Local inverse

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

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

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

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

(2) ${\displaystyle y_{0}}$. An approximate ${\displaystyle y}$ function 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 ${\displaystyle g}$ function 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 ${\displaystyle g}$ functions 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 ${\displaystyle y}$ function or a linear correction according to the prior knowledges about the ${\displaystyle y}$ function. This algorithm can be found in reference.[2]

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

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 the influence of the object function in outside. From this data it is easy to get correct solution,

or

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

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

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

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

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

On the above solution the result ${\displaystyle x_{0}}$ is related to the unknown vector ${\displaystyle y}$. Since ${\displaystyle y}$ can be any values, this way the result ${\displaystyle x_{0}}$ has very strong artifacts which is

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

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

${\displaystyle QB=0}$

Hence,

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

solve the above equation with Generalized inverse

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

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

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

This kind of matrix ${\displaystyle Q}$ is referred as transverse projection of matrix ${\displaystyle B}$

Here ${\displaystyle B^{+}}$ is the generalized inverse of the matrix ${\displaystyle B}$. ${\displaystyle B^{+}}$ satisfies

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

It can be proven 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

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

${\displaystyle A^{L}}$ is referred as the local inverse of Matrix ${\displaystyle {\begin{bmatrix}A&B\\C&D\end{bmatrix}}}$ . Using local inverse instead of generalized inverse or inverse can avoid the 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 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 bowl effect. Bowl effect does not related the unknown object ${\displaystyle y}$, it is only related the correct data ${\displaystyle f'}$

In case the contribution of ${\displaystyle [A]^{+}BB^{+}f'}$ to ${\displaystyle x}$ are 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 as bowl effect. This result is same as local tomography, hence local inverse is a direct extension of the concept of the 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 local inverse is a minimal L2 norm method subject to the condition that the influence of unknown object ${\displaystyle y}$ is ${\displaystyle 0}$. Hence the local inverse is also a direct extension of the concept of the generalized inverse.