The filtering method

Abma (1995) solved a constrained least-squares problem to separate signal from spatially uncorrelated noise:

$$\begin{eqnarray} \bf Nn &\approx& \bf 0 \nonumber \\ \bf \epsilon Ss &\approx& \bf 0 \\ \mbox{subject to} &\leftrightarrow& \bf d = s+n \nonumber\end{eqnarray}$$ (2)

The first equation defines mathematically the annihilation filter ${\bf N}$ whereas the second equation defines the annihilation filter ${\bf S}$. Minimizing in a least-squares sense the fitting goals in equation (2) with respect to ${\bf s}$ leads to the following expression for the estimated signal:  

$$\bf \hat{s} = \left( \bold N^T \bold N + \epsilon^2 \bold S^T \bold S \right)^{-1} \bold N^T \bold N \bold d.$$ (3)

$\left( \bold N^T \bold N + \epsilon^2 \bold S^T \bold S \right)^{-1} \bold N^T \bold N$ is a projection filter. I call it the filtering method because the noise components are filtered out by the PEF ${\bf N}$ in equation (3).