ROW PARTITIONED OPERATORS

A problem arises with partitioned operators. Here we are fitting observed data to theoretical data where there are two classes of model parameters $\bold m_1$ and $\bold m_2$.We seek to minimize the norm of the residual $\bold r$ defined by  

$$\bold 0 \quad\approx\quad \bold r \quad=\quad \left[ \alpha ... ..._1/\alpha \\ \bold m_2/\beta \end{array}\right] \ - \ \bold d$$ (9)

where $\alpha$ and $\beta$ are arbitrary scaling constants. The residual is independent of $\alpha$ and $\beta$,but when we ``solve" this system using (as we must) the idea that $\bold F^{-1}\approx \bold F'$,we see the result depends on $\alpha$ and $\beta$,namely it contains $\alpha^2$ and $\beta^2$ 

$$\left[ \begin{array} {l} \hat \bold m_1/\alpha \\ \hat \b... ...alpha \bold F' \\ \beta \bold B' \end{array}\right] \ \bold d$$ (10)

Let us find the best $\alpha$ and $\beta$.Inserting the image (10) into the residual (9) we get

$$\begin{eqnarray} \bold 0 &\approx& \bold r \quad=\quad \left[ \alpha \bold F \q... ...& \bold r \quad=\quad \alpha^2 \bold u + \beta^2 \bold v -\bold d\end{eqnarray}$$ (11) (12) (13)

which defines two vectors $\bold u$ and $\bold v$.

We find the best scaling factors by setting to zero the derivative of $\vert\vert\bold r\vert\vert^2$with respect to $\alpha^2$ and $\beta^2$

$$\bold 0 \quad=\quad \left[ \begin{array} {rr} \bold u'\bold... ...y} {c} \bold u'\bold d \\ \bold v'\bold d \end{array} \right]$$ (14)

solving gives  

$$\left[ \begin{array} {c} \alpha^2 \\ \beta^2 \end{array} \... ...y} {c} \bold u'\bold d \\ \bold v'\bold d \end{array} \right]$$ (15)

I believe it can be shown that the values $\alpha^2$ and $\beta^2$ are positive.

Recalling that $\bold u = \bold F \bold F' \bold d$ and $\bold v = \bold B \bold B' \bold d$,let us now define $\bold a = \bold F' \bold d$ and $\bold b = \bold B' \bold d$so $\bold u = \bold F \bold a$ and $\bold v = \bold B \bold b$.

In imaging applications we customarily ignore the scaling factor which is the common part of $\alpha$ and $\beta$,namely, the denominator determinant in (15). We have the proportions

$$\begin{eqnarray} \alpha^2 &\sim& \ \ (\bold b' \bold B' \bold B \bold b) ( \b... ...bold a) + (\bold a' \bold F' \bold F \bold a) ( \bold b' \bold b)\end{eqnarray}$$ (16) (17) (18)

A deeper problem of interest arises when we seek the best diagonal matrix scaling. Then we replace $\alpha^2$with two diagonal matrices, one before and one after $\bold F$.Likewise with $\beta$ and $\bold B$.We need help finding those four diagonal matrices.