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Shearlet

In applied mathematical analysis, shearlets are a multiscale framework which allows efficient encoding of anisotropic features in multivariate problem classes. Originally, shearlets were introduced in 2006 for the analysis and sparse approximation of functions f ∈ L 2 ( R 2 ) {\displaystyle f\in L^{2}(\mathbb {R} ^{2})} . They are a natural extension of wavelets, to accommodate the fact that multivariate functions are typically governed by anisotropic features such as edges in images, since wavelets, as isotropic objects, are not capable of capturing such phenomena.

Shearlets are constructed by parabolic scaling, shearing, and translation applied to a few generating functions. At fine scales, they are essentially supported within skinny and directional ridges following the parabolic scaling law, which reads length² ≈ width. Similar to wavelets, shearlets arise from the affine group and allow a unified treatment of the continuum and digital situation leading to faithful implementations. Although they do not constitute an orthonormal basis for L 2 ( R 2 ) {\displaystyle L^{2}(\mathbb {R} ^{2})} , they still form a frame allowing stable expansions of arbitrary functions f ∈ L 2 ( R 2 ) {\displaystyle f\in L^{2}(\mathbb {R} ^{2})} .

One of the most important properties of shearlets is their ability to provide optimally sparse approximations (in the sense of optimality in ) for cartoon-like functions f {\displaystyle f} . In imaging sciences, cartoon-like functions serve as a model for anisotropic features and are compactly supported in [ 0 , 1 ] 2 {\displaystyle [0,1]^{2}} while being C 2 {\displaystyle C^{2}} apart from a closed piecewise C 2 {\displaystyle C^{2}} singularity curve with bounded curvature. The decay rate of the L 2 {\displaystyle L^{2}} -error of the N {\displaystyle N} -term shearlet approximation obtained by taking the N {\displaystyle N} largest coefficients from the shearlet expansion is in fact optimal up to a log-factor:

‖ f − f N ‖ L 2 2 ≤ C N − 2 ( log ⁡ N ) 3 , N → ∞ , {\displaystyle \|f-f_{N}\|_{L^{2}}^{2}\leq CN^{-2}(\log N)^{3},\quad N\to \infty ,}

where the constant C {\displaystyle C} depends only on the maximum curvature of the singularity curve and the maximum magnitudes of f {\displaystyle f} , f ′ {\displaystyle f'} and f ″ . {\displaystyle f''.} This approximation rate significantly improves the best N {\displaystyle N} -term approximation rate of wavelets providing only O ( N − 1 ) {\displaystyle O(N^{-1})} for such class of functions.

Shearlets are to date the only directional representation system that provides sparse approximation of anisotropic features while providing a unified treatment of the continuum and digital realm that allows faithful implementation. Extensions of shearlet systems to L 2 ( R d ) , d ≥ 2 {\displaystyle L^{2}(\mathbb {R} ^{d}),d\geq 2} are also available. A comprehensive presentation of the theory and applications of shearlets can be found in.

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Definition

Continuous shearlet systems

The construction of continuous shearlet systems is based on parabolic scaling matrices

A a = [ a 0 0 a 1 / 2 ] , a > 0 {\displaystyle A_{a}={\begin{bmatrix}a&0\\0&a^{1/2}\end{bmatrix}},\quad a>0}

as a mean to change the resolution, on shear matrices

S s = [ 1 s 0 1 ] , s ∈ R {\displaystyle S_{s}={\begin{bmatrix}1&s\\0&1\end{bmatrix}},\quad s\in \mathbb {R} }

as a means to change the orientation, and finally on translations to change the positioning. In comparison to curvelets, shearlets use shearings instead of rotations, the advantage being that the shear operator S s {\displaystyle S_{s}} leaves the integer lattice invariant in case s ∈ Z {\displaystyle s\in \mathbb {Z} } , i.e., S s Z 2 ⊆ Z 2 . {\displaystyle S_{s}\mathbb {Z} ^{2}\subseteq \mathbb {Z} ^{2}.} This indeed allows a unified treatment of the continuum and digital realm, thereby guaranteeing a faithful digital implementation.

