Foreground component separation with generalised ILC

Abstract

The 'Internal Linear Combination' (ILC) component separation method has been
extensively used to extract a single component, the CMB, from the WMAP
multifrequency data. We generalise the ILC approach for separating other
millimetre astrophysical emissions. We construct in particular a
multidimensional ILC filter, which can be used, for instance, to estimate the
diffuse emission of a complex component originating from multiple correlated
emissions, such as the total emission of the Galactic interstellar medium. The
performance of such generalised ILC methods, implemented on a needlet frame, is
tested on simulations of Planck mission observations, for which we successfully
reconstruct a low noise estimate of emission from astrophysical foregrounds
with vanishing CMB and SZ contamination.

Full text

Page 1

Mon. Not. R. Astron. Soc. 000, 1–9 (2010)Printed 8 March 2011(MN LATEX style file v2.2)
Foreground component separation with generalised ILC
Mathieu Remazeilles?, Jacques Delabrouille†, Jean-Fran¸ cois Cardoso‡
APC 10, rue Alice Domon et L´ eonie Duquet, 75205 Paris Cedex 13, France
8 March 2011
ABSTRACT
The ‘Internal Linear Combination’ (ILC) component separation method has been
extensively used to extract a single component, the CMB, from the WMAP multi-
frequency data. We generalise the ILC approach for separating other millimetre astro-
physical emissions. We construct in particular a multidimensional ILC filter, which can
be used, for instance, to estimate the diffuse emission of a complex component origi-
nating from multiple correlated emissions, such as the total emission of the Galactic
interstellar medium. The performance of such generalised ILC methods, implemented
on a needlet frame, is tested on simulations of Planck mission observations, for which
we successfully reconstruct a low noise estimate of emission from astrophysical fore-
grounds with vanishing CMB and SZ contamination.
Key words: Cosmic Background Radiation – Methods: data analysis – ISM: general
1INTRODUCTION
The separation of emissions originating from distinct astro-
physical components in observations of the millimetre and
sub-millimetre sky is an important step in the scientific ex-
ploitation of such observational data. Various methods have
been developed to extract the emission of several compo-
nents out of multi-frequency Cosmic Microwave Background
(CMB) observations such as those of the WMAP and Planck
space missions (see, e.g., Delabrouille & Cardoso (2009) for
a review).
In many cases, such methods define components
through an (explicit or implicit) assumption that the ob-
servations are a linear mixture of unknown templates (or
sources) scaling rigidly with frequency. Such methods also
assume a fixed number of astrophysical emissions (e.g. CMB
anisotropies, thermal Sunyaev-Zel’dovich (SZ) effect, ther-
mal dust emission, synchrotron emission...).
Assuming that such a representation holds, blind com-
ponent separation methods such as the Spectral Match-
ing ICA (Delabrouille, Cardoso & Patanchon, 2003; Car-
doso et al., 2008), FastICA (Hyvarinen, 1999; Maino et al.,
2002), JADE (Cardoso, 1998), CCA (Bonaldi et al., 2006)
or GMCA (Bobin et al., 2008) are designed to solve the
problem of recovering the components of interest when their
mixing matrix (i.e., the matrix which specifies how much
each component contributes to a frequency observation) is
unknown. By exploiting the assumption of statistical inde-
pendence between the components, the mixing matrix can
?E-mail: remazeil@apc.univ-paris7.fr
† E-mail: delabrouille@apc.univ-paris7.fr
‡ E-mail: cardoso@enst.fr
be blindly estimated up to permutation and rescaling of its
columns. Once an estimate of the mixing matrix is available,
the components can be separated by inverting the linear
system, possibly taking into account the presence of instru-
mental noise. This has been investigated by a number of
authors (Tegmark & Efstathiou, 1996; Bouchet & Gispert,
1999; Hobson et al., 1998; Delabrouille, Patanchon & Audit,
2002).
However, in millimetre and sub-millimetre wave obser-
vations, some components cannot be correctly modelled as
a single template which would by simply scaled by mixing
coefficients (Tegmark, 1998). Emissions from the galactic in-
terstellar medium exhibit frequency scaling which depends
on local conditions (temperature, chemical composition) at
the location of emission, and hence are variable over the
celestial sphere.
Internal Linear Combination (ILC) methods do not as-
sume a particular parametrisation for foreground emission.
They offer a simple way to extract the map of a single
component of interest and have been used by several au-
thors in the analysis of the maps obtained by the WMAP
satellite to extract a CMB map (Bennett et al., 2003; Erik-
sen et al., 2004; Park, Park & Gott, 2007; Kim, Naselsky
& Christensen, 2008; Delabrouille et al., 2009). The tradi-
tional ILC, however, can only recover components for which
the emission scales rigidly with frequency (hence its use for
separating a CMB map). In addition, the ILC performs sat-
isfactorily only if the component of interest is not correlated
with the other emissions.
In a previous publication (Remazeilles, Delabrouille &
Cardoso, 2011), we introduced the Constrained ILC, which
extends the ILC to the case where there is more than one
component of interest (e.g. CMB and thermal SZ), and one
arXiv:1103.1166v1 [astro-ph.CO] 6 Mar 2011

