1. Implementation of simple ILC (Internal Linear Combination) method (see (1), (2), (3), also one can find a chapter about ILC in the book (8)).
2. Small example using CMB data from Planck experiment (see (4), (5)).
3. Visualization using HEALPix (Hierarchical Equal Area isoLatitude Pixelation of a sphere) python package (see (6)).

Separation of the main signal from noise is a challenging problem in analysis of cosmic microwave background (CMB) in WMAP and Planck experiments.

ILC method, alongside with classical methods such as Wavelet analysis, ICA (Independent Component Analysis) and many others, is one of the most widely used approaches in data analysis in Cosmology. The goal of this repository is to show that even simple version of ILC produces satisfactory results for the problem.

WMAP and Planck experiments created a full-sky map of CMB (electromagnetic radiation remnant from an early stage of the Universe) via multi-frequency observation.

The data consists of several maps in HEALPix format. There are separate files for different frequencies. Each file contains pixels with temperatures.

Supose that for some map in channel i at pixel p we have

where

- CMB contribution, which we want to find,

- diffuse Galactic foreground contribution, which influence we want to reduce,

- the experimental noise.

Example picture for one of the channels with applied mask:

The main assumption for ILC is that CMB contribution does not depend on the channel. Then the observed map is

Using these maps we need to estimate CMB contribution. One can search for a solution in the form of weighted linear combination, with the weights independent from p:

Also we imply the constraint

In this case, one can write the estimated component as

The weights have to minimize the variance:

The solution of this problem is

where C is covariance matrix:

1. Official CMB map by Planck experiment
2. Current code output map

1. https://arxiv.org/pdf/0806.0520.pdf A Statistical Analysis of the "Internal Linear Combination" Method in Problems of Signal Separation as in CMB Observations (R. Vio and P. Andreani, 2008)

2. https://arxiv.org/pdf/0811.4277.pdf "Internal Linear Combination" method for the separation of CMB from Galactic foregrounds in the harmonic domain (R. Vio and P. Andreani, 2008)

3. https://arxiv.org/pdf/astro-ph/0403098.pdf On Foreground Removal From The Wilkinson Microwave Anisotropy Probe Data By An Internal Linear Combination Method: Limitations And Implications (H. K. Eriksen et al, 2004)

4. https://lambda.gsfc.nasa.gov/product/map/dr5/ilc_map_info.cfm https://lambda.gsfc.nasa.gov/toolbox/tb_comp_separation.cfm Information about usage of ILC in WMAP experiment

5. https://irsa.ipac.caltech.edu/data/Planck/release_2/all-sky-maps/ Public download link for CMB maps

6. https://healpix.sourceforge.io/ https://github.com/healpy/healpy Hierarchical Equal Area isoLatitude Pixelation of a sphere project website

7. https://arxiv.org/pdf/0802.0400.pdf A Modified ICA Approach for Signal Separation in CMB Maps (R. Vio and P. Andreani, 2008)

8. https://www.springer.com/gp/book/9783540239727 Data Analysis in Cosmology (V.J. Martinez et al, Springer 2009)