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 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 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. 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.
Assume there is,, and that satisfies, Here is not equal to. is close to . is identical matrix. Examples of this kind of matrix are for example, filtered back-projection method for image reconstruction, the inverse with regularization. In this case an approximate solution can be found as following, and A better solution for can be found as following, In the above formula is useless, hence In the same way, there is In the above the solution is only divided to as two parts. is inside the ROI is at outside of ROI. f is inside of FOV 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
Assume ,,, are known matrices; and are unknown vectors; is known vector; is unknown vector. It is interested to know the vector x. What is the better solution?
Assume the above matrix inverse exists
E & F \\ G & H \end
Here or is close to. The local inverse algorithm is as follows: . An extrapolated function is obtained by . An approximate function is calculated by . A correction for is done by . A corrected function for is calculated by . An extrapolated function is obtained by . A local inverse solution is obtained In the above algorithm, there are two time extrapolations for functions which are used to overcome the data truncation problem. There is a correction for. This correction can be a constant correction to correct the DC values of function or a linear correction according to the prior knowledges about the function. This algorithm can be found in reference. In the example of reference, it is found that, here. In that example the constant correction is made. More complicated correction can be made, for example linear correction, which perhaps achieves better results.
is close to
Shuang-ren Zhao defined a Local inverse to solve the above problem. First consider the simplest solution. or Here 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 Here is a correct solution of the unknown, that means . In case that is not a square matrix or it has no inverse, generalized inverse can applied, Since is unknown, if it is set to, a approximate solution is obtained. On the above solution the result is related to the unknown vector. Since can be any values, this way the result has very strong artifacts which is 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 is considered, which satisfies Hence, solve the above equation with Generalized inverse Here is generalized inverse of the matrix. is a solution for. It is easy to find a matrix Q which satisfy, can be written as following: This kind of matrix is referred as transverse projection of matrix Here is the generalized inverse of the matrix. satisfies 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 is referred as the local inverse of Matrix . Using local inverse instead of generalized inverse or inverse can avoid the artifacts from unknown input data. Considering, Hence there is, Hence is only related the correct data. This kind error can be calculated as This kind error are called bowl effect. Bowl effect does not related the unknown object , it is only related the correct data In case the contribution of to are smaller than that of, or the local inverse solution is better than for this kind of inverse problem. Using instead of, 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 is. Hence the local inverse is also a direct extension of the concept of the generalized inverse.