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2Mathieu Remazeilles, Jacques Delabrouille, Jean-Fran¸ cois Cardoso
wishes to cancel out the contamination from one of them
into the recovered map of the other. In the present paper,
we generalise further the ILC and address the separation of
multidimensional components, which cannot be modelled as
one single template scaling with frequency according to a
universal emission law.
2CMB ESTIMATION BY STANDARD ILC
2.1Model of the measurement
In all of the following, we assume that all available maps
(Nobs maps) can be written, for all pixels p of the observed
maps, in the form
y(p) = as(p) + bz(p) + f(p) + n(p)(1)
where s(p) is the CMB template map, z(p) the thermal
Sunyaev-Zel’dovich effect, f(p) is the emission of the rest
of the foregrounds as they would be observed by the in-
strument in absence of anything else, and n(p) is the in-
strumental noise. Note that f(p) and n(p) are represented
with Nobs maps each, while the CMB and the SZ effect are
represented by one single map each, scaled across frequency
channels using CMB and SZ scaling coefficients, a and b,
which are assumed to be known.
Depending on the objective, any of as(p), bz(p) or f(p)
can be considered as ‘noise’ and implicitly included in the
noise term. Similarly, depending on the objectives of the
component separation, as(p) or bz(p) can be considered as
part of the total ‘foreground term’ i.e. implicitly included
in f(p).
2.2Extraction of the CMB
The ILC provides the estimate ˆ sILC of the CMB component
s by forming a linear combination of the Nobsobserved maps
which has unit response to the component of interest and
has minimum variance. Straightforward algebra leads to:
ˆ sILC =
at?R
−1
at ?R
−1a
y(2)
where?R is the empirical covariance matrix of the observa-
observation maps. This standard ILC can be used similarly
to recover an estimate ˆ zILC of the SZ effect (with a replaced
by b in Eq. (2)). Note that the quality of CMB reconstruc-
tion with an ILC depends on the accuracy with which a
is known. In presence of errors (for instance calibration er-
rors), there is no guarantee that the CMB is preserved (Dick,
Remazeilles & Delabrouille, 2010).
Assuming no correlations between the components, the
total covariance matrix R of the observations y can be writ-
ten as:
tions, a has dimension Nobs×1, and y is the Nobs×1 vector of
R = aatCCMB+ bbtCSZ+ RFG+ RN
(3)
2.3Wavelet space ILC
In its simplest implementation, the ILC is performed on the
complete maps, and one single global matrix?R is used for
the whole data set. This requires all maps to be at the same
resolution. It is possible, however, to decompose the orig-
inal maps as sums of different data subsets, covering each
a different region in pixel space or in harmonic space, to
apply independent versions of the ILC to the different data
subsets, and then to recompose a map from all these inde-
pendent results.
The main interest of such a decomposition is the possi-
bility to adapt the ILC filter to local contamination condi-
tions. Such localisation of the filter is useful in pixel space:
the galactic emissions are stronger in the galactic plane,
whereas noise dominates the total error at high galactic lat-
itudes. It is also useful in harmonic space, because contami-
nants do not all have the same angular power spectrum and
because of the channel-dependence of instrumental beams.
Note however that some care must be taken when sub-
dividing the original data into small subsets. The ILC in-
deed relies on the component of interest to be uncorrelated
with the contaminants (i.e. ?s(p)ni(p)? = 0, for all channels
of observation i). If this condition does not hold, the ILC
introduces a bias in the reconstruction. This has a conse-
quence on the minimum data size on which the ILC should
be implemented: too few independent data points results
in empirical correlations between the component of interest
and the contaminants, which generates a reconstruction bias
as described in the appendix of Delabrouille et al. (2009).
In the present paper, observations are decomposed us-
ing the spherical needlets discussed, in the context of CMB
data analysis, by several authors (see, e.g., Pietrobon, Balbi
& Marinucci (2006); Marinucci et al. (2008); Fa¨ y et al.
(2008); Guilloux, Fa¨ y & Cardoso (2009)). This needlet de-
composition provides localisation of the ILC filters both in
pixel and in harmonic space.
We define a set of spectral windows h(j)(?) such that,
over the useful range of ?, we have:
?
Maps of wavelet (needlet) coefficients are obtained, for each
observed map y(p), by inverse harmonic transform of the
associated map SHT coefficients y?mfiltered by the spectral
windows h(j)
?:
?
For each scale j, for each pixel p of the corresponding needlet
coefficients maps γ(j)
a
(one such map for each observation
a), the empirical covariance matrix?R used in equation (2) is
p and including some neighbouring pixels, of the product of
needlets coefficients. The ab entry is given by
?
In Delabrouille et al. (2009), the practical implementation
uses, as domains Dp, HEALPix ‘super-pixels’ obtained by
grouping 32×32 pixels of the needlet coefficient maps γ(j)(p)
(making use of the hierarchical definition of HEALPix pix-
els). Here, we use a slightly different prescription: we smooth
instead the product map γ(j)
Gaussian window in pixel space. This avoids artificial dis-
continuities at super-pixel edges.
j
?
h(j)
?
?2
= 1(4)
γ(j)(p) =
?
?
m
y?mh(j)
?
Y?m(p)(5)
computed from an average, in a domain Dpcentered at pixel
?Rab(p) =
1
Np
p?∈Dp
γ(j)
a (p?)γ(j)
b(p?)(6)
a (p)γ(j)
b(p) with a symmetric,