For ψ ∈ L 2 ( R 2 ) {\displaystyle \psi \in L^{2}(\mathbb {R} ^{2})} the continuous shearlet system generated by ψ {\displaystyle \psi } is then defined as

SH c o n t ⁡ ( ψ ) = { ψ a , s , t = a 3 / 4 ψ ( S s A a ( ⋅ − t ) ) ∣ a > 0 , s ∈ R , t ∈ R 2 } , {\displaystyle \operatorname {SH} _{\mathrm {cont} }(\psi )=\{\psi _{a,s,t}=a^{3/4}\psi (S_{s}A_{a}(\cdot -t))\mid a>0,s\in \mathbb {R} ,t\in \mathbb {R} ^{2}\},}

and the corresponding continuous shearlet transform is given by the map

f ↦ S H ψ f ( a , s , t ) = ⟨ f , ψ a , s , t ⟩ , f ∈ L 2 ( R 2 ) , ( a , s , t ) ∈ R > 0 × R × R 2 . {\displaystyle f\mapsto {\mathcal {SH}}_{\psi }f(a,s,t)=\langle f,\psi _{a,s,t}\rangle ,\quad f\in L^{2}(\mathbb {R} ^{2}),\quad (a,s,t)\in \mathbb {R} _{>0}\times \mathbb {R} \times \mathbb {R} ^{2}.}

Discrete shearlet systems

A discrete version of shearlet systems can be directly obtained from SH c o n t ⁡ ( ψ ) {\displaystyle \operatorname {SH} _{\mathrm {cont} }(\psi )} by discretizing the parameter set R > 0 × R × R 2 . {\displaystyle \mathbb {R} _{>0}\times \mathbb {R} \times \mathbb {R} ^{2}.} There are numerous approaches for this but the most popular one is given by

{ ( 2 j , k , A 2 j − 1 S k − 1 m ) ∣ j ∈ Z , k ∈ Z , m ∈ Z 2 } ⊆ R > 0 × R × R 2 . {\displaystyle \{(2^{j},k,A_{2^{j}}^{-1}S_{k}^{-1}m)\mid j\in \mathbb {Z} ,k\in \mathbb {Z} ,m\in \mathbb {Z} ^{2}\}\subseteq \mathbb {R} _{>0}\times \mathbb {R} \times \mathbb {R} ^{2}.}

From this, the discrete shearlet system associated with the shearlet generator ψ {\displaystyle \psi } is defined by

SH ⁡ ( ψ ) = { ψ j , k , m = 2 3 j / 4 ψ ( S k A 2 j ⋅ − m ) ∣ j ∈ Z , k ∈ Z , m ∈ Z 2 } , {\displaystyle \operatorname {SH} (\psi )=\{\psi _{j,k,m}=2^{3j/4}\psi (S_{k}A_{2^{j}}\cdot {}-m)\mid j\in \mathbb {Z} ,k\in \mathbb {Z} ,m\in \mathbb {Z} ^{2}\},}

and the associated discrete shearlet transform is defined by

f ↦ S H ψ f ( j , k , m ) = ⟨ f , ψ j , k , m ⟩ , f ∈ L 2 ( R 2 ) , ( j , k , m ) ∈ Z × Z × Z 2 . {\displaystyle f\mapsto {\mathcal {SH}}_{\psi }f(j,k,m)=\langle f,\psi _{j,k,m}\rangle ,\quad f\in L^{2}(\mathbb {R} ^{2}),\quad (j,k,m)\in \mathbb {Z} \times \mathbb {Z} \times \mathbb {Z} ^{2}.}

Examples

Let ψ 1 ∈ L 2 ( R ) {\displaystyle \psi _{1}\in L^{2}(\mathbb {R} )} be a function satisfying the discrete Calderón condition, i.e.,

