# Spherical wavelets and applications

### Multiscale analysis on the sphere

## Spherical wavelets

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.

## Component separation on a needlet frame

## Spectral estimation from heterogeneous data using needlets

## Testing the isotropy of UHE cosmic rays using needlets

© J. Delabrouille 2012 -
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