Curvelets are a non-adaptive technique for multi-scale object representation. Being an extension of the wavelet concept, they are becoming popular in similar fields, namely in image processing and scientific computing.

Motivation

Wavelets generalize the Fourier transform by using a basis that represents both location and spatial frequency. For 2D or 3D signals, directional wavelet transforms go further, by using basis functions that are also localized in orientation. A curvelet transform differs from other directional wavelet transforms in that the degree of localisation in orientation varies with scale. In particular, fine-scale basis functions are long ridges; the shape of the basis functions at scale j so the fine-scale bases are skinny ridges with a precisely determined orientation.

Curvelets are an appropriate basis for representing images (or other functions) which are smooth apart from singularities along smooth curves, where the curves have bounded curvature, i.e. where objects in the image have a minimum length scale. This property holds for cartoons, geometrical diagrams, and text. As one zooms in on such images, the edges they contain appear increasingly straight. Curvelets take advantage of this property, by defining the higher resolution curvelets to be more elongated than the lower resolution curvelets. However, natural images (photographs) do not have this property; they have detail at every scale. Therefore, for natural images, it is preferable to use some sort of directional wavelet transform whose wavelets have the same aspect ratio at every scale.

wavelets, and analysing the approximation error.

.

Curvelet construction

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

, i.e., it doubles in each second circular ring.

be the polar coordinates in the frequency domain.

We use the ansatz for the dilated basic curvelets in polar coordinates:

.

Then the admissibility yieldssee Window Functions for more information such that, . Then, it follows that

For a complete covering of the frequency plane including the region around zero, we need to define a low pass element with that is supported on the unit circle, and where we do not consider any rotation.

Applications

• Image processing

• Seismic exploration

• Fluid mechanics

• PDEs solving

• Compressed sensing

See also

• Bandelet transform

• Chirplet transform

• Contourlet transform

• Fresnelet transform

• Noiselet transform

• Scale space

• Shearlet transform

References

• 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.105120.

• 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

External links

• Curvelet.org homepage