∑ j ∈ Z | ψ ^ 1 ( 2 − j ξ ) | 2 = 1 , for a.e.  ξ ∈ R , {\displaystyle \sum _{j\in \mathbb {Z} }|{\hat {\psi }}_{1}(2^{-j}\xi )|^{2}=1,{\text{for a.e. }}\xi \in \mathbb {R} ,}

with ψ ^ 1 ∈ C ∞ ( R ) {\displaystyle {\hat {\psi }}_{1}\in C^{\infty }(\mathbb {R} )} and supp ⁡ ψ ^ 1 ⊆ [ − 1 2 , − 1 16 ] ∪ [ 1 16 , 1 2 ] , {\displaystyle \operatorname {supp} {\hat {\psi }}_{1}\subseteq [-{\tfrac {1}{2}},-{\tfrac {1}{16}}]\cup [{\tfrac {1}{16}},{\tfrac {1}{2}}],} where ψ ^ 1 {\displaystyle {\hat {\psi }}_{1}} denotes the Fourier transform of ψ 1 . {\displaystyle \psi _{1}.} For instance, one can choose ψ 1 {\displaystyle \psi _{1}} to be a Meyer wavelet. Furthermore, let ψ 2 ∈ L 2 ( R ) {\displaystyle \psi _{2}\in L^{2}(\mathbb {R} )} be such that ψ ^ 2 ∈ C ∞ ( R ) , {\displaystyle {\hat {\psi }}_{2}\in C^{\infty }(\mathbb {R} ),} supp ⁡ ψ ^ 2 ⊆ [ − 1 , 1 ] {\displaystyle \operatorname {supp} {\hat {\psi }}_{2}\subseteq [-1,1]} and

∑ k = − 1 1 | ψ ^ 2 ( ξ + k ) | 2 = 1 , for a.e.  ξ ∈ [ − 1 , 1 ] . {\displaystyle \sum _{k=-1}^{1}|{\hat {\psi }}_{2}(\xi +k)|^{2}=1,{\text{for a.e. }}\xi \in \left[-1,1\right].}

One typically chooses ψ ^ 2 {\displaystyle {\hat {\psi }}_{2}} to be a smooth bump function. Then ψ ∈ L 2 ( R 2 ) {\displaystyle \psi \in L^{2}(\mathbb {R} ^{2})} given by

ψ ^ ( ξ ) = ψ ^ 1 ( ξ 1 ) ψ ^ 2 ( ξ 2 ξ 1 ) , ξ = ( ξ 1 , ξ 2 ) ∈ R 2 , {\displaystyle {\hat {\psi }}(\xi )={\hat {\psi }}_{1}(\xi _{1}){\hat {\psi }}_{2}\left({\tfrac {\xi _{2}}{\xi _{1}}}\right),\quad \xi =(\xi _{1},\xi _{2})\in \mathbb {R} ^{2},}

is called a classical shearlet. It can be shown that the corresponding discrete shearlet system SH ⁡ ( ψ ) {\displaystyle \operatorname {SH} (\psi )} constitutes a Parseval frame for L 2 ( R 2 ) {\displaystyle L^{2}(\mathbb {R} ^{2})} consisting of bandlimited functions.6

Another example are compactly supported shearlet systems, where a compactly supported function ψ ∈ L 2 ( R 2 ) {\displaystyle \psi \in L^{2}(\mathbb {R} ^{2})} can be chosen so that SH ⁡ ( ψ ) {\displaystyle \operatorname {SH} (\psi )} forms a frame for L 2 ( R 2 ) {\displaystyle L^{2}(\mathbb {R} ^{2})} .78910 In this case, all shearlet elements in SH ⁡ ( ψ ) {\displaystyle \operatorname {SH} (\psi )} are compactly supported providing superior spatial localization compared to the classical shearlets, which are bandlimited. Although a compactly supported shearlet system does not generally form a Parseval frame, any function f ∈ L 2 ( R 2 ) {\displaystyle f\in L^{2}(\mathbb {R} ^{2})} can be represented by the shearlet expansion due to its frame property.

Cone-adapted shearlets

One drawback of shearlets defined as above is the directional bias of shearlet elements associated with large shearing parameters. This effect is already recognizable in the frequency tiling of classical shearlets (see Figure in Section #Examples), where the frequency support of a shearlet increasingly aligns along the ξ 2 {\displaystyle \xi _{2}} -axis as the shearing parameter s {\displaystyle s} goes to infinity. This causes serious problems when analyzing a function whose Fourier transform is concentrated around the ξ 2 {\displaystyle \xi _{2}} -axis.

