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Estimating the CMB from the Planck/HFI data

Schmitz Morgan & Sitbon Pascal

April 2016

1

Introduction

The estimation of the Cosmic Microwave Background from data gathered by

missions such as Planck is a problem of source separation of huge importance

to cosmology, since good estimates of the CMB in turn teach us about the early

Universe and allow us to determine constraints on cosmological parameters.

Our project was to perform such an estimation, using data simulated by mixing

three sources: the CMB itself, thermal emissions from dust, and the SunyaevZel’dovich effect. The blind source separation problem thus created is similar

(though much simpler) to that encountered with the Planck and/or the WMAP

data.

Often, and in our case, such problems are approached by assuming that the

observations are a linear mixture of the sources, in which case the multichannel

data X is modelled as:

X = AS + N

Where A is a mixing matrix, S’s rows contain the sources themselves, and N is

a Gaussian Noise.

2

2.1

Data

Resolution problems

In our case, the data X consisted of six different mixtures of three input sources

(see Figure 1). The input data was given to us at a 5 Arcmin resolution, which

is also the resolution at which we aim to reconstruct the CMB. So before we

can tackle the source separation problem itself, we must first ensure each of our

mixtures is at the same resolution. We will thus change the resolution of the

first two images, that are at a resolution of 10 and 7 Arcmin, respectively. Now,

let Xi be our initial images. The convolution kernels hi in the pixel domain are

given. If x

¯i is our data at infinite resolution, ni the additive gaussian noise, and

Xi the data we showed above, we have: Xi = hi ∗ x

¯i + ni We want to convert

1

˜ 1 , X˜2 so that they have the same resolution as, say, X3 . Since we

X1 , X2 to X

have the hi ’s at our disposal, we only have to do:

X˜i = F −1

F(h3 )

× F(Xi )

F(hi )

(1)

Where × is the element-wise multiplication, and the division is also performed

element-wise, and F, F −1 are the Fourier and the inverse Fourier transform,

respectively. Note that we can run into computational problems if F(hi ) has

elements equal to 0. However, we verified that this case only presents itself when

the element at the same position in F(h3 ) is also naught (i.e. the convolution

kernel is ”narrower”, which was to be expected because of the difference in

resolution). The first two images on Figure 2 are X˜1 , X˜2 , now at a resolution

Figure 1: The six mixtures (data) and the three input sources CMB, Dust and

SZ effect. Note all images are at a resolution of 5 Arcmin, apart from the first

two mixtures

of 5 Arcmin. Since the initial resolution was much poorer, it is to be expected

2

that at 5 Arcmin, the images would be quite degraded. Still, the results above

looked quite strange to us. We thought this might be due to the noise, which we

had not taken into account so far. An idea we had was to perform a preliminary

step of denoising before applying the change of resolution above. However, we

cannot do this, since the denoising might alternate our source separation model

X = AS. Instead, we turned to [1], where kernels are constructed precisely to

change the resolutions of images. There, they suggest adding a filtering step to

(1) :

F(h3 )

× F(Xi )

X˜i = F −1 f3

F(hi )

Where:

1

h

f3 (x) = x × 0.5 × 1 + cos π ×

0

if k ≤ kL

x−kL

kH −kL

i

if kL ≤ k ≤ kH

if k > kH

And:

2π

FWHM3

kL = 0.7 × kH

kH = κ3 ×

κ3 being an instrument- and wavelength-specific constant that lies in [1.08, 1.46].

Since we did not know the values for Planck, we simply used κ3 = 1.27. The two

images on the second row of Figure 2 are the images at 5 Arcmin thus obtained.

In what follows, we will only work with these resolution-corrected data.

3

Source separation

We are now ready to tackle the source separation problem itself. We began by

applying the two algorithms we were most familiar with, thanks to Practical

Work 3, ICA and GMCA.

