Local inverse

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

Calculation

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

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

$Ax=f-By$

the above solution of the equation is written as $x_{0}$ instead to distinguish it with the original vector $x$

$x_{0}=A^{-1}(f-By)$

or if $A$ is not a square matrix or it has no inverse, generalized inverse can applied,

$x_{0}=A^{+}(f-By)$

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

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

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:

$Q=I-BB^{+}$

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 A. Using local inverse instead of generalized inverse or inverse can avoid the artifacts from unknown input data.

It is clear that if $y=0$, a better solution than the above local inverse solution which is:

$x_{0}=[A]^{+}f_{0}$

where$f_{0}$ is where $f$ in the case $y=0$.

In this situation the difference of above solutions are

$\mathrm {error} _{1}=|x_{0}-x_{1}|=|[A]^{+}BB^{+}f_{0}|$

This kind error are called bowl effect. Bowl effect does not related the unknown object $y$

In case the contribution of $[A]^{+}BB^{+}f_{0}$ to $x$ are smaller than that of $[A]^{+}By$, or

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

the local inverse solution $x_{1}$ has a better solution compared to $x_{0}$ to this kind of inverse problem.

See also

• Formulation of the Jacobian conjecture
• Interior image reconstruction of CT scans