The table below lists a variety of the component separation methods used in CMB analysis. In most cases, the aim is to isolate the (full-sky) CMB signal, in contrast to the methods described for diffuse foreground components that isolate the different foregrounds. A few of these teams have made code publicly available, as indicated. This table is meant to summarize the variety of approaches to the problem of component separation that have been published in the literature.

Internal Linear Combination (ILC) | |
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Name | Description |

pyilc Code |
pyilc is a pure-python implementation of the needlet internal linear combination (NILC) algorithm for CMB component separation. Harmonic-space ILC is also implemented in the code. For details, see McCarthy & Hill (2023). The code repository also includes an inpainting code, diffusive_inpaint, that diffusively inpaints a masked region with the mean of the unmasked neighboring pixels. |

WMAP ILC | The pixel-based ILC method (Bennett et al., 2003) applied to WMAP data that forms linear combinations of the WMAP bands, minimizing the variance with the constraint that the cosmological component remains unchanged. Eriksen et al. (2004) modified this method to use Lagrange multipliers to compute the weights. |

Tegmark ILC |
Harmonic-space ILC (Tegmark et al. 2003) is similar to the pixel-space version but performed on the spherical harmonic coefficients instead of pixel values, which allows for a higher resolution result. |

NILC | The Needlet ILC is an implementation (Delabrouille et al. 2009) of internal linear combination (ILC) that works in the needlet (wavelet) domain. The input maps are decomposed into needlets at a number of different angular scales. The ILC solution for the CMB is produced by minimizing the variance at each scale. This has the advantage that the weights used to combine the data can vary with position on the sky and also with angular scale. The solutions are then combined to produce the final CMB map. |

MILCA | The Modified ILC Algorithm (Hurier et al. 2010) is generalized for the case of multiple astrophysical components with known spectra. For the Planck project, it was used to isolate both the Sunyaev-Zeldovich effect and the CO line emission. |

Spin-SILC | This is an internal linear combination method (Rogers et al. 2016) that uses spin wavelets to analyse the spin-2 polarisation signal P = Q +iU. T. Data Products available. |

Template Removal | |

Name | Description |

Delta-Map | This is a method to remove CMB foregrounds with spatially varying spectra from polarization maps (Ichiki et al. 2018). It extends the internal template foreground removal method by accounting for spatially varying spectral parameters such as the spectral indices of synchrotron and dust emission and the dust temperature. As the previous algorithm had to assume that the spectral parameters are uniform over the full sky (or some significant fraction of the sky), it resulted in a bias in the tensor-to-scalar ratio parameter r estimated from foreground-cleaned polarization maps of the cosmic microwave background (CMB). The new algorithm, "Delta-map method", accounts for spatially varying spectra to first order in perturbation. The only free parameters are the cosmological parameters such as r and the sky-averaged foreground parameters. We show that a cleaned CMB map is the maximum likelihood solution to first order in perturbation, and derive the posterior distribution of r and the sky-averaged foreground parameters using Bayesian statistics. |

WI-FIT | Wavelet based high resolution fitting of internal templates. The WI-FIT method (Hansen et al. 2006) computes CMB-free foreground plus noise templates from differences of the observations in different channels, and uses those to fit and subtract foregrounds from the CMB dominated channels in wavelet space. |

SEVEM | SEVEM ("Spectral Estimation Via Expectation Maximisation") is an implementation (Martínez-González et al. 2003, Leach et al. 2008, Fernández-Cobos et al. 2012) of the template cleaning approach to component separation that works in the map domain. Foreground templates are typically constructed by differencing pairs of maps from the low- and high-frequency channels. The differencing is done in order to null the CMB contribution to the templates. These templates are then used to clean each CMB-dominated frequency channel by finding a set of coefficients to minimize the variance of the map outside of a mask. Thus SEVEM produces multiple foreground-cleaned frequency channel maps. The final CMB map is produced by combining a number of the cleaned maps in harmonic space. |