3.1

ICA

Figure 3 shows the estimated sources obtained by applying FastICA. While two

sources have been clearly identified (Dust and CMB), the last estimated source

looked very much like noise to us. ICA is known not to be appropriate to deal

with noise, and because the SZ effect indeed appears to be very point-wise and

to contribute fairly little to the mixtures, we thought it made sense the additive

noise had a bigger contribution than the SZ source did. Therefore, ICA detected

noise as the third source instead of SZ.

3

Figure 2: 1st row: 5 Arcmin resolution for X˜1 , X˜2 after applying (1). 2nd row:

5 Arcmin resolution for X˜1 , X˜2 after applying (1) complete with a filtering step.

3rd row: original 10 and 7 Arcmin X1 , X2 , respectively

Interestingly, when we serendipitously tried to identify four sources instead of

the actual number of three using ICA, one of the estimated sources did look a

lot more like the input SZ (although still fairly noisy) - see Figure 4. We think

this observation is coherent with our hypothesis above.

3.2

GMCA

We now apply the GMCA algorithm to our data, with some slight variants,

namely whether and how we use the spectral signatures at our disposal.

First, however, we need to apply a transform to get a sparse representation of

our data. We chose to use the Starlet transform, as we know it is efficient for this

kind of astrophysical problem, and because we have experience using this particular transform from our practical work sessions. We use its implementation

4

Figure 3: Estimated and input sources using FastICA

Figure 4: Estimated and input sources using FastICA, imposing that 4 sources

be recovered

from the pyBSS module.

3.2.1

No spectral signature

We begin by running GMCA as it is implemented in the pyBSS script given to

us for Practical Work 3. That is, the user only specifies a maximum threshold factor, kmax , and a minimum threshold factor kmin (arguments maxts and

mints, respectively), which are in turn used to determine the threshold at each

iteration t ∈ {1, . . . , N }:

5

λt = kt × σ

ˆM AD

Where:

kt+1

k1 = kmax

kmax − kmin

= kt −

N −1

and σ

ˆM AD is the MAD estimator of the standard deviation of the contaminating

noise at current iteration. Note that one of the ideas we had for improvement

was to use the real noise standard deviation values, σi , that were at our disposal,

to select more efficient thresholds.

Interestingly, when using this method (even with various values for kmax and

kmin ), it seemed that only the Dust source was estimated, with all three reconstructed sources looking very much alike - see Figure 5. We never managed to

find an explanation for this result, and assume it is only due to an error on our

part. If that is indeed the case, however, we could not find it.

Figure 5: Estimated ”sources” when running GMCA on our mixtures without

using spectral signatures

3.2.2

Using spectral signatures

Since both the spectral signature of the CMB and that of the SZ effects were

given to us, we tried running GMCA while imposing that given columns of the

6

mixing matrix A stayed equal to them.

When imposing only the column corresponding to the CMB to be fixed to its

true value, we yielded the results shown in Figure 6, which were also the best we

managed to reach1 . The same observation we made for the ICA results holds

true here: the SZ effect was not at all recovered as one of the sources (while,

again, wrongly imposing an additional source be estimated made it appear, but

unsurprisingly hindered the quality of the CMB estimate).

Figure 6: Estimated sources when running GMCA on our mixtures when imposing a column of A be equal to the CMB’s spectral signature

Therefore, we had high hopes for our tweaked version of GMCA, which keeps

another column of A constant (equal to the spectral signature of the SZ effect).

However, it only led us to yet another perplexing result, with the CMB decently

recovered, but the other two sources equal to what looked very much like the

dust source (similar to what we showed in Figure 5). Again, we found no

satisfying explanation for these results: while it makes sense to us that a third

source would look like noise if the algorithm failed to capture the SZ effect’s

contribution to the mixtures, we cannot fathom why the same source would

appear several times.