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Foreground component separation with generalised ILC3
3FOREGROUND ESTIMATION BY
MULTIDIMENSIONAL ILC
We now address the problem of estimating the set of maps
f(p), i.e. a ‘catch-all’ foreground component comprising the
emission of the diffuse galactic interstellar medium (ISM),
and of numerous galactic and extragalactic compact sources.
The objective is to construct estimated maps?f(p) that ‘best
absence of CMB, SZ and noise (see Eq. 1).
Astrophysical emission originating from the Galactic
ISM and from numerous extragalactic sources is qualita-
tively different from the CMB and the SZ effect. Each of the
latter is somewhat special in the sense that its emission can
be modeled, with good accuracy, as a single template scaling
in a known way with frequency. The total foreground (FG)
emission f comprises contributions from several different
processes. In addition, we cannot even assume a priori that
a linear mixture model (in which each map constituting f
would be a linear superposition of well defined templates)
does hold.
For extracting such emissions from multi-frequency ob-
servations, we propose to generalize the ILC method to ad-
dress the case of such a ‘multidimensional component’.
match’ what would be observed by the instrument in the
3.1 Multidimensional components
Let RFG = ?fft? denote the covariance matrix of the ob-
served foregrounds in Nobs frequency channels. This Nobs×
Nobs matrix RFG will be refered to as the FG covariance
matrix.
Among astrophysical foregrounds included in the
‘catch-all’ component f, the ISM of our own galaxy is
the main contributor. It emits via the combination of sev-
eral processes (synchrotron, free-free, thermal dust, spinning
dust, molecular lines...). In previous work, some of these pro-
cesses have been individually modelled each by a fixed tem-
plate and an emission law. Bouchet & Gispert (1999), for
instance, assume that synchrotron emission scales with fre-
quency proportionally to ν−0.9, free-free proportionally to
ν−0.16, and dust proportionally to ν2Bν(T) with T = 18K.
Since the emission of the ISM in each channel is described
as a linear mixture of three templates, such a model pre-
dicts that the ISM covariance matrix (which we will denote
as RISM) is a rank 3 matrix. When the contribution of ex-
tragalactic compact sources is neglected (assuming bright
point sources are extracted from the maps, and faint ones
contribute a negligible amount of emission), the foreground
covariance matrix itself, RFG, is equal to RISM (the covari-
ance matrix of the galactic ISM emission), and is a rank 3
matrix.
Such a model is too crude in the context of the very
sensitive measurements performed by WMAP and Planck:
the emission laws of the galactic emissions vary as a function
of the direction on the sky. To make things even more com-
plex, the background of compact sources contributes emis-
sion that becomes significant for measurements such as those
of the Planck mission (Planck Collaboration et al., 2011),
and that can not be modelled at all as the sum of a few
independent components.
The question of the rank of the FG covariance matrix
is a crucial one for component separation. This matrix is
expected to be, strictly speaking, full rank. In practice how-
ever, the issue is slightly more subtle. Consider its eigen-
decomposition: RFG = VDVt, where V is an orthonormal
matrix and D is a diagonal matrix with eigen-values sorted
in decreasing order. While the three-component model of
Bouchet & Gispert (1999), predicts that only the first three
eigen-values are non-zero, a model with spatially varying
spectral indices, and numerous additional emission processes
(spinning dust, molecular lines, extragalactic source back-
ground) predicts that all the eigen-values are non-zero. How-
ever, if there is only a small variation of the spectral indices,
and if some components are very weak, it is expected, at
least in some regions of the sky or at some angular scales,
that the smallest eigen-values are very close to zero so that
RFG is ‘almost rank-deficient’ (see section 3.4 below for a
more rigorous statement).
In this paper we propose, as in Cardoso et al. (2008), to
model the FG covariance matrix as an Nobs× Nobs matrix
of rank NFG, not necessarily equal to Nobs. Loosely speak-
ing, NFG counts the number of different templates needed
to represent most of the emission of the FG in our data set.
In other words, we try to capture all foreground emission
as resulting from NFG (possibly correlated) templates. The
integer NFG is called the (effective) FG dimension and may
vary over the sky with respect to the pixel p, or with re-
spect to ? in harmonic space, or with respect to the needlets
domain considered for a decomposition of the maps on a
needlet frame.
3.2 The foreground subspace
The analysis in this paper is performed on a needlet frame.
The temperature map needlet coefficients are indexed by
(j,k), where j denote the scale and k the pixel.1
In a given needlet domain D(j)
trix RFG is a (symmetric, non negative) Nobs× Nobs matrix
of rank NFG, then foreground emission can be represented
as a superposition of NFG templates:
k, if the FG covariance ma-
f = F g(7)
where the Nobs × NFG matrix F is called the foreground
mixing matrix and where g is a vector of dimension NFG. It
follows that the FG covariance matrix is
RFG= ?fft? = F?ggt?Ft= FGFt
where G is a NFG× NFG full rank covariance matrix.
Note two important points. First, the templates g are
not expected to correspond to physical foregrounds. They
are just a basis of the NFG-dimensional subspace spanned
by f. We are not interested in recovering g. Our objective
with the method discussed here is to recover f (in addi-
tion to s and z). Secondly, matrix F and its number NFG
of columns may depend on the domain D(j)
instance, at high galactic latitude, it is quite possible that
our observations contain negligible emission from some of
(8)
k
considered. For
1Note that the methods described throughout the paper do not
require a needlet frame in particular and can be implemented
in pixel space as well, where domains D should be indexed by
pixels p, or in harmonic space with the domains indexed by (?,m)
coefficients.

