The wavelet transform is used in astrophysics for many applications. Its use is connected to different properties. The Time-Frequency analysis results from the two-dimensional feature of this transform. Some interesting applications were performed on nonstationary astrophysical signals. Many astrophysical results were obtained by this analysis, either on quasi regular variables, and on chaotic light curves. Solar time series have been also carefully analysed by the wavelet transform. New results have been obtained for series with identified periods (sunspots, diameter, irradiance, chromospheric oscillations) and for chaotic signals (magnetic activity).
Astronomers have exploited the wavelet transform for image compression. Many packages are proposed with significant gains. Some full sky surveys are available now with images compressed by the wavelet transform. Filtering and restorations are derived from this scale-space analysis. Some thresholding rules furnish adapted filtering. The restoration is connected to an approach for which we progressively extract the most energetic features. This may be related to the notion of multiscale support. Many applications were done for Hubble Space Telescope (HST) images or for astronomical aperture synthesis. The ability of the wavelet transform to localize an object in scale-space led also to applying this transform to the detection and to the analysis of astronomical sources. A multiscale vision model was developed by our group, which allows one to detect and to characterize all the sources of different sizes in an astronomical image. Many applications of image analysis were performed on different astrophysical sources, and specifically the ones having a power-law correlation, i.e. a fractal-like behaviour: molecular clouds, infrared cirrus, clumpy galaxies, comets, X-ray clusters, etc.