The Curvelet transform is a higher dimensional generalization of the
Wavelet transform designed to represent images at different scales and
different angles. Curvelets enjoy two unique mathematical properties,
Curved singularities can be well approximated with very few coefficients
and in a non-adaptive manner - hence the name "curvelets."
Curvelets remain coherent waveforms under the action of the wave equation
in a smooth medium.
More information can be found in the papers below. By releasing
the CurveLab toolbox, we hope to encourage the dissemination of curvelets
to image processing, inverse problems and scientific computing.
The Curvelet.org team: Emmanuel
Some links on curvelets:
Curvelets were introduced in 1999 by Candes and Donoho to address
the edge representation problem, see Curvelets
99. The definition they gave was based on windowed ridgelets --
it is a bit different from the one we now use in CurveLab. An early
implementation is presented here.
Further applications include inverse
problems with edges and curvilinear
Curvelet image denoising and image enhancement experiments based
on the curvelet implementation described in this
Component Analysis: Component Separation based on the curvelet transform,
and applications for texture separation and inpainting.
Curvelets in Astrophysics: Jean-Luc
Curvelets in Seismic Imaging at UBC:
Herrmann and his crew (Peyman,
Gilles). See this paper
and compression experiments using the curvelet transform on
non-equispaced synthetic seismic data.
Curvelets in Seismic Imaging: Martijn
de Hoop at Purdue, and Huub
Douma at the Colorado School of Mines. Recent work in collaboration
with Hart Smith, Gunter Uhlmann and Eric Dussaud.
Curvelets in Seismic Imaging: Mauricio Sacchi and his PhD student Mostafa Naghizadeh recently (2009) proposed seismic data interpolation using curvelets, in this paper.
Curvelets in Pure Math: Hart
Smith was the first to construct a tight frame of curvelets, in
the context of wave equations with low-regularity coefficients.
Curvelets in Plasma Physics: Bedros Afeyan at Polymath.
Contourlets: a curvelet-like transform based on filterbanks,
Do and Martin Vetterli.
Shearlets and wavelets with composite dilations: a curvelet-like
approach within the framework of affine systems, by Demetrio
Kutyniok and co-workers.
Waveatom.org: a webage on wave atoms, from the authors of CurveLab.
The reference for wavelets is Wavelet.org.
For the newest books on wavelets, check out: Books
Apologies to anybody who would feel left out
of this list - please drop me an email at laurent at curvelet.org.