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4Mathieu Remazeilles, Jacques Delabrouille, Jean-Fran¸ cois Cardoso
the galactic foregrounds, but not so at low galactic latitude.
The needlet implementation allows us to modulate the ef-
fective dimension of the multidimensional foreground com-
ponent both in pixel space and in harmonic space, i.e. vary
NFG across the sky regions and the physical scales.
We also stress that, unlike in the case of CMB and SZ re-
construction, where the mixing vectors a and b are assumed
fully known a priori, we do not assume here that the matrix
F is known. We will not resort to prior physical knowledge
about the components of the FG emission to determine ma-
trix F. In fact, as the basis templates g do not correspond to
anything physically meaningful, we are not even interested
in determining F itself but, for reconstruction purposes, only
the product f = Fg. It is only assumed that matrix RFGhas
a given rank NFG (which can be estimated from the data, if
needed).
Matrix F cannot be determined from the data only, that
is, without making use of some prior information or assump-
tion about g. Indeed, let T be some invertible NFG× NFG
matrix and consider the transformed matrices?F = FT−1and
completely equivalent, factorization of the FG covariance
matrix since, by construction, FGFt=?F?G?F
by any other linear combination Tg of them (provided the
linear combination is not degenerate, i.e. T is invertible).
However, as we shall see in section 3.3, the implemen-
tation of the ILC filter for estimating the total FG emis-
sion does not require the full knowledge of F. Indeed, the
expression (15) of that filter is strictly unchanged upon the
introduction of such an invertible factor T. In section 3.4, we
show how matrix F can be estimated up to multiplication by
a right factor T. It is worth stressing again that this indeter-
mination means that we are only concerned with estimating
the column space of matrix F. That NFG-dimensional space
can be called the ‘FG subspace’. Our working assumption
that RFG has rank NFG means that the FG data has a co-
variance structure which is unknown but is constrained to
live in the FG subspace.
Physically, accepting this indetermination amounts to
giving up, during the component separation stage discussed
here, distinction between processes of emission on the ba-
sis of physical criteria such as emission process or physical
origin. Obviously, this is not fully satisfactory from an as-
trophysicist’s point of view, since in the end we would like
to know what is the source of the observed emission. This
distinction among sources of FG emission, however, can be
made at a later stage of the data analysis, i.e. we can first
separate CMB and SZ from other foregrounds, and then put
physics into the interpretation of the reconstructed multidi-
mensional FG component and interpret it as the sum of
emissions from a number of physical emission processes.
?G = TGTt. These transformed matrices are an alternate,
means that the NFGunderlying templates g can be replaced
t. Physically, it
3.3Multidimensional ILC filter
Aiming at a direct estimation of the foregrounds, we gener-
alize the ILC method to address the case of a multidimen-
sional component (here NFG–dimensional, where NFG is the
number of components, i.e. the rank of the foreground co-
variance matrix). In a given needlet domain, we model the
observation maps, collected into the Nobs× 1 vector y, as
y = Ax + n,(9)
where n is the Nobs× 1 vector of instrumental noise and
A =?
Here the (NFG+ 1) × 1 signal vector x contains the CMB
emission s as first entry and the NFG×1 vector g which col-
lects the emission of the NFG components needed to model
the total foreground emission. The Nobs×(NFG+1) mixing
matrix A contains, as a first column, the Nobs× 1 vector
a giving the frequency scaling of the CMB component. The
other columns correspond to the Nobs×NFGforeground mix-
ing matrix F, i.e. they span the foreground subspace.Note
that this assumes that a itself cannot be obtained by linear
combinations of the columns of F (more about this later).
As a refinement, it can be useful to single out both
the CMB and the SZ, in which case the second column in A
explicitly appears as the frequency scaling vector b of the SZ
component (and the SZ can be considered as excluded from
the rest of the foregrounds). We get back to this refinement
in sections 3.5 and 4.
Eq. (9) assumes that all observations are at the same
resolution, which is needed to implement the ILC filter (for
practical implementation, maps are put to the same res-
olution by partial deconvolution in harmonic space). The
localisation in harmonic space allows dropping out some of
the channels at high ? if needed by reason of insufficient
resolution.
We consider the estimation of f by a linear operation
a
F
?,x =
?
s
g
?
.(10)
?f = By,(11)
where, as in standard (one-dimensional) ILC, the Nobs×Nobs
ILC weight matrix B is designed to offer unit gain to the
foregrounds while minimizing the total variance of the vec-
tor estimate?f. In other words, matrix B is the minimizer of
B thus solves the following constrained variance minimiza-
tion problem
BF=Ftr?BRBt?,
where R is the covariance matrix of the observations y. That
problem can be solved by introducing a Lagrange multiplier
matrix Λ and the Lagrangian
L(B,Λ) = tr?BRBt?− tr?Λt(BF − F)?.
By differentiating (13) with respect to B, one finds that
∂L(B,Λ)/∂B = 0 entails
2BR = ΛFt.
E(||By||2) under the constraint BF = F. The weights matrix
min
(12)
(13)
(14)
By imposing the constraint BF = F on (14), one then finds
that Λ = 2F(FtR−1F)−1. Hence, the solution of the problem
(12) is the foreground ILC weight matrix given by
B = F?FtR−1F?−1FtR−1.
Comparing Eq. (15) to Eq. (2), multi-dimensional ILC ap-
pears as a direct generalization of one-dimensional ILC.2
One can immediately notice that expression (15) for B
(15)
2TheILCestimateoftheCMBvectorofemissionin