SMICA | SMICA (Spectal Matching Independent Component Analysis) is a non-parametric method (Delabrouille et al. 2003, Cardoso et al., 2008) that works in the spherical harmonic domain. Foregrounds are modelled as a small number of templates with arbitrary frequency spectra, arbitrary power spectra and arbitrary correlation between the components. The solution is obtained by minimizing the mismatch of the model to the auto- and cross-power spectra of the frequency channel maps. From the solution, a set of weights is derived to combine the frequency maps in the spherical harmonic domain to produce the final CMB map. Maps of the total foreground emission in each frequency channel can also be produced. In the analysis performed for the 2013 release (Planck Collaboration XII 2014), SMICA was the method that performed best on the simulated temperature data. |

Monte-Carlo | |

Name | Description |

ForeGroundBuster Code |
ForeGroundBuster ('fgbuster') is a library of functions designed to perform separation of a suite of frequency maps into astrophysical components. It utilizes a maximum likelihood algorithm for parametric component separation. The algorithm, described in Stompor et al (2009) , is a modified version of the two-step MCMC approach discussed by Eriksen et al. (2006). A framework is also provided to estimate the level of residuals remaining in the resultant CMB map and forecast the impact on cosmological parameter recovery. |

Commander3 (BeyondPlanck release) Code |
Commander3 is the public release of the most recent version of Commander (Commander is an Optimal Monte-carlo Markov chAiN Driven EstimatoR) code for fast and efficient end-to-end CMB posterior exploration through Gibbs sampling. |

Commander2 Code |
Commander is a Bayesian parametric method (Eriksen et al., 2006, Eriksen et al., 2008) that works in the map domain. Both the CMB and foregrounds are modelled using a physical parameterization in terms of amplitudes and frequency spectra, so the method is well suited to perform astrophysical component separation in addition to CMB extraction (Planck Collaboration X 2016). The joint solution for all components is obtained by sampling from the posterior distribution of the parameters given the likelihood and a set of priors. To produce a high-resolution CMB map, the separation is performed at multiple resolutions with different combinations of input channels. The final CMB map is obtained by combining these solutions in the spherical harmonic domain. |

Maximum Entropy | |

Name | Description |

FastMEM | The FastMEM is a harmonic-space maximum entropy method that estimates (Hobson et al., 1998, Stolyarov et al., 2002) component maps given frequency scaling models and external foreground power spectra (and crosspower spectra) with adjustable prior weight. It is a nonblind, non-linear approach, which assumes a maximum-entropy prior probability distribution for the underlying components. |

Blind | |

Name | Description |

GMCA Code |
Generalised Morphological Component Analysis (Bobin et al., 2007) is a semi-blind source separation method which disentangles the components by assuming that each of them is sparse in a fixed appropriate waveform dictionary such as wavelets. As demonstrated in (Leach et al., 2008), GMCA can be used in two ways: GMCA-blind to optimize the separation of the CMB component, and GMCA-model to optimize the separation of Galactic components. This one is different from all the others in that its application to CMB was almost incidental; the method is far more generic. |

FastICA | The Independent Component Analysis (ICA) algorithm implemented in FastICA (Maino et al., 2002) is aimed at recovering both the spatial pattern and the frequency scalings of the emissions from statistically independent astrophysical processes along the line-of-sight using multi-frequency observations. It requires no a priori knowledge of the components except that they are statistically independent and all, except possibly one, have non-Gaussian distributions. These assumptions hold in the case of Gaussian CMB fluctuations and non-Gaussian foregrounds. The main advantage of this approach is that the algorithm is able to learn how to reconstruct independent components and their frequency scalings from the input maps. |

CCA | Correlated Component Analysis (Bedini et al., 2005) starts with an estimation of the mixing matrix on patches of sky by exploiting spatial correlations in the data, supplemented by constraints from external templates and foreground scaling modeling. The estimated parameters are then used to reconstruct the components by Wiener filtering in the harmonic domain. |