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Curvelet
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<h2 id="curvelet-construction">Curvelet construction</h2> <p>To construct a basic curvelet ϕ {\displaystyle \phi } and provide a tiling of the 2-D frequency space, two main ideas should be followed: </p> <ol><li>Consider polar coordinates in frequency domain</li> <li>Construct curvelet elements being locally supported near wedges</li></ol> <p>The number of wedges is N j = 4 ⋅ 2 ⌈ j 2 ⌉ {\displaystyle N_{j}=4\cdot 2^{\left\lceil {\frac {j}{2}}\right\rceil }} at the scale 2 − j {\displaystyle 2^{-j}} , i.e., it doubles in each second circular ring. </p><p>Let ξ = ( ξ 1 , ξ 2 ) T {\displaystyle {\boldsymbol {\xi }}=\left(\xi _{1},\xi _{2}\right)^{T}} be the variable in frequency domain, and r = ξ 1 2 + ξ 2 2 , ω = arctan ξ 1 ξ 2 {\displaystyle r={\sqrt {\xi _{1}^{2}+\xi _{2}^{2}}},\omega =\arctan {\frac {\xi _{1}}{\xi _{2}}}} be the polar coordinates in the frequency domain. </p><p>We use the <a href="/facts/Ansatz/3epoB3Rg">ansatz</a> for the <i>dilated basic curvelets</i> in polar coordinates: ϕ ^ j , 0 , 0 := 2 − 3 j 4 W ( 2 − j r ) V ~ N j ( ω ) , r ≥ 0 , ω ∈ [ 0 , 2 π ) , j ∈ N 0 {\displaystyle {\hat {\phi }}_{j,0,0}:=2^{\frac {-3j}{4}}W(2^{-j}r){\tilde {V}}_{N_{j}}(\omega ),r\geq 0,\omega \in [0,2\pi ),j\in N_{0}} </p><p>To construct a basic curvelet with compact support near a ″basic wedge″, the two windows W {\displaystyle W} and V ~ N j {\displaystyle {\tilde {V}}_{N_{j}}} need to have compact support. Here, we can simply take W ( r ) {\displaystyle W(r)} to cover ( 0 , ∞ ) {\displaystyle (0,\infty )} with dilated curvelets and V ~ N j {\displaystyle {\tilde {V}}_{N_{j}}} such that each circular ring is covered by the translations of V ~ N j {\displaystyle {\tilde {V}}_{N_{j}}} . </p><p>Then the admissibility yields ∑ j = − ∞ ∞ | W ( 2 − j r ) | 2 = 1 , r ∈ ( 0 , ∞ ) . {\displaystyle \sum _{j=-\infty }^{\infty }\left|W(2^{-j}r)\right|^{2}=1,r\in (0,\infty ).} see <i><a href="/facts/Window_function/5Xj7h2ah">Window Functions</a></i> for more information For tiling a circular ring into N {\displaystyle N} wedges, where N {\displaystyle N} is an arbitrary positive integer, we need a 2 π {\displaystyle 2\pi } -periodic nonnegative window V ~ N {\displaystyle {\tilde {V}}_{N}} with support inside [ − 2 π N , 2 π N ] {\displaystyle \left[{\frac {-2\pi }{N}},{\frac {2\pi }{N}}\right]} such that ∑ l = 0 N − 1 V ~ N 2 ( ω − 2 π l N ) = 1 {\displaystyle \sum _{l=0}^{N-1}{\tilde {V}}_{N}^{2}\left(\omega -{\frac {2\pi l}{N}}\right)=1} , for all ω ∈ [ 0 , 2 π ) {\displaystyle \omega \in \left[0,2\pi \right)} , V ~ N {\displaystyle {\tilde {V}}_{N}} can be simply constructed as 2 π {\displaystyle 2\pi } -periodizations of a scaled window V ( N ω 2 π ) {\displaystyle V\left({\frac {N\omega }{2\pi }}\right)} . Then, it follows that ∑ l = 0 N j − 1 | 2 3 j 4 ϕ ^ j , 0 , 0 ( r , ω − 2 π l N j ) | 2 = | W ( 2 − j r ) | 2 ∑ l = 0 N j − 1 V ~ N j 2 ( ω − 2 π l N ) = | W ( 2 − j r ) | 2 {\displaystyle \sum _{l=0}^{N_{j}-1}\left|2^{\frac {3j}{4}}{\hat {\phi }}_{j,0,0}\left(r,\omega -{\frac {2\pi l}{N_{j}}}\right)\right|^{2}=\left|W(2^{-j}r)\right|^{2}\sum _{l=0}^{N_{j}-1}{\tilde {V}}_{N_{j}}^{2}\left(\omega -{\frac {2\pi l}{N}}\right)=\left|W(2^{-j}r)\right|^{2}} </p><p>For a complete covering of the frequency plane including the region around zero, we need to define a low pass element ϕ ^ − 1 := W 0 ( | ξ | ) {\displaystyle {\hat {\phi }}_{-1}:=W_{0}(\left|\xi \right|)} with W 0 2 ( r ) 2 := 1 − ∑ j = 0 ∞ W ( 2 − j r ) 2 {\displaystyle W_{0}^{2}(r)^{2}:=1-\sum _{j=0}^{\infty }W(2^{-j}r)^{2}} that is supported on the unit circle, and where we do not consider any rotation. </p> <h2 id="applications">Applications</h2> <ul><li><a href="/facts/Image_processing/zenalNCf">Image processing</a></li> <li>Seismic exploration</li> <li><a href="/facts/Fluid_mechanics/NIy4tRfE">Fluid mechanics</a></li> <li><a href="/facts/Partial_differential_equation/DyPsBlv7">PDEs</a> solving</li> <li><a href="/facts/Compressed_sensing/aR5DQm3n">Compressed sensing</a></li></ul> <h2 id="see-also">See also</h2> <ul><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/Contourlet/FfF5TWaJ">Contourlet transform</a></li> <li><a href="/facts/Fresnel_transform/lxyq42M4">Fresnelet transform</a></li> <li><a href="/facts/Noiselet/o5JLTFRK">Noiselet transform</a></li> <li><a href="/facts/Scale_space/XJNvTH1N">Scale space</a></li> <li><a href="/facts/Shearlet/g9uasWPy">Shearlet transform</a></li></ul> <ul><li><a href="/facts/Emmanuel_Cand%25C3%25A8s/bR3yxvxp">E. Candès</a> and D. Donoho, "Curvelets – a surprisingly effective nonadaptive representation for objects with edges." In: A. Cohen, C. Rabut and L. Schumaker, Editors, <i>Curves and Surface Fitting</i>: Saint-Malo 1999, Vanderbilt University Press, Nashville (2000), pp. 105–120.</li> <li>Majumdar Angshul <a href="http://jprr.org/index.php/jprr/article/view/27">Bangla Basic Character Recognition using Digital Curvelet Transform</a> Journal of Pattern Recognition Research (<a href="http://www.jprr.org">JPRR</a>), Vol 2. (1) 2007 p. 17-26</li> <li>Emmanuel Candes, Laurent Demanet, David Donoho and Lexing Ying <a href="http://www.curvelet.org/papers/FDCT.pdf">Fast Discrete Curvelet Transforms</a></li> <li>Jianwei Ma, <a href="/facts/Gerlind_Plonka/PfN7Nzrs">Gerlind Plonka</a>, <i>The Curvelet Transform</i>: IEEE Signal Processing Magazine, 2010, 27 (2), 118-133.</li> <li>Jean-Luc Starck, Emmanuel J. Candès, and David L. Donoho, <i>The Curvelet Transform for Image Denoising,</i>: IEEE Transactions on Image Processing, Vol. 11, No. 6, June 2002</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://curvelet.org/">Curvelet.org homepage</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Candès, Emmanuel; Demanet, Laurent; Donoho, David; Ying, Lexing (Jan 2006). "Fast Discrete Curvelet Transforms". Multiscale Modeling & Simulation. 5 (3): 861–899. doi:10.1137/05064182X. ISSN 1540-3459. <a href="http://epubs.siam.org/doi/10.1137/05064182X" target="_blank">http://epubs.siam.org/doi/10.1137/05064182X</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>