The local inverse is a kind of inverse function or matrix inverse used in image and signal processing, as well as in other general areas of mathematics.
The concept of a local inverse came from interior reconstruction of CT[clarification needed] images.
One interior reconstruction method first approximately reconstructs the image outside the ROI (region of interest), and then subtracts the re-projection data of the image outside the ROI from the original projection data; then this data is used to make a new reconstruction.
This idea can be widened to a full inverse.
Instead of directly making an inverse, the unknowns outside of the local region can be inverted first.
Recalculate the data from these unknowns (outside the local region), subtract this recalculated data from the original, and then take the inverse inside the local region using this newly produced data for the outside region.
This concept is a direct extension of the local tomography, generalized inverse and iterative refinement methods.
It is used to solve the inverse problem with incomplete input data, similarly to local tomography.
However this concept of local inverse can also be applied to complete input data.
are the filtered back-projection method for image reconstruction and the inverse with regularization.
is useless, hence In the same way, there is In the above the solution is divided into two parts,
The two parts can be extended to many parts, in which case the extended method is referred to as the sub-region iterative refinement method [1] Assume
A local inverse solution is obtained In the above algorithm, there are two time extrapolations for
which are used to overcome the data truncation problem.
or a linear correction according to prior knowledge about
[2] In the example of the reference,[3] it is found that
the constant correction is made.
A more complicated correction can be made, for example a linear correction, which might achieve better results.
Shuang-ren Zhao defined a Local inverse[2] to solve the above problem.
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, Here
is a correct(or exact) solution for the unknown
In order to minimize the above artifacts in the solution, a special matrix
It is easy to find a matrix Q which satisfies
: and hence Hence Q is also the generalized inverse of Q That means Hence, or The matrix is referred to as the local inverse of the matrix
This kind error can be calculated as This kind error are called the bowl effect.
The bowl effect is not related to the unknown object
, the truncation artifacts are replaced by the bowl effect.
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