Assume the inverse of the above matrix exists

${\displaystyle {\begin{bmatrix}A&B\\C&D\end{bmatrix}}{\begin{bmatrix}E&F\\G&H\end{bmatrix}}=J.}$

Here ${\displaystyle J}$ is or is close to ${\displaystyle I}$. The local inverse algorithm is as follows:

(1) ${\displaystyle g_{ex}}$. An extrapolated version of ${\displaystyle g}$ is obtained by

${\displaystyle g_{ex}|_{\partial G}=f|_{\partial F}.}$

(2) ${\displaystyle y_{0}}$. An approximate version of ${\displaystyle y}$ is calculated by

${\displaystyle y_{0}=Hg_{ex}.}$

(3) ${\displaystyle y'}$. A correction for ${\displaystyle y}$ is done by

${\displaystyle y'=y_{0}+y_{\text{co}}.}$

(4) ${\displaystyle f'}$. A corrected function for ${\displaystyle f}$ is calculated by

${\displaystyle f'=f-By'.}$

(5) ${\displaystyle g_{1ex}}$. An extrapolated function for ${\displaystyle g}$ is obtained by

${\displaystyle g_{1ex}|_{\partial G}=f'|_{\partial F}.}$

(6) ${\displaystyle x_{1}}$. A local inverse solution is obtained

${\displaystyle x_{1}=Ef'+Fg_{1ex}.}$

In the above algorithm, there are two time extrapolations for ${\displaystyle g}$ which are used to overcome the data truncation problem. There is a correction for ${\displaystyle y}$. This correction can be a constant correction to correct the DC values of ${\displaystyle y}$ or a linear correction according to prior knowledge about ${\displaystyle y}$. This algorithm can be found in the following reference:.^[2]

In the example of the reference,^[3] it is found that ${\displaystyle y'=y_{0}+y_{\text{co}}=ky_{0}}$, here ${\displaystyle k=1.04}$ the constant correction is made. A more complicated correction can be made, for example a linear correction, which might achieve better results.

A^+ B is close to 0

Shuang-ren Zhao defined a Local inverse^[2] to solve the above problem. First consider the simplest solution

${\displaystyle f=Ax+By,}$

or

${\displaystyle Ax=f-By=f'.}$

Here ${\displaystyle f'=f-By}$ is the correct data in which there is no influence of the outside object function. From this data it is easy to get the correct solution,

${\displaystyle x'=A^{-1}f'.}$

Here ${\displaystyle x'}$ is a correct(or exact) solution for the unknown ${\displaystyle x}$, which means ${\displaystyle x'=x}$. In case that ${\displaystyle A}$ is not a square matrix or has no inverse, the generalized inverse can applied,

${\displaystyle x'=A^{+}(f-By)=A^{+}f'.}$

Since ${\displaystyle y}$ is unknown, if it is set to ${\displaystyle 0}$, an approximate solution is obtained.

${\displaystyle x_{0}=A^{+}f.}$

In the above solution the result ${\displaystyle x_{0}}$ is related to the unknown vector ${\displaystyle y}$. Since ${\displaystyle y}$ can have any value the result ${\displaystyle x_{0}}$ has very strong artifacts, namely

${\displaystyle \mathrm {error} _{0}=|x_{0}-x'|=|A^{+}By|}$.

These kind of artifacts are referred to as truncation artifacts in the field of CT image reconstruction. In order to minimize the above artifacts in the solution, a special matrix ${\displaystyle Q}$ is considered, which satisfies

${\displaystyle QB=0,}$

and thus satisfies

${\displaystyle QAx=Qf-QBy=Qf.}$

Solving the above equation with Generalized inverse gives

${\displaystyle x_{1}=[QA]^{+}Qf=[A]^{+}Q^{+}Qf.}$

Here ${\displaystyle Q^{+}}$ is the generalized inverse of ${\displaystyle Q}$, and ${\displaystyle x_{1}}$ is a solution for ${\displaystyle x}$. It is easy to find a matrix Q which satisfies ${\displaystyle QB=0}$, specifically ${\displaystyle Q}$ can be written as the following:

${\displaystyle Q=I-BB^{+}.}$

This matrix ${\displaystyle Q}$ is referred as the transverse projection of ${\displaystyle B}$, and ${\displaystyle B^{+}}$ is the generalized inverse of ${\displaystyle B}$. The matrix ${\displaystyle B^{+}}$ satisfies

${\displaystyle BB^{+}B=B,}$

from which it follows that

${\displaystyle QB=[I-BB^{+}]B=B-BB^{+}B=B-B=0.}$

It is easy to prove that ${\displaystyle QQ=Q}$:

${\displaystyle {\begin{aligned}QQ&=[I-BB^{+}][I-BB^{+}]=I-2BB^{+}+BB^{+}BB^{+}\\&=I-2BB^{+}+BB^{+}=I-BB^{+}=Q,\end{aligned}}}$

and hence

${\displaystyle QQQ=(QQ)Q=QQ=Q.}$

Hence Q is also the generalized inverse of Q

That means

${\displaystyle Q^{+}Q=QQ=Q.}$

Hence,

${\displaystyle x_{1}=A^{+}[Q]^{+}Qf=A^{+}Qf}$

or

${\displaystyle x_{1}=[A]^{+}[I-BB^{+}]f.}$

The matrix

${\displaystyle A^{L}=[A]^{+}[I-BB^{+}]}$

is referred to as the local inverse of the matrix ${\displaystyle {\begin{bmatrix}A&B\\C&D\end{bmatrix}}.}$ Using the local inverse instead of the generalized inverse or the inverse can avoid artifacts from unknown input data. Considering,

${\displaystyle [A]^{+}[I-BB^{+}]f'=[A]^{+}[I-BB^{+}](f-By)=[A]^{+}[I-BB^{+}]f.}$

Hence there is

${\displaystyle x_{1}=[A]^{+}[I-BB^{+}]f'.}$

Hence ${\displaystyle x_{1}}$ is only related to the correct data ${\displaystyle f'}$. This kind error can be calculated as

${\displaystyle \mathrm {error} _{1}=|x_{1}-x'|=|[A]^{+}BB^{+}f'|.}$

This kind error are called the bowl effect. The bowl effect is not related to the unknown object ${\displaystyle y}$, it is only related to the correct data ${\displaystyle f'}$.

In case the contribution of ${\displaystyle [A]^{+}BB^{+}f'}$ to ${\displaystyle x}$ is smaller than that of ${\displaystyle [A]^{+}By}$, or

${\displaystyle \mathrm {error} _{1}\ll \mathrm {error} _{0},}$

the local inverse solution ${\displaystyle x_{1}}$ is better than ${\displaystyle x_{0}}$ for this kind of inverse problem. Using ${\displaystyle x_{1}}$ instead of ${\displaystyle x_{0}}$, the truncation artifacts are replaced by the bowl effect. This result is the same as in local tomography, hence the local inverse is a direct extension of the concept of local tomography.

It is well known that the solution of the generalized inverse is a minimal L2 norm method. From the above derivation it is clear that the solution of the local inverse is a minimal L2 norm method subject to the condition that the influence of the unknown object ${\displaystyle y}$ is ${\displaystyle 0}$. Hence the local inverse is also a direct extension of the concept of the generalized inverse.