f0 + f1 χB }, 2 3 3 3 where fi ∈ C (R ) with supp fi ⊂ [0, 1] and B ⊂ [0, 1] with ∂B a closed C 2-surface whose principal curvatures are bounded by ν.

Theorem (K, Lemvig, Lim; 2010): 2 3 Let (ψλ)λ ⊂ L (R ). Allowing only polynomial depth search, the 2 optimal asymptotic approximation error of f ∈E3 (ν) is 2 −1 f − fN 2 = O(N ), N →∞.

Deﬁnition: α,β Let 1 < α, β ≤ 2. The set of 3D images E3 (ν) is deﬁned by α,β 2 R3 E3 = {f ∈ L ( ) : f = f0 + f1 χB },

β β 3 3 3 where fi ∈ C ∩ H (R ), supp fi ⊂ [0, 1] and B ⊂ [0, 1] with ∂B a closed C α-surface whose principal curvatures are bounded by ν.

Deﬁnition: α,β Let 1 < α, β ≤ 2. The set of 3D images E3 (ν) is deﬁned by α,β 2 R3 E3 = {f ∈ L ( ) : f = f0 + f1 χB },

β β 3 3 3 where fi ∈ C ∩ H (R ), supp fi ⊂ [0, 1] and B ⊂ [0, 1] with ∂B a closed C α-surface whose principal curvatures are bounded by ν.

Theorem (K, Lemvig, Lim; 2010): 2 3 Let (ψλ)λ ⊂ L (R ). Allowing only polynomial depth search, the α,β optimal asymptotic approximation error of f ∈E3 (ν) is

2 − min{α/2,2β/3} f − fN 2 = O(N ), N →∞.

◮ Anisotropic scaling Aj :

2j 0 0 j/2 Aj =  0 2 0  , 0 0 2j/2   ◮ Shearing Sk , k = (k1, k2) (direction parameter ↔ rotations):

1 k1 k2 Sk = 0 1 0  0 0 1  

◮ Anisotropic scaling Aj (α ∈ (1, 2]):

2jα/2 0 0 j/2 Aj =  0 2 0  , 0 0 2j/2   ◮ Shearing Sk , k = (k1, k2) (direction parameter ↔ rotations):

1 k1 k2 Sk = 0 1 0  0 0 1  

Spatial Domain :

Frequency Domain :

2D Shearlets :

3D Shearlets :

Deﬁnition: The pyramid-adapted shearlet system SH(φ, ψ, ψ,˜ ψ˘; c) generated by φ ∈ L2(R3) and ψ, ψ,˜ ψ˘ ∈ L2(R3) is

{φ(− cm) : m ∈ Z3}

j ∪{2 ψ(Sk Aj −cm) : (j, k, m) ∈ Λpyramid } j ∪{2 ψ˜(S˜k A˜ j −cm) : (j, k, m) ∈ Λpyramid } j ∪{2 ψ˘(S˘k A˘j −cm) : (j, k, m) ∈ Λpyramid }, where

j/2 2 Λpyramid = {(j, k, m) : j ≥ 0, |k1|, |k2| ≤ ⌈2 ⌉, m ∈ Z }, c > 0.

Deﬁnition: The pyramid-adapted shearlet system SH(φ, ψ, ψ,˜ ψ˘; c; α) generated by φ ∈ L2(R3) and ψ, ψ,˜ ψ˘ ∈ L2(R3) is

{φ(− cm) : m ∈ Z3}

j ∪{2 ψ(Sk Aj −cm) : (j, k, m) ∈ Λpyramid } j ∪{2 ψ˜(S˜k A˜ j −cm) : (j, k, m) ∈ Λpyramid } j ∪{2 ψ˘(S˘k A˘j −cm) : (j, k, m) ∈ Λpyramid }, where

(α−1)j/2 2 Λpyramid = {(j, k, m) : j ≥ 0, |k1|, |k2| ≤ ⌈2 ⌉, m ∈ Z }, c > 0.

