curvelets

Curvelets are a multiscale, directional transform for representing two-dimensional signals, especially images, with edge-like singularities along smooth curves. A curvelet transform decomposes a function into coefficients associated with location, scale, and orientation, using a redundant dictionary of atoms localized in space and oriented in angle. In two dimensions, curvelets are elongated and highly anisotropic, with parabolic scaling: the width of a curvelet is proportional to the square root of its length. This geometry allows curvelets to capture curved edges sparsely, in contrast to wavelets, which struggle with curved discontinuities.

Curvelets tile the Fourier domain by wedge-shaped regions at multiple scales and orientations. Each atom is localized both in space and frequency, enabling efficient representation of smooth curves with a small number of significant coefficients. The transform typically forms a near tight frame, ensuring stable reconstruction from coefficients.

Curvelets were introduced by Candès and Donoho in the late 1990s as part of geometric multiresolution analysis for optimal sparse approximation of images. They extend wavelets by adding directionality and anisotropy, and they underpin subsequent directional systems such as ridgelets and contourlets.

Applications include image denoising and compression, deconvolution, tomography, seismic data processing, and medical imaging. Curvelets are also used for edge detection and feature extraction where curved edges are prominent.

Fast algorithms for the discrete curvelet transform exist, and implementations are available in scientific computing libraries, enabling practical use for large-scale data analysis.