Local inverse

The local inverse is a kind of inverse function or matrix inverse used in image and signal processing, as well as other general areas of mathematics.

The concept of local inverse comes from interior reconstruction of the CT image. This concept is a direct extensions of local tomography and generalized inverse. It is used to solve the inverse problem with incomplete input data. Like local tomography, local inverse solution eliminate the truncation artifacts but it create other errors: bowl effect. Like generalized inverse local inverse offers a minimal norm solution, but local inverse's solution subject to a condition which leads to complete eliminate all truncation artifacts. It is often the bowl effect are less severe than truncation artifacts. Hence local inverse solution are better than a generalized inverse solution for incomplete input data situation. The bowl effect of local inverse solution are also less severe than local tomography.

Local inverse for full field of view system or over-determined system

${\begin{bmatrix}f\\g\end{bmatrix}}={\begin{bmatrix}A&B\\C&D\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}$

Assume there is $E$, $F$, $G$ and $H$ that satisfies,

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

Here $J$ is not equal to $I$. $J$ is close to $I$. $I$ is identical matrix. In this case a approximate solution can be found as following,

and

${\begin{bmatrix}x_{0}\\y_{0}\end{bmatrix}}={\begin{bmatrix}E&F\\G&H\end{bmatrix}}{\begin{bmatrix}f\\g\end{bmatrix}}$

A better solution for $x_{1}$ can be found as following,

${\begin{bmatrix}x_{1}\\y_{1}\end{bmatrix}}={\begin{bmatrix}E&F\\G&H\end{bmatrix}}{\begin{bmatrix}f-By_{0}\\g-Dy_{0}\end{bmatrix}}$

In the above formula $y_{1}$ is useless, hence

$x_{1}=E(f-By_{0})+F(g-Dy_{0})$

In the same way, there is

$y_{1}=G(f-Ax_{0})+H(g-Cx_{0})$

In the above the solution is only divided to as two parts. $x$ is inside the ROI(region of Interest) $y$ is at outside of ROI. f is inside of FOV(field of view) y is outside of FOV.

The two parts can be extended to many parts, in this case, the extended method is referred as the sub-region iterative refinement method method

Local inverse for Limited field of view system or under-determined system

a) $A^{+}B$ is close to $0$

${\begin{bmatrix}f\\g\end{bmatrix}}={\begin{bmatrix}A&B\\C&D\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}$

Assume $A$, $B$, $C$, $D$ are known matrices; $x$ and $y$ are unknown vectors; $f$ is known vector; $g$ is unknown vector. It is interested to know the vector x. What is the best solution? Shuang-ren Zhao defined a Local inverse to solve the above problem. First consider the simplest solution.

$f=Ax+By$

or

$Ax=f-By=f'$

Here $f'=f-By$ 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

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

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

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

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

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

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

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

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 $Q$ is considered, which satisfies

$QB=0$

Hence,

$QAx=Qf-QBy=Qf$

solve the above equation with Generalized inverse

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

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

$Q=I-BB^{+}$

This kind of matrix $Q$ is referred as transverse projection of matrix $B$

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

$BB^{+}B=B$

It can be proven that

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

It is easy to prove that $QQ=Q$

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

and hence

$QQQ=(QQ)Q=QQ=Q$

Hence Q is also the generalized inverse of Q

That means

$Q^{+}Q=QQ=Q$

Hence,

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

or

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

The matrix

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

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

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

Hence there is,

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

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

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

This kind error are called bowl effect. Bowl effect does not related the unknown object $y$, it is only related the correct data $f'$

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

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

the local inverse solution $x_{1}$ is better than $x_{0}$ for this kind of inverse problem. Using $x_{1}$ instead of $x_{0}$, 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 $y$ is $0$. Hence the local inverse is also an direct extension of the concept of the generalized inverse.

See also

• Formulation of the Jacobian conjecture
• Interior image reconstruction of CT scans
• interior reconstruction
• extrapolation
• matrix inverse
• generalized inverse
• iterative refinement
• Local tomography