7 The results

We selected four decompositions for our experiments:

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KL: The basic KL expansion;

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SOBI2-16: SOBI in the 1D Fourier space, with 16 correlation matrices;

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SOBI4-8: SOBI in the 2D direct space, with 8 correlation matrices;

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FastICA: FastICA with the deflation algorithm with $g(y)=\tanh(y)$ associated function.

Table 2 contains SMI values of each selected BSS.

 

 

Table 2: Source mutual information of the selected decompositions

Method	SMI
KL	0.771
SOBI2-16	0.581
SOBI4-8	0.805
FastICA	0.584

FastICA and SOBI2-16 give a similar SMI, significantly lower than KL and SOBI4-8 ones. In Fig. 3 the resulting sources of four selected decompositions are shown.

Figure 3: The sources obtained from the four selected BSSs on the 3C 120 HST images. From top to bottom respectively the Karhunen-Loève expansion, SOBI2 algorithm with 16 cross-correlation matrices, SOBI4 algorithm with 8 cross-correlation matrices and FastICA algorithm with a deflation algorithm and $g(y)=\tanh(y)$

The last three decompositions provide similar sources, different from the KL expansion. In Fig. 4 the corresponding source filters are drawn.

Figure 4: The source filters obtained from the four selected BSSs on the 3C 120 HST images. From top to bottom respectively KL, SOBI2-16, SOBI4-8 and FastICA. The line thickness is decreasing with the source order, the thickest one corresponds to source 1. The lines are also shifted -10 units on ordinate, from a source to the following one, in order to clarify the diagrams

We find some similarities between the resulting filters, even if the plots are sometimes reversed due to the sign of the demixing coefficients. We note that the sources and the mixture matrix are defined with scalars $\alpha_{j}$. Even if the source variance is equal to 1, we have an uncertainty about the signs. The diagrams in Fig. 4 are built from the combination of the filter transmissions with the demixing coefficients. It is obvious that it is not possible to obtain source filters which extract well defined spectral regions only by combinations of the original filters because we have not all the required monochromatic images. BSSs carried out the best linear image combinations, namely those having the maximum independence.

In Table 3 we indicated the percentage of energy contained in each source for each BSS. We remark that this break-down is different from one decomposition to another.

 

 

Table 3: Source energy of the selected decompositions

Method	1	2	3	4
KL	0.942	0.034	0.015	0.008
SOBI2-16	0.386	0.449	0.030	0.134
SOBI4-8	0.779	0.177	0.029	0.015
FastICA	0.295	0.626	0.038	0.041

The KL expansion is, of course, the decomposition that leads to the greatest energy concentration.

Interpretation of the source images.

If we exclude the KL expansion, the source decomposition from the three other selected BSSs are similar, as commented on below. Each source displays interesting features:

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Source 1: This source displays mainly the central galaxy region. For SOBI4-8 decomposition, we got the maximum of energy, but for the other BSSs the energy was seriously reduced. We note a trace of structures associated with source 2. The source filters look similar. They correspond to a difference of the mean flux after 7000 Å and between 4500 to 6000 Å. The H$_{\alpha}$ region is excluded, but not the [OIII] one, explaining the trace of source 2;

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Source 2: This is the most interesting source for physical insight. On the source filters we can see that the continuum after 7000 Å was extracted. The other part is at the opposite end, in a region containing [OIII] lines. For SOBI2-16, H$_{\alpha}$ line plays a faint role, while it has no role for the other BSSs. This source corresponds to the ionized [OIII] regions surrounding the galaxy. It corresponds also to a large part of the energy and for two decompositions the maximum value is obtained. If we compare the source decomposition obtained with KL to the other ones, we note that for the BSSs an important part of the energy was clearly transferred from source 1 to source 2. The first KL source roughly corresponds to a mean image, and the energy from ionized regions was simply averaged. The energy from KL source 2 only corresponds to the variations from one filter to the other. For the other BSSs, the rotations allow one to partially recover the energy from the mean;

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Source 3: This corresponds clearly to a set of rings around the nucleus. Its energy is relatively very constant from one BSS to another. These rings could be seen in the KL sources, but the other BSSs display a cleaner result. Taking into account their size, these rings do not correspond to an Airy pattern associated to a simple mirror of HST size. For the three BSSs [OIII] and H$_{\alpha}$ lines play the major part in the source filters, in an opposite manner. We compared the source 3 images to the point spread function (PSF) of WFPC2 at 675 nm obtained with the Tiny Tim program (Krist & Hook 2000) (Fig. 5).

Figure 5: The point spread function at the wavelength 675 nm of the WFPC2 camera

Even if the contrasts were not the same, the ring patterns correspond to that function. The main energy in the lines comes from H$_{\alpha}$, and this is the reason why the structure looks like its corresponding PSF at the central position. It is amazing that BSS allows us to display a real physical phenomenon, which is due to the fact that a large part of the energy in image F675W is emitted in a spectral line in a small region, which is not resolved with the WFPC2. For this line we have the image of its PSF, which displays rings due to the central occultation of the telescope. Due to the large wavelength range, the PSF corresponding to the spectral continuum region does not display these rings. As we can see in Fig. 4, the H$_{\alpha}$ line is excluded from source 1. BSSs played their role by extracting the H$_{\alpha}$ PSF as a specific source;

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Source 4: This is the less informative source. For SOBI2 it looks like a smooth version of source 2. The main energy comes from the region between 4500 and 5200 Å. The influence of the emission lines is faint, but not equal to 0.

Discussion.

BSS algorithms extracted three sources which seemed to correspond to three independent physical components. Even if the sources are orthogonal, some traces of the structures which are well-defined in each source are seen in the other sources. A physical mixture can still exist, but it is faint compared to the one in the original images. The source 4 may be interpreted as the residue of the separation.

We can note that a second order blind identification algorithm, based on correlations in a large region around a pixel, carries out quite a similar decomposition to a blind identification related to local high order statistics. That fact brings some confidence to the obtained decomposition.

This statistical analysis tool allows one to get a simple model of the galaxy 3C 120 with two components:

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A very bright nucleus which is not resolved in these exposures. The central pixels are saturated, a linear artifact due to the CCD blooming exists in each image. In Fig. 5 we can notice the large extension of the PSF at a given wavelength (6750 Å). This size is similar to the extension of source 1. This is because this source can be only due to an unresolved nucleus, like the source 3 as we discussed above. Sources 1 and 3 correspond to the same physical component. Obviously we can see other features in the image sources, BSS does not provide a perfect physical model, but a decomposition which is optimal for a given statistical criterion. There is no reason that the physical reality corresponds exactly to this criterion;

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A gaseous region surrounding the galaxy as shown in source 2. Stellar objects can be associated with this component, but the main structures are described by the ionized [OIII] regions.

This resulting description is not new, but it cannot be seen directly in the observed images or in the KL sources. BSS failed to display the optical counterpart of the radio jet as a specific source. On the one hand, we did not process enough images allowing us to find more components. On the other hand, the radio jet does not necessarily correspond to a source with a pixel distribution independent from the ones of the other image components. Images obtained with more filters are needed to improve the decomposition.