curvelet.org

What is CurveLab?

CurveLab is the software toolbox released by the Curvelets.org team for working with the curvelet transform, a higher-dimensional generalization of the wavelet transform designed to represent images at different scales and angles. According to the site, the newest version is CurveLab 2.1.2 (April 2008), which includes mirror-extended (ME) curvelets.

The curvelet transform is aimed at data where edges and curved singularities matter: it is built so that curved features can be approximated well with very few coefficients, and in a non-adaptive manner. The site also notes that curvelets remain coherent waveforms under the action of the wave equation in a smooth medium. The team behind the release — Emmanuel Candes, Laurent Demanet, David Donoho and Lexing Ying — state that they released the toolbox to encourage the dissemination of curvelets to image processing, inverse problems and scientific computing.

Practical uses and audiences

• Image processing and analysis: representing images with edges and curves more efficiently than traditional wavelets.
• Inverse problems: sparsity-promoting regularisation where curved features are present.
• Scientific computing: numerical work involving wave propagation in smooth media, given the coherence property above.

A practical next step

If you are deciding whether CurveLab fits your work, check whether your data has curved or edge-like structure and whether you need a multiscale, multi-angle representation. For a quick test, compare a curvelet-based sparse approximation of one of your images against a wavelet-based one and look at how many coefficients each needs to capture the edges. If curved singularities dominate, CurveLab is the natural starting point. The related wave atom transform has its own site at Waveatom.org.

How does the curvelet transform differ from the wavelet transform?

The curvelet transform is a higher-dimensional extension of the wavelet transform built to handle images and other 2D or 3D data. Wavelets work well for point-like features, but they struggle with edges and curves. Curvelets add orientation to scale, so they can represent curved singularities with far fewer coefficients, and they do so non-adaptively—no need to learn a dictionary from the data.

Key differences

What this means in practice

If you are denoising a photograph with long, smooth contours, wavelets tend to produce ringing or require many coefficients to capture the edges. Curvelets can represent those contours with a compact set of coefficients, which often helps in image processing, inverse problems and scientific computing. The trade-off is added complexity: curvelet frames are redundant and the implementation is more involved than a standard wavelet filter bank.

Next step

A concrete way to decide is to test both on the same task. For a piecewise-smooth image with curved boundaries, compare reconstruction error at a fixed number of coefficients. Curvelets should win on edge-heavy data; wavelets remain simpler and often sufficient for texture or point-like features. The CurveLab toolbox from Curvelets Homepage is the reference implementation for experimenting with the transform. For the related wave atom transform, see Wave Atoms.

What are the practical applications of curvelets in image processing?

Curvelets are best understood as a representation designed for images that contain edges and curves, not just smooth regions or point-like details. Their practical value in image processing comes from two properties stated on the Curvelets Homepage: curved singularities can be approximated with very few coefficients in a non-adaptive way, and curvelets stay coherent under the wave equation in a smooth medium. In plain terms, a curvelet-like transform spends its coefficients on the curved edges that matter, rather than on many small pieces needed to imitate a curve.

H3 Practical uses in image processing

• Edge-preserving denoising. Because edges are captured compactly, thresholding curvelet coefficients can suppress noise while keeping boundaries sharper than methods that blur across edges.
• Image reconstruction and inpainting. When pixels are missing or measurements are incomplete, a sparse representation of edges helps fill in or recover the image. This is the "inverse problems" area the Curvelets Homepage names as a motivation for releasing the CurveLab toolbox.
• Compression and representation. Fewer significant coefficients for curved contours means a more efficient description of images dominated by smooth regions and strong edges.
• Feature and contour analysis. The transform separates information by scale and angle, which is useful when the orientation of a structure is itself the signal of interest.
• Seismic and wave-propagation imaging. The homepage notes that curvelets remain coherent under the wave equation in a smooth medium, which is why they appear in geophysical imaging and other wave-based reconstruction work.

H3 Who benefits, and the trade-offs

The natural audience is researchers and engineers in image processing, inverse problems and scientific computing, as the Curvelets Homepage itself frames the toolbox's purpose. Compared with wavelets, curvelets add directional sensitivity at fine scales, which is an advantage for curved edges but also means more parameters and more complex implementations. Compared with learned or patch-based methods, they are mathematically transparent and non-adaptive, which is attractive when you need predictable behavior or a theoretical guarantee rather than a trained model.

A useful decision rule: if your images are dominated by smooth areas and long, curved boundaries, and you care about sparse, interpretable representations, curvelets are worth testing. If your content is mostly texture, fine-grained noise, or point-like features, a wavelet or other transform may be simpler and just as effective.

For a next step, start from the CurveLab toolbox and the papers linked on the Curvelets Homepage, and compare a curvelet thresholding pipeline against a wavelet baseline on one concrete task, such as denoising a set of images with strong curved edges. The sister page for the related wave atom transform is at Waveatom.org, which is relevant if your structures are more oscillatory than curved.

