Curvelet

To construct a basic curvelet ${\displaystyle \phi }$ and provide a tiling of the 2-D frequency space, two main ideas should be followed:

1. Consider polar coordinates in frequency domain
2. Construct curvelet elements being locally supported near wedges

The number of wedges is ${\displaystyle N_{j}=4\cdot 2^{\left\lceil {\frac {j}{2}}\right\rceil }}$ at the scale ${\displaystyle 2^{-j}}$, i.e., it doubles in each second circular ring.

Let ${\displaystyle {\boldsymbol {\xi }}=\left(\xi _{1},\xi _{2}\right)^{T}}$ be the variable in frequency domain, and ${\displaystyle r={\sqrt {\xi _{1}^{2}+\xi _{2}^{2}}},\omega =\arctan {\frac {\xi _{1}}{\xi _{2}}}}$ be the polar coordinates in the frequency domain.

We use the ansatz for the dilated basic curvelets in polar coordinates:
${\displaystyle {\hat {\phi }}_{j,0,0}:=2^{\frac {-3j}{4}}W(2^{-j}r){\tilde {V}}_{N_{j}}(\omega ),r\geq 0,\omega \in [0,2\pi ),j\in N_{0}}$

To construct a basic curvelet with compact support near a ″basic wedge″, the two windows ${\displaystyle W}$ and ${\displaystyle {\tilde {V}}_{N_{j}}}$need to have compact support. Here, we can simply take ${\displaystyle W(r)}$ to cover ${\displaystyle (0,\infty )}$with dilated curvelets and ${\displaystyle {\tilde {V}}_{N_{j}}}$ such that each circular ring is covered by the translations of ${\displaystyle {\tilde {V}}_{N_{j}}}$ .

Then the admissibility yields
${\displaystyle \sum _{j=-\infty }^{\infty }\left|W(2^{-j}r)\right|^{2}=1,r\in (0,\infty ).}$_{see Window Functions for more information}

For tiling a circular ring into ${\displaystyle N}$ wedges, where ${\displaystyle N}$ is an arbitrary positive integer, we need a ${\displaystyle 2\pi }$-periodic nonnegative window ${\displaystyle {\tilde {V}}_{N}}$ with support inside ${\displaystyle \left[{\frac {-2\pi }{N}},{\frac {2\pi }{N}}\right]}$ such that
${\displaystyle \sum _{l=0}^{N-1}{\tilde {V}}_{N}^{2}\left(\omega -{\frac {2\pi l}{N}}\right)=1}$,
for all ${\displaystyle \omega \in \left[0,2\pi \right)}$, ${\displaystyle {\tilde {V}}_{N}}$ can be simply constructed as ${\displaystyle 2\pi }$-periodizations of a scaled window ${\displaystyle V\left({\frac {N\omega }{2\pi }}\right)}$.

Then, it follows that
${\displaystyle \sum _{l=0}^{N_{j}-1}\left|2^{\frac {3j}{4}}{\hat {\phi }}_{j,0,0}\left(r,\omega -{\frac {2\pi l}{N_{j}}}\right)\right|^{2}=\left|W(2^{-j}r)\right|^{2}\sum _{l=0}^{N_{j}-1}{\tilde {V}}_{N_{j}}^{2}\left(\omega -{\frac {2\pi l}{N}}\right)=\left|W(2^{-j}r)\right|^{2}}$

For a complete covering of the frequency plane including the region around zero, we need to define a low pass element
${\displaystyle {\hat {\phi }}_{-1}:=W_{0}(\left|\xi \right|)}$ with
${\displaystyle W_{0}^{2}(r)^{2}:=1-\sum _{j=0}^{\infty }W(2^{-j}r)^{2}}$
that is supported on the unit circle, and where we do not consider any rotation.

• Image Processing
• Seismic Exploration
• Fluid Mechanics
• PDEs Solving
• Compressed Sensing

• Shearlet transform
• Bandelet transform
• Contourlet transform
• Fresnelet transform
• Chirplet transform
• Noiselet transform
• Scale space

• E. Candès and D. Donoho, "Curvelets – a surprisingly effective nonadaptive representation for objects with edges." In: A. Cohen, C. Rabut and L. Schumaker, Editors, Curves and Surface Fitting: Saint-Malo 1999, Vanderbilt University Press, Nashville (2000), pp. 105–120.
• Majumdar Angshul Bangla Basic Character Recognition using Digital Curvelet Transform Journal of Pattern Recognition Research (JPRR), Vol 2. (1) 2007 p. 17-26
• Emmanuel Candes, Laurent Demanet, David Donoho and Lexing Ying Fast Discrete Curvelet Transforms
• Jianwei Ma, Gerlind Plonka, The Curvelet Transform: IEEE Signal Processing Magazine, 2010, 27 (2), 118-133.
• Jean-Luc Starck, Emmanuel J. Candès, and David L. Donoho, The Curvelet Transform for Image Denoising,: IEEE Transactions on Image Processing, Vol. 11, No. 6, June 2002

• Curvelet.org homepage