Theorem (K, Lemvig, Lim; 2010): Let α, β ∈ (1, 2]. Let φ, ψ, ψ,˜ ψ˘ ∈ L2(R3) be compactly supported. For ψ suppose that: ˆ δ 3 −γ (i) |ψ(ξ)|≤ C min(1, |ξ1| ) i=1 min(1, |ξi | ) −γ ∂ ˆ |ξ2| (ii) ∂ξ ψ(ξ) ≤ |h(ξ1)| 1+ |ξ | , 2 1 −γ ∂ ˆ |ξ3| (iii) ∂ξ ψ(ξ) ≤ |h(ξ1)| 1+ |ξ | , 3 1 1 where δ > 6+ β, γ ≥ 4, h ∈ L (R), and similar for ψ˜ and ψ˘. Further, suppose that SH(φ, ψ, ψ,˜ ψ˘; c; α) forms a frame for 2 R3 α,β L ( ). Then, for f ∈E3 (ν),

2 − min{α/2,2β/3} 2 f − fN 2 = O(N (log N) ), N →∞.

Heuristic Argument: 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)

Is our model suﬃcient? ◮ Typically point, curve and surface singularities are present.

Theorem (K, Lemvig, Lim; 2010): The optimal sparse approximation result extends to a class of cartoon-like 3D images with only piecewise smooth C α boundaries separating C β ∩ Hβ functions.

Surprise 1: ◮ The optimal rate is still the same when introducing 0- and 1-dimensional singularities.

Surprise 2: ◮ Pyramid-adapted shearlets do achieve the optimal approximation rate. ◮ ‘Needle-like’ shearlets are not necessary for optimal sparse approximation in 3D.

Dream: ◮ Generators of the form

ψ(x1, x2, x3)= η(x1)φ(x2)φ(x3)

for the pyramid-adapted shearlets, where η is a 1D wavelet and φ is a scaling/bump function. ◮ These generators support fast decomposition algorithms.

Theorem (K, Lim; 2010): Let φ, ψ, ψ,˜ ψ˘ ∈ L2(R3) and for ψ suppose that:

3 δ −γ |ψˆ(ξ)|≤ C min(1, |ξ1| ) min(1, |ξi | ), i=1 where δ > 2γ > 6 and

ˆ T j 2 |ψ(Sk A ξ)| ≥ A > 0, j,k

and similar for ψ˜ and ψ˘.

Then there exists a sampling constant c0 > 0 such that 2 3 SH(φ, ψ, ψ,˜ ψ˘; c; α) forms a frame for L (R ) for c ≤ c0.

Gitta Kutyniok, Jakob Lemvig, & Wang-Q Lim Compactly Supported Shearlets Applied Harmonic Analysis and Imaging Sciences Compactly Supported Shearlet Systems Denoising Applications Image Separation Conclusions Recent Approaches to Fast 2D Shearlet Transforms

◮ Filter-based implementation (Easley, Labate, Lim; 2009) ◮ Separable Shearlet Transform (Lim; 2009) −→ www.ShearLab.org ◮ Decomposition based on Adaptive Subdivision Schemes (K, Sauer; 2009) ◮ Shearlet Unitary Extension Principle (Han, K, Shen; 2009) ◮ Rationally designed Digital Shearlet Transform (Donoho, K, Shahram, Zhuang; 2010) −→ www.ShearLab.org

Original Noisy (20.17dB)

Curvelets (28.70dB / 7.22 sec) Shearlets (29.20dB / 5.56 sec)

(L. Demanet, L. Ying; CurveLab 2.1.2, 2008)

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(Source : Brandt, K, Lim, S¨undermann; 2010) Theory: Geometric Separation (Donoho, K; 2009)

Gitta Kutyniok, Jakob Lemvig, & Wang-Q Lim Compactly Supported Shearlets Applied Harmonic Analysis and Imaging Sciences Compactly Supported Shearlet Systems Applications Conclusions What to take Home...?

◮ Anisotropic features in multivariate data require special eﬃcient encoding strategies. ◮ The shearlet theory is perfectly suited to this problem and is already very well established in the band-limited case. ◮ One main advantage of shearlets is that they provide a uniﬁed treatment of the continuum and digital setting. ◮ We introduced a theory for compactly supported shearlets for 2D and 3D data encompassing: ◮ Compactly supported shearlet frames, ◮ Explicit estimates for frame bounds, ◮ Optimal sparse approximation of cartoon-like images.

Gitta Kutyniok, Jakob Lemvig, & Wang-Q Lim Compactly Supported Shearlets Applied Harmonic Analysis and Imaging Sciences Compactly Supported Shearlet Systems Applications Conclusions

THANK YOU!

References available at:

www.analysis.uos.de, www.shearlet.org and www.shearlab.org

Applied Analysis Group University of Osnabrück

Gitta Kutyniok, Jakob Lemvig, & Wang-Q Lim Compactly Supported Shearlets