What Is the Curvelet Transform and How Does It Differ from the Wavelet Transform?

Updated

The curvelet transform is a higher-dimensional generalization of the wavelet transform, designed to represent images at different scales and different angles. Where wavelets describe a signal mainly by scale and position, curvelets add an orientation component, which lets them approximate curved edges with very few coefficients and in a non-adaptive way — that is, without first detecting where the edges are. The main reference implementation distributed from curvelet.org is CurveLab 2.1.2 (April 2008), which includes mirror-extended (ME) curvelets.

The core idea

The name "curvelet" comes from the transform's central strength: curved singularities can be well approximated with very few coefficients, and this happens in a non-adaptive manner. A natural image is mostly smooth, interrupted by edges, and those edges are usually curved. A representation that can match curved structure directly needs fewer terms to describe it accurately.

Curvelets also have a second, more physical property: they remain coherent waveforms under the action of the wave equation in a smooth medium. That makes them relevant beyond still-image coding — for instance in problems tied to wave propagation and inverse problems.

How it differs from the wavelet transform

Aspect Wavelet transform Curvelet transform
Dimensionality Typically 1-D, extended to 2-D Higher-dimensional generalization
Organizing parameters Scale and position Scale, position, and angle
Edge handling Isotropic elements; curved edges need many coefficients Anisotropic, oriented elements; curved edges captured with few coefficients
Adaptivity Fixed transform Non-adaptive — no prior edge detection needed
Wave-equation behavior — Remains coherent under the wave equation in a smooth medium

The practical consequence is efficiency at representing edges. A wavelet basis uses roughly isotropic building blocks, so a smooth curve has to be assembled from many small pieces. Curvelets are elongated and directional, so a curved edge aligns with relatively few oriented elements.

Where it is used

According to curvelet.org, the CurveLab toolbox was released to encourage the dissemination of curvelets to:

  • image processing,
  • inverse problems,
  • scientific computing.

The same site points to a sister webpage for the wave atom transform at Waveatom.org, and notes that more detail is available in the published papers.

Getting the software

The newest version listed is CurveLab 2.1.2 (April 2008), which includes mirror-extended (ME) curvelets. The homepage does not state pricing or licensing terms, so check the site directly before assuming any conditions of use.

Who maintains it

The Curvelet.org team is listed as Emmanuel Candes, Laurent Demanet, David Donoho, and Lexing Ying.