Post by Jason McEwen

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**Why wavelets?**

It can be insightful to analyse data, or “signals”, in different domains. For example, the frequency spectrum of music is often studied, where contributions to the base and treble are more clearly visible.

Music in time (lower panel) and frequency (upper panel). These representations can be computed on your iPhone.

In the Fourier domain, we can probe the frequency content of signals but lose information about the time localisation of signal structure. In the time domain, the reverse is true: we can probe the time content of signals but lose information about the frequency localisation of signal structure. Wavelets overcome this problem by looking at signal content in time and frequency (scale) simultaneously.

Often, we may be interested in signals that are defined on domains other than time. For example, a standard image is a two-dimensional signal defined on a spatial domain. Nevertheless, for such signals it can also be insightful to view the signal content in the frequency domain, rather than the spatial domain, or in the wavelet domain.

Many physical processes are manifest on particular physical scales, while also spatially localised. Wavelets are therefore a powerful analysis tool for extracting the fingerprint of a physical process of interest when it is embedded in some background signal.

**Wavelet analysis of the CMB**

Wavelets have now become a standard analysis technique for studying the anisotropies of the cosmic microwave background (CMB).

In this setting, the signal of interest (the CMB) is defined on the celestial sphere. We therefore need wavelet transforms defined on the sphere.

Temperature anisotropies of the CMB defined on the celestial sphere. [Credit: WMAP]

A number of wavelet transforms have been defined on the sphere. The construction of many of these analysis methods has been motivated directly by the desire to study the CMB but these techniques are of general use for studying signals on the sphere, such as observations made in geophysics and computer graphics.
Wavelets defined on the sphere are now a prevalent analysis technique for studying the CMB. In fact, many of the cosmological studies performed in the 2013 analysis and release of Planck data used wavelet methods .

**Exact wavelet on the ball for studying LSS**

The large-scale structure (LSS) of the Universe, as traced by the distribution of galaxies, is another powerful cosmological probe. Observations tracing the LSS are made in three-dimensions, with the radial dimension measuring redshift. These observations therefore also live in spherical space and are made on the ball, i.e. on the sphere augmented with depth information.

Observations tracing the LSS defined on the ball. [Credit: SDSS]

Recently, we have developed wavelet methods defined on the ball for the purpose of extracting cosmological information from observations tracing the LSS. More technical details on wavelets on the ball will appear in a future post.

We hope that these types of methods can prove as useful for studying the LSS as they have for studying the CMB.

We’re now busy applying them for various cosmological analyses and will keep you posted!

**Related reading**

B. Leistedt, J. D. McEwen, Exact Wavelets on the Ball

J. D. McEwen, B. Leistedt, Fourier-Laguerre transform, convolution and wavelets on the ball

F. Lanusse, A. Rassat, J.-L. Starck, Spherical 3D isotropic wavelets

B. Leistedt, H. V. Peiris, J. D. McEwen, Flaglets for studying the large-scale structure of the Universe

Y. Wiaux, J. D. McEwen, P. Vandergheynst, O. Blanc, Exact reconstruction with directional wavelets on the sphere

B. Leistedt, J. D. McEwen, P. Vandergheynst, Y. Wiaux, S2LET: A code to perform fast wavelet analysis on the sphere,

D. Marinucci, D. Pietrobon, A. Balbi, P. Baldi, P. Cabella, G. Kerkyacharian, P. Natoli, D. Picard, N. Vittorio, Spherical needlets for CMB data analysis

J.-L. Starck, Y. Moudden, P. Abrial, M. Nguyen, Wavelets, ridgelets and curvelets on the sphere

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