EMPTY BINS AND PRECONDITIONING

There are at least three ways to fill empty bins. They seem to be all equivalent, though that is not as obvious as I would like it to be.

The original way in Chapter is to restore missing data by ensuring that the restored data, after specified filtering, has minimum energy, say $\bold A\bold m\approx \bold 0$.Introduce the selection mask operator $\bold K$, a diagonal matrix with ones on the known data and zeros elsewhere (on the missing data). Thus $\bold A(\bold I-\bold K+\bold K)\bold m\approx \bold 0$ or  

$$\bold A (\bold I-\bold K) \bold m \quad \approx \quad - \bold A \bold K \bold m \quad =\quad-\bold A \bold m_k\;,$$ (27)

where we have defined $\bold m_k$ to be the data with missing values set to zero by $\bold m_k=\bold K\bold m$.

A second way to find missing data is with the set of goals  

$$\begin{array} {ccc} \ \ \bold K \bold m & \approx & \bold m_k\ \\ \epsilon \bold A \bold m & \approx & 0\end{array}$$ (28)

and take the limit as the scalar $\epsilon \rightarrow 0$.At that limit, we should have the same result as equation (27).

A third way to find missing data is to precondition equation (28), namely, try the substitution $\bold m = \bold A^{-1} \bold p$. 

$$\begin{array} {rrr} \bold K \bold A^{-1} \bold p & \approx & \bold m_k \\ \epsilon \bold p & \approx & 0 \ \ \end{array}$$ (29)

I think (hope) it is proven later that if we start from $\bold p = 0$ and if we are interested in the limit $\epsilon \rightarrow 0$we can simply forget about the fitting goal $\epsilon\bold p \approx 0$.