Separation of Stellar Spectra Based on Non-negativity and Parametric Modelling of Mixing Operator

Abstract

In this chapter, we propose blind source separation methods to extract stellar spectra from hyperspectral images. The presented work particularly concerns astrophysical data cubes from the multi-unit spectroscopic explorer (MUSE) European project. These data cubes here consist of a spectrally and spatially transformed version of the light signals emitted by dense fields of stars, due to the influence of atmosphere and of the MUSE instrument, which is modelled by the so-called point spread function (PSF). The spectrum associated with each pixel of these data cubes can thus result from contributions of different star signals. We first derive the associated mixing model, taking into account the PSF properties. We then propose two separation methods based on the non-negativity of observed data, source spectra and mixing parameters. The first method is based on non-negative matrix factorization (NMF). The second one, which is the principal contribution in this chapter, uses the proposed parametric modelling of the mixing operator and performs alternate estimation steps for the star spectra and PSF parameters. Several versions of this method are proposed, in order to eventually combine good estimation accuracy and reduced computational load. Tests are performed with simulated but realistic data. The separation method based on parametric modelling gives very satisfactory performance.

Appendix

We can write

$$\begin{aligned} \dfrac{\partial J}{\partial \alpha }= & {} \sum _{k} \sum _i ~ \dfrac{\partial J}{\partial M_{ki} } \dfrac{\partial M_{ki}}{\partial \alpha } \end{aligned}$$

(6.29)

$$\begin{aligned} \dfrac{\partial J}{\partial \beta }= & {} \sum _{k} \sum _i ~ \dfrac{\partial J}{\partial M_{ki} } \dfrac{\partial M_{ki}}{\partial \beta }. \end{aligned}$$

(6.30)

Derivation of (6.14) with respect to matrix $\varvec{M}$ gives

$$\begin{aligned} \dfrac{\partial J}{\partial M_{ki}} = \left( y_{k} - (\varvec{Mx})_{k} \right) (-x_i) = - \left[ (\varvec{y} - \varvec{Mx})\varvec{x}^T \right] _{ki}. \end{aligned}$$

(6.31)

Calculation with respect to $\alpha$ then reads

$$\begin{aligned} \dfrac{\partial J}{\partial \alpha }= & {} \sum _k \sum _i ~ \left( y_k - (\varvec{Mx})_k \right) (-x_i) \times \dfrac{\partial M_{ki}}{\partial \alpha } \nonumber \\= & {} - \sum _k ~ \left( y_k - (\varvec{Mx})_k \right) \sum _i \dfrac{\partial M_{ki}}{\partial \alpha } x_i \nonumber \\= & {} - \sum _k ~ \left[ \varvec{y} - \varvec{Mx} \right] _k \left[ \dfrac{\varvec{\partial M}}{\varvec{\partial \alpha }} \varvec{x} \right] _k \nonumber \\= & {} - \left[ \dfrac{\varvec{\partial M}}{\varvec{\partial \alpha } } \varvec{x} \right] ^T (\varvec{y} - \varvec{Mx}). \end{aligned}$$

(6.32)

The same type of expression is obtained for $\beta$:

$$\begin{aligned} \dfrac{\partial J}{\partial \beta } = - \left[ \dfrac{\varvec{\partial M}}{\varvec{\partial \beta } } \varvec{x} \right] ^T (\varvec{y} - \varvec{Mx}). \end{aligned}$$

(6.33)

To compute $\dfrac{\varvec{\partial M}}{\varvec{\partial \alpha }}$ and $\dfrac{\varvec{\partial M}}{\varvec{\partial \beta }}$, we use the expression of $M_{ki}$ as a function of $(\alpha ,\beta )$, given by Eq. (6.10). To simplify notations, as the star positions do not depend on $(\alpha ,\beta )$, we introduce $Z = ||p_k-z_i|| _{F}^2$ in expression (6.10).

This gives, for $\alpha$:

$$\begin{aligned} \dfrac{\partial M_{ki}}{\partial \alpha }= & {} \dfrac{\beta - 1}{\pi } \times \dfrac{-2}{\alpha ^3}\left( 1 + \dfrac{Z}{\alpha ^2} \right) ^{- \beta } \nonumber \\&+ \dfrac{\beta - 1}{\pi \alpha ^2} \times (-\beta )\left( 1 + \dfrac{Z}{\alpha ^2} \right) ^{- \beta - 1}\times \dfrac{-2}{\alpha ^3}\times ~ Z \nonumber \\= & {} \dfrac{2 ( \beta - 1)}{\pi \alpha ^3}\left( 1 + \dfrac{Z}{\alpha ^2} \right) ^{- \beta -1} \left( \dfrac{\beta }{ \alpha ^2}~ Z - \left( 1+\dfrac{Z}{\alpha ^2}\right) \right) \nonumber \\= & {} \dfrac{2 ( \beta - 1)}{\pi \alpha ^3}\left( 1 + \dfrac{Z}{\alpha ^2} \right) ^{- \beta -1} \left( \dfrac{\beta -1}{ \alpha ^2}~ Z - 1 \right) \end{aligned}$$

(6.34)

And calculation for $\beta$ reads¹⁰

$$\begin{aligned} \dfrac{\partial M_{ki}}{\partial \beta }= & {} \dfrac{ 1}{\pi \alpha ^2} \left( 1 + \dfrac{Z}{\alpha ^2} \right) ^{- \beta } - \dfrac{\beta - 1}{\pi \alpha ^2}\left( 1 + \dfrac{Z}{\alpha ^2} \right) ^{- \beta } \times \ln \left( 1 + \dfrac{Z}{\alpha ^2} \right) \nonumber \\= & {} \dfrac{ 1}{\pi \alpha ^2} \left( 1 + \dfrac{Z}{\alpha ^2} \right) ^{- \beta } \times \left( 1 - (\beta - 1)\ln \left( 1 + \dfrac{Z}{\alpha ^2} \right) \right) \end{aligned}$$

(6.35)