Wavelets are a useful tool for the analysis of data that is both correlated and non stationary. In an interdisciplinary collaboration involving applied mathematicians and astrophysicists, we have developed multiscale tools for the analysis of intensity and polarisation data on the sphere, using decompositions on a frame of functions called needlets.
Needlets, or spherical wavelets, are ideal tools for implementing component separation in the analysis of multifrequency observations that are non stationary, can be part sky, and have various resolution. They can be used also for spectral estimation, data fusion, and statistical analysis on the sphere.
The figure below illustrates a typical processing pipeline using a needlet decomposition. An original map (top left) is analysed into a set of needlet coefficients by convolution with analysis functions. The needlet coefficients are represented by maps corresponding to various scales. The needlet coefficients are processed (e.g. scaled, thresholded, masked) to yield modified needlet coefficients (bottom right), which are then used to reconstruct a set of maps by convolution by synthesis functions (which are usually the same as the anaylysis functions). The final processed map is obtained by summation of the maps reconstructed in this way.