4 The BSS experimented tools

SOBI and its derived versions.

SOBI (Second Order Blind Identification, Belouchrani et al. 1997) is an efficient second order algorithm. It depends on the number of spatial shifts p of sources with themselves and their values s_i, $i\in (1,p)$. After the data whitening, a set of pcovariance matrices is computed. Cross-correlation terms are minimized, thus diagonal terms are maximized. A joint diagonalization criterion of several p covariance matrices improves its robustness (Cardoso & Souloumiac 1996). We adapted the algorithm from the temporal field to the 2D one.

In its original version (we call it SOBI1), we compute the cross-correlation matrices in the direct space: a vector signal is constructed by concatenation of the pixel rows.

For analyzing Nuclear Magnetic Resonance (NMR) spectrograms, Nuzillard (1999) modified SOBI by computing the cross-correlations of the Fourier transforms, which are easily estimated in the direct space. This can be viewed as an alternative correlation choice. SOBI1 takes into account the correlation at short distances, while the correlations at short frequency distances play the main role in SOBI2. NMR spectrograms display narrow peaks (or lines), so that the correlation rapidly decreases, and therefore SOBI2 provides better separation. For our current data this argument is not so clear. The cross-correlation between two vectors A and B is expressed as:

SOBI1 In the direct space:

$$% R_{AB}\left(\xi\right)={1\over L}\sum_k a_k\cdot b^*_{k-\xi}$$ (6)

where $\xi$ is the spatial shift and L the number of samples, a_k and b_k are elements of A and B, the sampling step being 1.

SOBI2 In the Fourier space:

$$% R_{AB}\left(n\right)=\sum_k a_k\cdot b_k^*\cdot {\rm e}^{-2i\pi({kn\over L})}$$ (7)

where n is the spatial frequency shift. More precisely ncorresponds to a shift of a cross-correlation computed on the Fourier transform of the vector A and B.

SOBI3 and SOBI4 are respectively SOBI2 and SOBI1 adaptations to 2D images. The algorithms take into account the correlation matrices between two images in different directions: rows, columns, diagonal, etc.

SOBI3 In the Fourier transform space:

$$% R_{AB}\left(n,m\right)=\sum_k\sum_l a_{kl}\cdot b_{kl}^*\cdot {\rm e}^{-2i\pi({nk\over L_1}+{ml\over L_2})}$$ (8)

where n and m are the spatial frequency shifts, L₁ and L₂ the row and column numbers and a_kl and b_kl pixel values of images A and B.

SOBI4 In the direct space:

$$% R_{AB}\left(\xi,\eta\right)={1\over L_1L_2}\sum_k\sum_l a_{kl}\cdot b^*_{k-\xi,l-\eta}$$ (9)

where $\xi$ and $\eta$ are also the spatial shifts.

JADE and FastICA algorithms.

A contrast is a mapping from the set of PDFs p_x to the real set ${\rm I \kern-0.2em R}$, x being a random vector of length N with values in ${{\rm I \kern-0.2em R}}^N$. A contrast satisfies the following requirements:

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f(Pp_x)=f(p_x), where P is any permutation matrix;

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$f(P_{\Lambda x})=f(p_x)$, $\Lambda$ is diagonal and can be inverted;

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$f(p_{Ax})\le f(p_x)$, if x has independent components, where A can be inverted.

In JADE (Joint Approximate Diagonalization of Eigen-matrices, Cardoso & Souloumiac 1993), the statistical independence of the sources is obtained through the joint maximization of the fourth order cumulants since these fourth order terms behave as contrast functions (Comon 1994).

In FastICA (Hyvärinen & Oja 1997) non-Gaussianity is measured by a fixed-point algorithm using an approximation of negentropy through a neural network. FastICA was first introduced using the kurtosis as a contrast function; then it was extended for general contrast functions such as:

$$% J_{\rm G}(y)=\vert Ey{G(y)}-E\nu{G(\nu)}\vert^p$$ (10)

where $\nu$ corresponds to the Gaussian variable which has the same mean and the same variance as y. Generally p is equal to 2. G(y) is used in its derivative form, g(y), in the fixed point algorithm. The couple (G(y),g(y)) can be chosen among the following functions:

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G(y)=y⁴, g(y)= y³;

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$G(y)=\log(\cosh(a y))$, $g(y)=\tanh(ay)$;

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$G(y)={\rm e}^{-{ay^2\over 2}}$, $g(y)=y{\rm e}^{-{ay^2\over 2}}$.

The algorithm works in two different ways: a symmetric solution is used for which the sources are computed simultaneously, or a deflation one for which the sources are extracted successively.