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.
Contents
- Local inverse for full field of view system or over determined system
- Local inverse for Limited field of view system or under determined system
- Assume the above matrix inverse exists E F G H displaystyle beginbmatrixEFGHendbmatrix
- A B displaystyle AB is close to 0 displaystyle 0
- References
The concept of local inverse came from interior reconstruction of the CT image. One of the interior reconstruction methods was done through that first approximately reconstructs the image outside the ROI (region of interest) and then subtracts the re-projection data of the image at outside the ROI from the original projection data; then the above created data are used to make a new reconstruction. This idea can be widened to inverse. Instead of directly making an inverse, the unknowns at the outside of the local region can be first inverted. Recalculate the data from these unknowns (at outside the local region). Subtract this recalculated data from the original data, then the inverse for the unknowns inside the local region is done through the above newly produced data.
This concept is a direct extension of local tomography, generalized inverse and iterative refinement method. It is used to solve the inverse problem with incomplete input data, similar to local tomography. However this concept of local inverse also can be applied to complete input data.
Local inverse for full field of view system or over-determined system
Assume there is
Here
and
A better solution for
In the above formula
In the same way, there is
In the above the solution is only divided to as two parts.
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
Assume
Assume the above matrix inverse exists [ E F G H ] {\displaystyle {\begin{bmatrix}E&F\\G&H\end{bmatrix}}}
Here
(1)
(2)
(3)
(4)
(5)
(6)
In the above algorithm, there are two time extrapolations for
In the example of reference, it is found that
A + B {\displaystyle A^{+}B} is close to 0 {\displaystyle 0}
Shuang-ren Zhao defined a Local inverse to solve the above problem. First consider the simplest solution.
or
Here
or
Here
Since
On the above solution the result
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
Hence,
solve the above equation with Generalized inverse
Here
This kind of matrix
Here
It can be proven that
It is easy to prove that
and hence
Hence Q is also the generalized inverse of Q
That means
Hence,
or
The matrix
Hence there is,
Hence
This kind error are called bowl effect. Bowl effect does not related the unknown object
In case the contribution of
the local inverse solution
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