Nonetheless, because of this incapacity of recovering the SZ effect, even when

using its actual spectral signature, we once again tried to tweak the number of

sources we asked GMCA to recover, and launched it for only two sources. Our

1 Since visual inspection of the estimated CMB seemed of a similar quality for several

different methods, eg. ICA and GMCA with only one column of A constrained, we compared

them by plotting the residuals between the input and the estimated CMB. See Figure 7, or

the accompanying notebook, for examples.

7

reasoning was that if the SZ effect really accounted for less than the noise, then

it made sense to consider it as such. This led yet again to two distinct estimated

sources ressembling the input Dust.

3.3

SMICA (Spectral Matching Independant Component

Analysis

Since GMCA yielded the best results (in terms of visual inspection of the CMB

residuals), we thought of trying to implement the L-GMCA (Local Generalized

Morphological Component Analysis) method. However, it is our understanding

that one of the key appeals of L-GMCA is that it does not (unlike SMICA)

require to use post-processing techniques such as inpainting to get a complete

CMB map. In our case, however, since there is no contamination by the Galactic

Center, we would not need to use interpolation of any sort even if we used

SMICA. Moreover, the results yielded by ICA seemed fairly decent considering

it is a fact here that noise contamination is not negligible here, and SMICA was

designed specifically to work in such cases (unlike regular ICA).

We therefore chose to try and implement the SMICA method instead, but failed

to do so for several reasons. One of them was that, if RX˜ is the estimated power

˜X

˜ T ), then using

spectrum of our mixtures in the Fourier domain (i.e. RX˜ = X

different frequency bands Ω would always yield:

RX˜ (Ω) = ARS˜ (Ω)AT

Our plan was therefore to perform the joint diagonalisation of RX˜ (Ω) for several different frequency bands Ω, which would yield an estimate of A. We then

planned to determine S directly (which we thought seemed simpler than, for instance, the EM algorithm proposed in the original SMICA paper by Delabrouille

et al). However, since our matrixes RX˜ were of size 6 × 6, they each had 6 real

eigenvalues, but of course since we know we only have three contributing sources,

we would have wanted A to be of size 6 × 3.

An idea we had to overcome this was to perform a PCA of sorts on RX˜ [Ω]

(i.e. truncate the eigenvectors matrix to keep only the three corresponding to

the most significant eigenvalues). This heuristic made sense to us in that the

eigenvalues are directly linked to the input sources; therefore it made sense the

most significant ones were associated with the sources, while the others were

only non-zero because of contamination. Another intuitive way to consider it

was that, no matter the number of mixtures, since X = AS, if the model was

perfectly true (and there was no noise!), the rank of X would be determined by

the number of input sources, and adding more mixtures would therefore only

add 0 as eigenvalues to RX .

We never got that far, however, since another trouble we stumbled upon was

that the RX˜ [Ω] were complex, therefore the estimated eigenvectors corresponding to A would also have been. We know properly chosen frequency bands can

yield real matrixes RX˜ [Ω], but we were not sure how to construct them, nor how

8

to actually perform the joint diagonalization. Due to lack of time, we stopped

there and never managed to implement a working version of SMICA.

4

Conclusion

A look at the residual (that is, the difference between the estimated and the

input CMB images), as shown on Figure 7, seems to indicate that our best

estimate of the CMB is, in the end, not too bad. However, we never managed

to properly recover all three input sources, the SZ effect remaining undetected

unless we forced our algorithms to estimate more sources than were actually

used to generate our data.

In the end, said best result was obtained using a fairly simple variant of the

GMCA algorithm. We were hoping to implement a completely different method

(SMICA), a more advanced and state-of-the-art variant of the GMCA (L-GMCA),

or at least to perform a smarter selection of the thresholds used by maybe using

the σi values at our disposal.

One of the reason for our lack of time, we think, might be because we spent a

lot of it on things that probably seemed a lot more complicated to us (due to

our lack of experience in basic signal processing practical problems) than they

ought to be.

9

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