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Shearlet
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<h2 id="definition">Definition</h2> <h3>Continuous shearlet systems</h3> <p>The construction of continuous shearlet systems is based on <i>parabolic scaling matrices</i> </p> 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} <p>as a mean to change the resolution, on <i>shear matrices</i> </p> S s = [ 1 s 0 1 ] , s ∈ R {\displaystyle S_{s}={\begin{bmatrix}1&s\\0&1\end{bmatrix}},\quad s\in \mathbb {R} } <p>as a means to change the orientation, and finally on translations to change the positioning. In comparison to <a href="/facts/Curvelet/MEdv1vyr">curvelets</a>, shearlets use shearings instead of rotations, the advantage being that the shear operator S s {\displaystyle S_{s}} leaves the <a href="/facts/Integer_lattice/h4FP4pjm">integer lattice</a> 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. </p><p>For ψ ∈ L 2 ( R 2 ) {\displaystyle \psi \in L^{2}(\mathbb {R} ^{2})} the <i>continuous shearlet system</i> generated by ψ {\displaystyle \psi } is then defined as </p> 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}\},} <p>and the corresponding <i>continuous shearlet transform</i> is given by the map </p> 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}.} <h3>Discrete shearlet systems</h3> <p>A discrete version of shearlet systems can be directly obtained from SH c o n t ( ψ ) {\displaystyle \operatorname {SH} _{\mathrm {cont} }(\psi )} by <a href="/facts/Discretization/JvjSzwyk">discretizing</a> 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 </p> { ( 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}.} <p>From this, the <i>discrete shearlet system</i> associated with the shearlet generator ψ {\displaystyle \psi } is defined by </p> 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}\},} <p>and the associated <i>discrete shearlet transform</i> is defined by </p> 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}.} <h2 id="examples">Examples</h2> <p>Let ψ 1 ∈ L 2 ( R ) {\displaystyle \psi _{1}\in L^{2}(\mathbb {R} )} be a function satisfying the <i>discrete Calderón condition</i>, i.e., </p> ∑ 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} ,} <p>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 <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a> of ψ 1 . {\displaystyle \psi _{1}.} For instance, one can choose ψ 1 {\displaystyle \psi _{1}} to be a <a href="/facts/Meyer_wavelet/bd11PhqV">Meyer wavelet</a>. 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 </p> ∑ 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].} <p>One typically chooses ψ ^ 2 {\displaystyle {\hat {\psi }}_{2}} to be a smooth <a href="/facts/Bump_function/ExdBtMHN">bump function</a>. Then ψ ∈ L 2 ( R 2 ) {\displaystyle \psi \in L^{2}(\mathbb {R} ^{2})} given by </p> 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},} <p>is called a <i>classical shearlet</i>. It can be shown that the corresponding discrete shearlet system SH ( ψ ) {\displaystyle \operatorname {SH} (\psi )} constitutes a <a href="/facts/Parseval_frame/xn70xhDg">Parseval frame</a> for L 2 ( R 2 ) {\displaystyle L^{2}(\mathbb {R} ^{2})} consisting of <a href="/facts/Bandlimited/LrYmxzlA">bandlimited</a> functions.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p><p>Another example are <a href="/facts/Compact_space/d0cgXJH7">compactly</a> <a href="/facts/Support_(mathematics)/BTuMGbsb">supported</a> 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 <a href="/facts/Frame_of_a_vector_space/xn70xhDg">frame</a> for L 2 ( R 2 ) {\displaystyle L^{2}(\mathbb {R} ^{2})} .<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a><a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a><a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> 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. </p> <h2 id="cone-adapted-shearlets">Cone-adapted shearlets</h2> <p>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. </p> <p>To deal with this problem, the frequency domain is divided into a low-frequency part and two conic regions (see Figure): </p> 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}}} <p>The associated <i>cone-adapted discrete shearlet system</i> 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 <i>lattice <a href="/facts/Sampling_(signal_processing)/xxck7aLN">sampling</a></i> factor c = ( c 1 , c 2 ) ∈ ( R > 0 ) 2 : {\displaystyle c=(c_{1},c_{2})\in (\mathbb {R} _{>0})^{2}:} </p> 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),} <p>where </p> 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}}} <p>with </p> 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}}} <p>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 <i>scaling function</i> ϕ {\displaystyle \phi } is associated with the low-frequency part R {\displaystyle {\mathcal {R}}} . </p> <h2 id="applications">Applications</h2> <ul><li><a href="/facts/Image_processing/zenalNCf">Image processing</a> and <a href="/facts/Computer_sciences/lN3pEyPz">computer sciences</a><a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> <ul><li><a href="/facts/Image_denoising/9jrDQnAN">Denoising</a></li> <li><a href="/facts/Inverse_problems/Fe01pQ9E">Inverse problems</a></li> <li><a href="/facts/Image_enhancement/l5LvzVBa">Image enhancement</a></li> <li><a href="/facts/Edge_detection/9VplpIcS">Edge detection</a></li> <li><a href="/facts/Inpainting/5fHD96N8">Inpainting</a></li> <li>Image separation</li></ul></li> <li><a href="/facts/Partial_differential_equation/DyPsBlv7">PDEs</a><a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> <ul><li>Resolution of the <a href="/facts/Wavefront_set/As2j99Tn">wavefront set</a></li> <li><a href="/facts/Transport_equations/GVav2Vet">Transport equations</a></li></ul></li> <li><a href="/facts/Coorbit_theory/K9S90tJR">Coorbit theory</a>, characterization of <a href="/facts/Besov_space/MdUkSCel">smoothness spaces</a><a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a></li> <li><a