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Foreground component separation with generalised ILC5
is invariant if F is changed into FT for any invertible matrix
T. Hence, as already mentioned in Section 3.2, implement-
ing the foreground ILC filter (15) only requires that the
foreground mixing matrix F be known up to right multipli-
cation by an invertible factor. Again, in other words, the
only meaningful and mandatory quantity for implementing
a multi-dimensional ILC is the column space of F.
3.4Estimation of the foreground subspace
In this section, we propose a method for estimating the fore-
ground subspace locally, that is, in each needlet domain.
We consider only the case where the model accounts for the
CMB, an NFG-dimensional foreground component and noise
at a known level:
R = CCMBaat+ FGFt+ RN
(16)
and we want to estimate the foreground subspace Col(F)
from an estimate?R of R. Define the Nobs×(NFG+1) matrix:
L =aC1/2
?
CMB| FG1/2?
(17)
so that the signal part of the covariance matrix is LLt:
R = LLt+ RN.
Our procedure for estimating Col(F) is in two steps. In a first
step, we obtain an estimate of L up to right multiplication by
a rotation matrix (and an estimate for the dimension NFGof
the foreground subspace) using the knowledge of the noise
covariance matrix. In a second step, we use the fact that
the first column of L is known (up to scale) to obtain an
estimate of Col(F). That is described next.
Denote the eigenvalue decomposition of the noise-
whitened signal covariance matrix R−1/2
N
LLtR−1/2
N
as
R−1/2
N
LLtR−1/2
N
= U∆Ut.
where U is orthonormal: UUt= I, and ∆ is diagonal. Now,
R−1/2
N
RR−1/2
N
= R−1/2
N
= R−1/2
N
= U∆Ut+ UUt
= U[∆ + I]Ut,
(LLt+ RN)R−1/2
N
LLtR−1/2
N
+ I
showing that R−1/2
same eigen-vectors but that the former has its eigenvalues
shifted by 1 with respect to the latter. Further, if L has rank
NFG+ 1 then its eigen-structure actually is
?
where Us has (NFG+1) columns, Un has Nobs− (NFG+ 1)
columns, and ∆Sis a (NFG+1)×(NFG+1) diagonal matrix.
N
RR−1/2
N
and R−1/2
N
LLtR−1/2
N
share the
U∆Ut= [UsUn]
∆S
0
0
0
?
[UsUn]t
(18)
each frequency channel is obtained by applying the filter
a?atR−1a?−1atR−1(i.e., the filter of eq. (2) multiplied on the
left by the vector a).
3.4.1 Estimation of NFG and L
In the needlet domain considered, given an estimate?R of R,
R−1/2
N
?RR−1/2
corresponding to the eigenvalues that are larger than (1 + ε)
and by?US the corresponding subset of columns of?U. Here
nated by instrumental noise (see section 4 for the choice of
the threshold). This threshold condition thus provides an
estimate for the dimension NFG of the foreground subspace
in the needlet domain given the dimension (NFG+1) of the
sub-block?DS which fulfills the threshold condition.
signal subspace, (NFG + 1), depends on the level of noise
in the needlet domain considered, so the signal subspace is
estimated locally, both in space and in scale. This processing
thus locally performs a rank reduction of the observations
covariance matrix allowing the reduction of the instrumental
noise in the reconstruction.
By construction, the matrix
we compute the eigen-decomposition:
N
=?U?D?U
t
(19)
and, similar to eq. (18), we denote by?DSthe sub-block of?D
ε is a threshold above which the observation is not domi-
In this preprocessing, the dimension of the estimated
?M = R1/2
?M?M
N?US
N
??DS− I
is close to R−1/2
?1/2
(20)
is such that R−1/2
?R is close to R. That property, in turn, implies that?MO
completing the first step of our estimation procedure.
N
tR−1/2
N
LLtR−1/2
N
if
is close to L for some (undetermined) rotation matrix O,
3.4.2Estimation of Col(F)
In the second step, we note that the rotation matrix O
should be such that?MO is close to L. However, only the
O as O = [v |V] where v is a unit norm vector and V is an
(NFG + 1) × NFG matrix. The only available constraint is
thus that?Mv should be close to the first column of L. How-
equal to aC1/2
that Col(?M) does not necessarily contain a (as would be the
that?Mv is equal to the projection of aC1/2
the projection is
??M
Let us then denote by ? a the vector
? a =
The projection of aC1/2
and vector v is therefore given by v = ? aC1/2
v = ? a/|? a|
is a rotation matrix uniquely determines V up to right mul-
tiplication by a rotation factor. However, nothing more is
first column of L is known, up to scale. Hence, we partition
ever, we cannot expect to find a v such that?Mv is strictly
case for?R = R). The best we can do is to determine v such
The orthogonal projection matrix is?M(?M
?M
??M
CMBbecause?M is estimated from the data so
CMBonto Col(?M).
t?M)−1?M
tso that
t?M
?−1?M
taC1/2
CMB.
t?M
?−1?M
ta.(21)
CMBonto Col(?M) then is?M ? aC1/2
CMB
CMB. Recall that
v is a unit norm vector so we must have:
andCCMB = 1/|? a|2.(22)
Once vector v is determined, the constraint that O = [v |V]