To deal with this problem, the frequency domain is divided into a low-frequency part and two conic regions (see Figure):

R = { ( ξ 1 , ξ 2 ) ∈ R 2 ∣ | ξ 1 | , | ξ 2 | ≤ 1 } , C h = { ( ξ 1 , ξ 2 ) ∈ R 2 ∣ | ξ 2 / ξ 1 | ≤ 1 , | ξ 1 | > 1 } , C v = { ( ξ 1 , ξ 2 ) ∈ R 2 ∣ | ξ 1 / ξ 2 | ≤ 1 , | ξ 2 | > 1 } . {\displaystyle {\begin{aligned}{\mathcal {R}}&=\left\{(\xi _{1},\xi _{2})\in \mathbb {R} ^{2}\mid |\xi _{1}|,|\xi _{2}|\leq 1\right\},\\{\mathcal {C}}_{\mathrm {h} }&=\left\{(\xi _{1},\xi _{2})\in \mathbb {R} ^{2}\mid |\xi _{2}/\xi _{1}|\leq 1,|\xi _{1}|>1\right\},\\{\mathcal {C}}_{\mathrm {v} }&=\left\{(\xi _{1},\xi _{2})\in \mathbb {R} ^{2}\mid |\xi _{1}/\xi _{2}|\leq 1,|\xi _{2}|>1\right\}.\end{aligned}}}

The associated cone-adapted discrete shearlet system consists of three parts, each one corresponding to one of these frequency domains. It is generated by three functions ϕ , ψ , ψ ~ ∈ L 2 ( R 2 ) {\displaystyle \phi ,\psi ,{\tilde {\psi }}\in L^{2}(\mathbb {R} ^{2})} and a lattice sampling factor c = ( c 1 , c 2 ) ∈ ( R > 0 ) 2 : {\displaystyle c=(c_{1},c_{2})\in (\mathbb {R} _{>0})^{2}:}

SH ⁡ ( ϕ , ψ , ψ ~ ; c ) = Φ ( ϕ ; c 1 ) ∪ Ψ ( ψ ; c ) ∪ Ψ ~ ( ψ ~ ; c ) , {\displaystyle \operatorname {SH} (\phi ,\psi ,{\tilde {\psi }};c)=\Phi (\phi ;c_{1})\cup \Psi (\psi ;c)\cup {\tilde {\Psi }}({\tilde {\psi }};c),}

where

Φ ( ϕ ; c 1 ) = { ϕ m = ϕ ( ⋅ − c 1 m ) ∣ m ∈ Z 2 } , Ψ ( ψ ; c ) = { ψ j , k , m = 2 3 j / 4 ψ ( S k A 2 j ⋅ − M c m ) ∣ j ≥ 0 , | k | ≤ ⌈ 2 j / 2 ⌉ , m ∈ Z 2 } , Ψ ~ ( ψ ~ ; c ) = { ψ ~ j , k , m = 2 3 j / 4 ψ ( S ~ k A ~ 2 j ⋅ − M ~ c m ) ∣ j ≥ 0 , | k | ≤ ⌈ 2 j / 2 ⌉ , m ∈ Z 2 } , {\displaystyle {\begin{aligned}\Phi (\phi ;c_{1})&=\{\phi _{m}=\phi (\cdot {}-c_{1}m)\mid m\in \mathbb {Z} ^{2}\},\\\Psi (\psi ;c)&=\{\psi _{j,k,m}=2^{3j/4}\psi (S_{k}A_{2^{j}}\cdot {}-M_{c}m)\mid j\geq 0,|k|\leq \lceil 2^{j/2}\rceil ,m\in \mathbb {Z} ^{2}\},\\{\tilde {\Psi }}({\tilde {\psi }};c)&=\{{\tilde {\psi }}_{j,k,m}=2^{3j/4}\psi ({\tilde {S}}_{k}{\tilde {A}}_{2^{j}}\cdot {}-{\tilde {M}}_{c}m)\mid j\geq 0,|k|\leq \lceil 2^{j/2}\rceil ,m\in \mathbb {Z} ^{2}\},\end{aligned}}}

