1 Multispectral analysis of astronomical images

The spectral distribution of celestial sources carries essential information on the physical processes which take place in these objects. Multispectral images are currently produced using CCD detectors and color filters which delimit well-defined spectral bands. From the flux measurements for each filter, astrophysicists compute color indexes from which they deduce the parameter values related to stars or non-stellar sources.

The color indexes allow one to classify the pixels. This is enough if the pixel intensity comes from only one source. But often pixel classification suffers from the drawback of the pixel being in fact a mixture: each pixel value results from the contribution of different objects. If we admit that this mixing is linear, each image X_i, a mixture of physically independent sources S_j, may be written:

$$% X_i=\sum a_{ij} S_j +N_i$$ (1)

where the matrix A=[a_ij] is called the mixing matrix. N_iis the noise of image X_i. This model is called the cocktail party in the framework of communication theory (Hyvärinen 1999). Solving Eq. (1), in which Aand S are the unknowns, is the goal of all BSS methods.

The spectral and spatial energy distributions are determined by the transfer radiation equation. Non-linearities result from this transfer. Consequently a set of colored sky images cannot be exactly written as a weighted linear combination of independent sources. But since Eq. (1), being linear, is far more tractable, it is important to examine the ability of Blind Source Separation (BSS) methods to display interesting features, which could help astrophysicists to improve their description of celestial sources.

The Karhunen-Loève (KL) expansion constitutes the oldest approach to decorrelate signals. The cross-correlation matrix of the images is first computed and a singular value decomposition is carried out. The sources are computed as the weighted means of the images, the weights being the eigenvector components. This approach was applied many times for astrophysical problems: data compression (Bijaoui 1974; Pelat 1974), identification of variable phenomena (Bijaoui & Doazan 1979), visualization of a large set of images, computation of energy spectrum (Vogeley & Szalay 1996; Tegmak et al. 1997) and classification (Connolly et al. 1995).

Let us consider the probability density function (PDF) of the observed images. In the case of Gaussian PDFs uncorrelated pixel values are equivalent to independent ones, and the KL expansion is sufficient to separate sources. But generally, the PDFs are not Gaussian and a set of blind source separation methods was developed in order to take account of higher order statistics leading to what has been called Independent Component Analysis (ICA). An alternative approach, based on temporal (or spatial) source correlations has also been developed (Belouchrani et al. 1997).

First we present the data set on which we have carried out the experiments. We give then a short overview of BSS methods. These applied methods provide slightly different separations. We describe some criteria to evaluate their quality. In particular, mutual information is considered. These results are discussed in the light of the physical nature of the celestial object. We conclude on the perspectives opened up by these new tools.