Curvelet

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.

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 is 2 ^{− j} by 2 ^{− j / 2} 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 skinnier 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.

When the image is of the right type, curvelets provide a representation that is considerably sparser than other wavelet transforms. This can be quantified by considering the best approximation of a geometrical test image that can be represented using only n wavelets, and analysing the approximation error as a function of n. For a Fourier transform, the error decreases only as O(1 / n^{1 / 2}). For a wide variety of wavelet transforms, including both directional and non-directional variants, the error decreases as O(1 / n). The extra assumption underlying the curvelet transform allows it to achieve O((log(n))³ / n²).

Efficient numerical algorithms exist for computing the curvlet transform of discrete data. The computational cost of a curvlet transform is approximately 10–20 times that of an FFT, and has the same dependence of O(n²log(n)) for an image of size $n \times n$.

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. 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

External links

• Curvelet.org homepage