with

A ~ a = [ a 1 / 2 0 0 a ] , a > 0 , S ~ s = [ 1 0 s 1 ] , s ∈ R , M c = [ c 1 0 0 c 2 ] , and M ~ c = [ c 2 0 0 c 1 ] . {\displaystyle {\begin{aligned}&{\tilde {A}}_{a}={\begin{bmatrix}a^{1/2}&0\\0&a\end{bmatrix}},\;a>0,\quad {\tilde {S}}_{s}={\begin{bmatrix}1&0\\s&1\end{bmatrix}},\;s\in \mathbb {R} ,\quad M_{c}={\begin{bmatrix}c_{1}&0\\0&c_{2}\end{bmatrix}},\quad {\text{and}}\quad {\tilde {M}}_{c}={\begin{bmatrix}c_{2}&0\\0&c_{1}\end{bmatrix}}.\end{aligned}}}

The systems Ψ ( ψ ) {\displaystyle \Psi (\psi )} and Ψ ~ ( ψ ~ ) {\displaystyle {\tilde {\Psi }}({\tilde {\psi }})} basically differ in the reversed roles of x 1 {\displaystyle x_{1}} and x 2 {\displaystyle x_{2}} . Thus, they correspond to the conic regions C h {\displaystyle {\mathcal {C}}_{\mathrm {h} }} and C v {\displaystyle {\mathcal {C}}_{\mathrm {v} }} , respectively. Finally, the scaling function ϕ {\displaystyle \phi } is associated with the low-frequency part R {\displaystyle {\mathcal {R}}} .

Applications

Generalizations and extensions

  • 3D-Shearlets 1415
  • α {\displaystyle \alpha } -Shearlets 16
  • Parabolic molecules 17
  • Cylindrical Shearlets 1819

See also

References

  1. Guo, Kanghui, Gitta Kutyniok, and Demetrio Labate. "Sparse multidimensional representations using anisotropic dilation and shear operators." Wavelets and Splines (Athens, GA, 2005), G. Chen and MJ Lai, eds., Nashboro Press, Nashville, TN (2006): 189–201. "PDF" (PDF). /wiki/Gitta_Kutyniok ↩

  2. Donoho, David Leigh. "Sparse components of images and optimal atomic decompositions." Constructive Approximation 17.3 (2001): 353–382. "PDF". CiteSeerX 10.1.1.379.8993. /wiki/CiteSeerX_(identifier) ↩

  3. Guo, Kanghui, and Demetrio Labate. "Optimally sparse multidimensional representation using shearlets." SIAM Journal on Mathematical Analysis 39.1 (2007): 298–318. "PDF" (PDF). http://www3.math.tu-berlin.de/numerik/mt/mt/www.shearlet.org/papers/OSMRuS.pdf ↩

  4. Kutyniok, Gitta, and Wang-Q Lim. "Compactly supported shearlets are optimally sparse." Journal of Approximation Theory 163.11 (2011): 1564–1589. "PDF" (PDF). /wiki/Gitta_Kutyniok ↩

  5. Kutyniok, Gitta, and Demetrio Labate, eds. Shearlets: Multiscale analysis for multivariate data. Springer, 2012, ISBN 0-8176-8315-1 /wiki/Gitta_Kutyniok ↩

  6. Kutyniok, Gitta, and Demetrio Labate, eds. Shearlets: Multiscale analysis for multivariate data. Springer, 2012, ISBN 0-8176-8315-1 /wiki/Gitta_Kutyniok ↩

  7. Kutyniok, Gitta, and Wang-Q Lim. "Compactly supported shearlets are optimally sparse." Journal of Approximation Theory 163.11 (2011): 1564–1589. "PDF" (PDF). /wiki/Gitta_Kutyniok ↩

  8. Kittipoom, Pisamai, Gitta Kutyniok, and Wang-Q Lim. "Construction of compactly supported shearlet frames." Constructive Approximation 35.1 (2012): 21–72. Kittipoom, P.; Kutyniok, G.; Lim, W. (2010). "PDF". arXiv:1003.5481 [math.FA]. /wiki/Gitta_Kutyniok ↩