Page 6

6 Mathieu Remazeilles, Jacques Delabrouille, Jean-Fran¸ cois Cardoso
required to determine the foreground subspace, as already
stressed. Our procedure is therefore complete and can be
summarized by the following steps:
• Compute the eigen-decomposition (19) of the noise
whitened covariance matrix.
• Form matrix ?M by eq. (20), compute vector ? a by
• Compute an (NFG+1)×NFGmatrix V such that matrix
[v |V] is a rotation.
• Obtain a basis of the foreground subspace as?F =?MV.
?B =?F
3.5Projecting foregrounds orthogonally to both
thermal SZ and CMB
eq. (21) and get v by normalization (22).
• Compute the NFG-dimensional ILC filter
??F
tR−1?F
?−1?F
tR−1
The foreground multidimensional ILC filter can be gener-
alised further. Thermal SZ emission can be singled out in
the same way as the CMB, in which case we may require
that there is no thermal SZ residual in the reconstructed
foreground map. This is doable because the emission law of
the SZ component, like that of the CMB, is known. We then
write the model of emissions as:
A =?
ab
F
?,x =


s
z
g

.(23)
where we have explicitly distinguished the thermal SZ emis-
sion z from the other foregrounds through its frequency scal-
ing vector b (emission law). We may then generalize the
processing developed in Sec. 3.4. In that spirit, the FG mix-
ing matrix F can then be estimated in the needlet domain
considered from the set of NFG columns orthogonal to both
the projection of the CMB scaling vector and the projec-
tion of the thermal SZ scaling vector onto the estimated
(NFG+2)–dimensional signal subspace. This guarantees that
the foreground map reconstructed by the multidimensional
ILC now contains neither SZ nor CMB (with, however, the
usual caveat that the statistics used to compute the covari-
ance matrices must be accurate enough). In addition, the
rank-reduction procedure (restriction of the observations to
the (NFG+ 2)-dimensional signal subspace) in each needlet
region reduces the level of instrumental noise locally in the
reconstructed foregrounds.
3.6 Discussion of special cases
3.6.1Less channels than foreground dimension
In the discussion above, we assumed that the signal sub-
space is the direct sum of two subspaces: the CMB subspace
which is one-dimensional (because of the rigid scaling of the
CMB with frequency) and the foreground subspace which is
NFG-dimensional. The former is not included in the latter
if no combination of foreground emission has the same scal-
ing as the CMB across available frequencies. Of course, this
property requires enough properly chosen frequency chan-
nels.
When there are more components than observations,
then unless the foreground emissions are either fully
correlated or very faint (below noise), then we have
NFG+ 1 > Nobs and the CMB cannot be perfectly sepa-
rated from the foregrounds. When there are enough inde-
pendent observations (i.e. a large number of channels), then
NFG < Nobs, and in general the CMB subspace is not con-
tained in the (larger) foreground subspace. Separation is
then possible up to finite-sample size errors in the deter-
mination of the appropriate subspaces.
Note in the passing that the kinetic SZ cannot be sep-
arated from the CMB on the basis of its emission law.
Throughout this paper the CMB, distinguished solely by its
known emission law a, also includes the kinetic SZ effect.
3.6.2 CMB subtraction as an (Nobs− 1)–dimensional ILC
A foreground estimation has been obtained in Ghosh et al.
(2010) by subtracting the CMB-ILC estimate from the ob-
servations data,?f = y − a? s. It is interesting to note that the
dimensional ILC filtering, i.e. the particular multidimen-
sional ILC filtering where the dimension of the foreground
subspace is assumed to be constant over the whole sky and
the whole range of scales, and equal to (Nobs− 1). Indeed,
the CMB subtracted estimate expands as follows
CMB subtraction procedure is equivalent to an (Nobs− 1)–
?f = y − aatR−1y
atR−1a,
I − Wa?(Wa)tWa?−1(Wa)t?
= W−1(I − P1)Wy,
where W = R−1/2denotes the inverse square root of the
data covariance matrix.
Matrix P1 = Wa?(Wa)tWa?−1(Wa)tis an orthog-
Span(Wa) (one-dimensional ‘whitened’ CMB subspace).
Itimpliesthat
I − P1 = PH
the (Nobs− 1)–dimensional hyperplane H = [Span(Wa)]⊥,
which is orthogonal and complementary to the one-
dimensional whitened CMB subspace. Let us denote
v = Wa/|Wa| and consider an Nobs× (Nobs− 1) matrix V
such that [v |V] is a rotation. Then PH = V?VtV?−1Vtand,
?f = W−1PHWy,
= F?FtR−1F?−1FtR−1y,
which completes the proof since F is full rank (Nobs− 1).
Here, it is interesting to notice that the (Nobs− 1)–
dimensional ILC can be obtained without even knowing the
mixing matrix F since the procedure becomes equivalent to
the estimation obtained by subtracting the CMB-ILC esti-
mate from the observations data.
This equivalence means that the CMB subtraction
procedure does not take advantage of the fact that the
foreground mixing matrix can be almost rank-deficient in
some regions of the sky or at some scales (for instance at
small scale where the instrumental noise is dominant). The
(Nobs− 1)–dimensional subspace Col(F) reconstructed here
thus includes both noise and foregrounds components. Con-
sequently, such a foreground reconstruction is noisy. The
NFG–dimensional ILC procedure described in section 3.4
= W−1?
Wy,
(24)
onal projection (P2
1= P1 and Pt
1= P1) onto the line
istheprojectiononto
by denoting F = W−1V, we get
= W−1V?VtV?−1VtWy,
(25)