Can CurveLab be used for inverse problems and scientific computing?

Yes. The CurveLab toolbox is explicitly released to encourage the spread of curvelets into image processing, inverse problems, and scientific computing, so those applications are part of its intended scope rather than an afterthought. The stated mathematical motivation is that curved singularities are approximated with very few coefficients in a non-adaptive way, and that curvelets stay coherent under the wave equation in a smooth medium — both properties that matter when you solve PDEs, propagate waves, or regularize ill-posed reconstruction problems.

What that means in practice

• Inverse problems: curvelet sparsity is a natural fit for regularization, e.g. denoising, deblurring, inpainting, or limited-angle and sparse-view tomography, where the unknown image has edges along curves.
• Scientific computing: wave propagation in smooth media, seismic and medical imaging, and other settings where you want a representation aligned with directional, curved features instead of axis-aligned wavelets.
• Image processing: the general-purpose entry point — analysis, denoising, compression-style experiments.

Who it suits

Researchers and graduate students in applied mathematics, signal processing, geophysics, and computational imaging who are comfortable with MATLAB-style toolboxes and reading the underlying papers. It is a research codebase, not a turnkey solver: you supply the forward model, the noise model, and the solver loop.

A concrete first step

If you have a linear inverse problem b = A x + noise, start by testing whether a curvelet-domain sparsity prior beats a wavelet prior on a small synthetic phantom. Transform, threshold or solve a short iterative scheme, then invert and compare reconstruction error at equal sparsity. That single experiment tells you whether curvelets are worth the extra implementation effort for your operator.

Decision criteria

Choose CurveLab when your data has curved or directional structure and you can tolerate research-grade code. Prefer wavelets when features are point-like or the pipeline must be simple. If your structures are more oscillatory or texture-like, the related wave atom transform may be a better match.

Related resources

• Wave Atom Transform
• Curvelets Homepage

What are the key mathematical properties of curvelets?

Curvelets are a higher-dimensional extension of wavelets designed to represent images at different scales and different angles. Their two defining mathematical properties are:

• Sparse approximation of curved singularities. Curved edges and other smooth curved features can be approximated with very few coefficients, and this happens in a non-adaptive way — no need to first detect where the edges are. That efficiency is the reason for the name "curvelets."
• Stability under the wave equation. Curvelets remain coherent waveforms when propagated through the wave equation in a smooth medium, so they behave predictably as waves travel rather than scrambling into unrelated coefficients.

The practical payoff of the first property is compression and denoising: a natural image with smooth contours needs fewer significant curvelet coefficients than a wavelet representation would, so you can threshold aggressively and still reconstruct clean edges. The second property matters for wave-propagation problems, where you want a transform whose elements stay organized along wavefronts.

A useful way to place them:

If you want to test this rather than take it on faith, a concrete next step is to take a test image with strong curved contours, run both a wavelet and a curvelet decomposition, and compare how many coefficients you must keep to reach a given reconstruction error. The gap you observe is the sparsity advantage in practice. For oscillatory or texture-dominated data, the related wave-atom transform is the more appropriate comparison point; see Wave Atom Transform for that sister project.

How can I download and install CurveLab 2.1.2?

CurveLab 2.1.2 is distributed through the Curvelets Homepage at Curvelets Homepage, which is the project's own site. The page identifies this as the newest version (April 2008) and notes that it includes mirror-extended (ME) curvelets. Downloading and installing it is a manual, research-software process rather than a one-click package install.

How to get and set it up

1. Go to the official page. Open Curvelets Homepage and look for the CurveLab download section or link.
2. Get the archive. The toolbox is distributed as a compressed archive containing the MATLAB and C source code. Save it to a local folder.
3. Check for build instructions. CurveLab has historically shipped with a README or installation file describing how to compile the MEX interfaces. Read that file before doing anything else.
4. Compile the MEX files. The transform routines are written in C and called from MATLAB, so you need a working C compiler configured for your MATLAB version (mex -setup). Run the provided make or install script, or compile the MEX sources listed in the documentation.
5. Add the folder to your MATLAB path. Once compiled, add the CurveLab directories to the MATLAB path so the functions are callable.
6. Verify with an example. Run one of the included demo or test scripts on a small image to confirm the forward and inverse transforms work.

Practical notes

• Platform matters. Because this is 2008-era C/MEX code, expect friction on modern 64-bit systems and recent MATLAB releases. Older compilers and matching MATLAB versions tend to work more smoothly.
• Where to look for help. The site points to the project's papers for the underlying mathematics; the sister page for the related wave atom transform is Waveatom.org if you are comparing transforms.
• Who this is for. This suits researchers in image processing, inverse problems and scientific computing who need curvelets in MATLAB and are comfortable compiling code. If you want a maintained, prebuilt package, check whether a current toolbox or library already wraps the curvelet transform before investing time in the build.

A useful next step: after downloading, open the README first and confirm your compiler and MATLAB versions are compatible before running any install script.