href="/facts/Differential_geometry/c04XmyLu">Differential geometry</a>: <a href="/facts/Manifold_learning/1lL1Y0vN">manifold learning</a></li></ul> <h2 id="generalizations-and-extensions">Generalizations and extensions</h2> <ul><li>3D-Shearlets <a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a><a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a></li> <li> α {\displaystyle \alpha } -Shearlets <a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a></li> <li>Parabolic molecules <a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a></li> <li>Cylindrical Shearlets <a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a><a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a></li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Wavelet/3XcX4uIP">Wavelet transform</a></li> <li><a href="/facts/Curvelet/MEdv1vyr">Curvelet transform</a></li> <li><a href="/facts/Contourlet/FfF5TWaJ">Contourlet transform</a></li> <li><a href="/facts/Bandelet/95QS5SVX">Bandelet transform</a></li> <li><a href="/facts/Chirplet_transform/CKQP3Imn">Chirplet transform</a></li> <li><a href="/facts/Noiselet/o5JLTFRK">Noiselet transform</a></li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://www.math.tu-berlin.de/fachgebiete_ag_modnumdiff/angewandtefunktionalanalysis/v-menue/mitarbeiter/kutyniok/v-menue/home/parameter/en/">Homepage of Gitta Kutyniok</a></li> <li><a href="http://www.math.uh.edu/~dlabate/">Homepage of Demetrio Labate</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>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). <a href="/wiki/Gitta_Kutyniok" target="_blank">/wiki/Gitta_Kutyniok</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>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. <a href="/wiki/CiteSeerX_(identifier)" target="_blank">/wiki/CiteSeerX_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Guo, Kanghui, and Demetrio Labate. "Optimally sparse multidimensional representation using shearlets." SIAM Journal on Mathematical Analysis 39.1 (2007): 298–318. "PDF" (PDF). <a href="http://www3.math.tu-berlin.de/numerik/mt/mt/www.shearlet.org/papers/OSMRuS.pdf" target="_blank">http://www3.math.tu-berlin.de/numerik/mt/mt/www.shearlet.org/papers/OSMRuS.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Kutyniok, Gitta, and Wang-Q Lim. "Compactly supported shearlets are optimally sparse." Journal of Approximation Theory 163.11 (2011): 1564–1589. "PDF" (PDF). <a href="/wiki/Gitta_Kutyniok" target="_blank">/wiki/Gitta_Kutyniok</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Kutyniok, Gitta, and Demetrio Labate, eds. Shearlets: Multiscale analysis for multivariate data. Springer, 2012, ISBN 0-8176-8315-1 <a href="/wiki/Gitta_Kutyniok" target="_blank">/wiki/Gitta_Kutyniok</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Kutyniok, Gitta, and Demetrio Labate, eds. Shearlets: Multiscale analysis for multivariate data. Springer, 2012, ISBN 0-8176-8315-1 <a href="/wiki/Gitta_Kutyniok" target="_blank">/wiki/Gitta_Kutyniok</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Kutyniok, Gitta, and Wang-Q Lim. "Compactly supported shearlets are optimally sparse." Journal of Approximation Theory 163.11 (2011): 1564–1589. "PDF" (PDF). <a href="/wiki/Gitta_Kutyniok" target="_blank">/wiki/Gitta_Kutyniok</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>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]. <a href="/wiki/Gitta_Kutyniok" target="_blank">/wiki/Gitta_Kutyniok</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>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]. <a href="/wiki/Gitta_Kutyniok" target="_blank">/wiki/Gitta_Kutyniok</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>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. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Kutyniok, Gitta, and Demetrio Labate, eds. Shearlets: Multiscale analysis for multivariate data. Springer, 2012, ISBN 0-8176-8315-1 <a href="/wiki/Gitta_Kutyniok" target="_blank">/wiki/Gitta_Kutyniok</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Kutyniok, Gitta, and Demetrio Labate, eds. Shearlets: Multiscale analysis for multivariate data. Springer, 2012, ISBN 0-8176-8315-1 <a href="/wiki/Gitta_Kutyniok" target="_blank">/wiki/Gitta_Kutyniok</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Kutyniok, Gitta, and Demetrio Labate, eds. Shearlets: Multiscale analysis for multivariate data. Springer, 2012, ISBN 0-8176-8315-1 <a href="/wiki/Gitta_Kutyniok" target="_blank">/wiki/Gitta_Kutyniok</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>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]. <a href="/wiki/Gitta_Kutyniok" target="_blank">/wiki/Gitta_Kutyniok</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>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). <a href="http://www3.math.tu-berlin.de/numerik/mt/www.shearlet.org/papers/shear_construction.pdf" target="_blank">http://www3.math.tu-berlin.de/numerik/mt/www.shearlet.org/papers/shear_construction.pdf</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>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]. <a href="/wiki/Gitta_Kutyniok" target="_blank">/wiki/Gitta_Kutyniok</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Grohs, Philipp and Kutyniok, Gitta. "Parabolic molecules." Foundations of Computational Mathematics (to appear) Grohs, Philipp; Kutyniok, Gitta (2012). "PDF". arXiv:1206.1958 [math.FA]. <a href="/wiki/Gitta_Kutyniok" target="_blank">/wiki/Gitta_Kutyniok</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>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. <a href="https://link.springer.com/article/10.1007/s12220-020-00493-0" target="_blank">https://link.springer.com/article/10.1007/s12220-020-00493-0</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>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. <a href="https://link.springer.com/article/10.1007/s10444-019-09736-3" target="_blank">https://link.springer.com/article/10.1007/s10444-019-09736-3</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> </ol>