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Foreground component separation with generalised ILC7
Figure 1. Simulated Planck observations. A 12.5◦× 12.5◦patch of the simulated sky located at low Galactic latitude, around
Galactic coordinates of (l,b) = (72◦,−8◦). From left to right: observed map, foreground map, and thermal SZ map at 70 GHz. All maps
are at the resolution of the 70 GHz channel (14 arc-minutes).
performs a cleaner foreground reconstruction (in terms of
signal to noise ratio) because the effective rank NFG of
the foreground subspace and the foreground mixing matrix
(with reduced rank) are estimated locally in each needlet
domain. In effect, this boils down to performing at the same
time both component separation, and denoising by thresh-
olding the needlet coefficients.
4PLANCK SIMULATIONS AND RESULTS
We now turn to illustrating this discussion with examples
based on simulated data sets. We apply our multidimen-
sional FG ILC filter on a frame of needlets. For each needlet
domain considered, we both project the data onto the ‘full
rank’ foreground subspace (equivalent to simple CMB-ILC
subtraction), and onto a ‘reduced rank’ foreground subspace.
For the latter, we reject the eigenvalues smaller than 1.25
times the noise level, i.e. values for which the instrumental
noise contributes more than 80% of the total emission. This
is an arbitrary choice, selected for illustration purposes. In
practice, this threshold must be fixed considering the trade-
off between rejecting some low-level foreground emission and
letting in the final map too much instrumental noise. It also
depends on the size of the needlet domains used to com-
pute the statistics, which sets the accuracy with which the
eigenvalues of the covariance matrix are estimated.
Our investigations are carried out on sky simulations
generated with the Planck Sky Model (PSM) version 1.6.6.
Sky simulations include Gaussian CMB generated assuming
a theoretical angular spectrum fitting the WMAP obser-
vations, thermal and kinetic SZ effect, four components of
galactic ISM emission including thermal and spinning dust,
synchrotron, and free-free, and emission from point sources
(radio and infrared). The resolution and noise level of the
observations correspond to nominal mission parameters as
described in the Planck “Blue Book”. Details about PSM
simulations can be found in Leach et al. (2008) and Betoule
et al. (2009).
Figure 1 shows the ‘observed’ 70 GHz map, the input
foreground map at 70 GHz, and the thermal SZ map, all at
the resolution of the 70 GHz channel. Our maps are centred
on an interesting region which is both at low galactic lati-
tude, around (l,b) = (72◦,−8◦) and close to a set of bright
galaxy clusters. The 70 GHz reconstructed foregrounds, re-
covered by multidimensional ILC filtering, are shown in the
same region of the sky on the top panels of figure 2. The
corresponding reconstruction error (difference between re-
constructed output and original input) is displayed on the
bottom row of the same figure. The (Nobs−1)–dimensional