  9. Kutyniok, Gitta, Jakob Lemvig, and Wang-Q Lim. "Optimally sparse approximations of 3D functions by compactly supported shearlet frames." SIAM Journal on Mathematical Analysis 44.4 (2012): 2962–3017. Kutyniok, Gitta; Lemvig, Jakob; Lim, Wang-Q (2011). "PDF". arXiv:1109.5993 [math.FA]. /wiki/Gitta_Kutyniok ↩

  10. Purnendu Banerjee and B. B. Chaudhuri, “Video Text Localization using Wavelet and Shearlet Transforms”, In Proc. SPIE 9021, Document Recognition and Retrieval XXI, 2014 (doi:10.1117/12.2036077).Banerjee, Purnendu; Chaudhuri, B. B. (2013). "Video text localization using wavelet and shearlet transforms". In Coüasnon, Bertrand; Ringger, Eric K (eds.). Document Recognition and Retrieval XXI. Vol. 9021. pp. 90210B. arXiv:1307.4990. doi:10.1117/12.2036077. S2CID 10659099. /wiki/ArXiv_(identifier) ↩

  11. Kutyniok, Gitta, and Demetrio Labate, eds. Shearlets: Multiscale analysis for multivariate data. Springer, 2012, ISBN 0-8176-8315-1 /wiki/Gitta_Kutyniok ↩

  12. Kutyniok, Gitta, and Demetrio Labate, eds. Shearlets: Multiscale analysis for multivariate data. Springer, 2012, ISBN 0-8176-8315-1 /wiki/Gitta_Kutyniok ↩

  13. Kutyniok, Gitta, and Demetrio Labate, eds. Shearlets: Multiscale analysis for multivariate data. Springer, 2012, ISBN 0-8176-8315-1 /wiki/Gitta_Kutyniok ↩

  14. Kutyniok, Gitta, Jakob Lemvig, and Wang-Q Lim. "Optimally sparse approximations of 3D functions by compactly supported shearlet frames." SIAM Journal on Mathematical Analysis 44.4 (2012): 2962–3017. Kutyniok, Gitta; Lemvig, Jakob; Lim, Wang-Q (2011). "PDF". arXiv:1109.5993 [math.FA]. /wiki/Gitta_Kutyniok ↩

  15. Guo, Kanghui, and Demetrio Labate. "The construction of smooth Parseval frames of shearlets." Mathematical Modelling of Natural Phenomena 8.01 (2013): 82–105. "PDF" (PDF). http://www3.math.tu-berlin.de/numerik/mt/www.shearlet.org/papers/shear_construction.pdf ↩

  16. Kutyniok, Gitta, Jakob Lemvig, and Wang-Q Lim. "Optimally sparse approximations of 3D functions by compactly supported shearlet frames." SIAM Journal on Mathematical Analysis 44.4 (2012): 2962–3017. Kutyniok, Gitta; Lemvig, Jakob; Lim, Wang-Q (2011). "PDF". arXiv:1109.5993 [math.FA]. /wiki/Gitta_Kutyniok ↩

  17. Grohs, Philipp and Kutyniok, Gitta. "Parabolic molecules." Foundations of Computational Mathematics (to appear) Grohs, Philipp; Kutyniok, Gitta (2012). "PDF". arXiv:1206.1958 [math.FA]. /wiki/Gitta_Kutyniok ↩

  18. Easley, Glenn R.; Guo, Kanghui; Labate, Demetrio; Pahari, Basanta R. (2020-08-10). "Optimally Sparse Representations of Cartoon-Like Cylindrical Data". The Journal of Geometric Analysis. 39 (9): 8926–8946. doi:10.1007/s12220-020-00493-0. S2CID 221675372. Retrieved 2022-01-22. https://link.springer.com/article/10.1007/s12220-020-00493-0 ↩

  19. Bernhard, Bernhard G.; Labate, Demetrio; Pahari, Basanta R. (2019-10-29). "Smooth projections and the construction of smooth Parseval frames of shearlets". Advances in Computational Mathematics. 45 (5–6): 3241–3264. doi:10.1007/s10444-019-09736-3. S2CID 210118010. Retrieved 2022-01-22. https://link.springer.com/article/10.1007/s10444-019-09736-3 ↩