ILC reconstruction (left panels), equivalent to a simple sub-
traction of the ILC estimate of the CMB map to the ob-
servation map, is clearly noisy (as expected from the dis-
cussion of section 3.6.2). The reduction of the noise in the
foreground reconstruction is achieved by performing a NFG–
dimensional ILC reconstruction (middle panels), where NFG
is the local dimension of the FG subspace depending both on
the needlet scale and on the pixel. We observe the leakage of
a thermal SZ emission in the FG reconstruction on the left
and middle panels of figure 2. Using the modified ‘reduced
rank’ ILC introduced in section 3.5, we obtain instead the
reconstruction displayed on the right panels of figure 2 with
no visible contamination by SZ emission.
For completeness the same results are shown on full
sky maps in figure 3 and we have plotted the corresponding
power spectra in figure 4. The suppression of the noise con-
tamination is clearly visible on the spectrum at high ? when
a NFG–dimensional ILC method is employed.
5CONCLUSION
In this article, we have shown how the standard ILC proce-
dure, originally dedicated to the CMB extraction, can be
extended for the reconstruction of complex astrophysical
emissions, beyond the CMB alone. We have developed gener-
alised ILC filters (multidimensional ILC) to reconstruct the
diffuse emission of a complex multidimensional component

Page 8

8Mathieu Remazeilles, Jacques Delabrouille, Jean-Fran¸ cois Cardoso
Figure 2. 70GHz foreground multidimensional ILC reconstruction at low galactic latitude around (l, b) = (72◦, −8◦).
Top: CMB-orthogonal (Nobs− 1)–dimensional ILC map, CMB-orthogonal NFG–dimensional ILC map, (CMB+SZ)-orthogonal NFG–
dimensional ILC map. Bottom: error (difference input-output) maps respectively for CMB-orthogonal (Nobs− 1)–dimensional ILC,
CMB-orthogonal NFG–dimensional ILC, and (CMB+SZ)-orthogonal NFG–dimensional ILC.
originating from multiple correlated emissions, such as the
total Galactic foreground emission. Similar, though pixel-
based extensions have been also implemented in a fastICA-
based code, AltICA, as used in Leach et al. (2008) and are in-
tegrated in the Planck LFI Data Processing Center pipeline
(C. Baccigalupi, R.Stompor, private communication). Our
estimators were implemented on a needlet frame and tested
on simulations of Planck observations. This new ILC filter-
ing successfully reconstructs the foreground emission, ex-
empt from both the CMB and the SZ emission, and with a
reduced level of instrumental noise.
ACKNOWLEDGEMENTS
We thank Tuhin Ghosh, Radek Stompor and Carlo Bacci-
galupi for useful conversations related to this work. Some
of the results in this paper have been derived using the
HEALPix package (G´ orski et al., 2005). We also acknowl-
edge the use of the Planck Sky Model, developed by
the Component Separation Working Group (WG2) of the
Planck Collaboration.
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Foreground component separation with generalised ILC9
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