Interior reconstruction

In iterative reconstruction in digital imaging, interior reconstruction (also known as limited field of view (LFV) reconstruction) is a technique to correct truncation artifacts caused by limiting image data to a small field of view. The reconstruction focuses on an area known as the region of interest (ROI). Although interior reconstruction can be applied to dental or cardiac CT images, the concept is not limited to CT. It is applied with one of several methods.
Methods.
The purpose of each method is to solve for vector formula_1 in the following problem:
Let formula_3 be the region of interest (ROI) and formula_4 be the region outside of formula_3.
Assume formula_6, formula_7, formula_8, formula_9 are known matrices; formula_1 and formula_11 are unknown vectors of the original image, while formula_12 and formula_13 are vector measurements of the responses (formula_12 is known and formula_13 is unknown). formula_1 is inside region formula_3, (formula_18) and formula_11, in the region formula_4, (formula_21), is outside region formula_3. formula_12 is inside a region in the measurement corresponding to formula_3. This region is denoted as formula_25, (formula_26), while formula_13 is outside of the region formula_25. It corresponds to formula_4 and is denoted as formula_30, (formula_31).
For CT image-reconstruction purposes, formula_32.
To simplify the concept of interior reconstruction, the matrices formula_6, formula_7, formula_8, formula_9 are applied to image reconstruction instead of complex operators.
The first interior-reconstruction method listed below is extrapolation. It is a local tomography method which eliminates truncation artifacts but introduces another type of artifact: a bowl effect. An improvement is known as the adaptive extrapolation method, although the iterative extrapolation method below also improves reconstruction results. In some cases, the exact reconstruction can be found for the interior reconstruction. The local inverse method below modifies the local tomography method, and may improve the reconstruction result of the local tomography; the iterative reconstruction method can be applied to interior reconstruction. Among the above methods, extrapolation is often applied.
Extrapolation method.
formula_6, formula_7, formula_8, formula_9 are known matrices; formula_1 and formula_11 are unknown vectors; formula_12 is a known vector, and formula_13 is an unknown vector. We need to know the vector formula_1. formula_1 and formula_11 are the original image, while formula_12 and formula_13 are measurements of responses. Vector formula_1 is inside the region of interest formula_3, (formula_18). Vector formula_11 is outside the region formula_3. The outside region is called formula_4, (formula_21) and formula_12 is inside a region in the measurement corresponding to formula_3. This region is denoted formula_25, (formula_26). The region of vector formula_13 (outside the region formula_25) also corresponds to formula_4 and is denoted as formula_30, (formula_31).
In CT image reconstruction, it has
To simplify the concept of interior reconstruction, the matrices formula_6, formula_7, formula_8, formula_9 are applied to image reconstruction instead of a complex operator.
The response in the outside region can be a guess formula_30; for example, assume it is formula_73
A solution of formula_1 is written as formula_76, and is known as the extrapolation method. The result depends on how good the extrapolation function formula_73 is. A frequent choice is
at the boundary of the two regions.
The extrapolation method is often combined with "a priori" knowledge, and an extrapolation method which reduces calculation time is shown below.
Adaptive extrapolation method.
Assume a rough solution, formula_76 and formula_80, is obtained from the extrapolation method described above. The response in the outside region formula_81 can be calculated as follows:
The reconstructed image can be calculated as follows:
It is assumed that 
at the boundary of the interior region; formula_85 solves the problem, and is known as the adaptive extrapolation method.
formula_86 is the adaptive extrapolation function.
Iterative extrapolation method.
It is assumed that a rough solution, formula_76 and formula_80, is obtained from the extrapolation method described below:
or
The reconstruction can be obtained as
Here formula_86 is an extrapolation function, and it is assumed that
formula_85 is one solution of this problem.
Local tomography.
Local tomography, with a very short filter, is also known as lambda tomography.
Local inverse method.
The local inverse method extends the concept of local tomography. The response in the outside region can be calculated as follows:
Consider the generalized inverse formula_96 satisfying
Define 
so that
Hence,
The above equation can be solved as
considering that
formula_104 is the generalized inverse of formula_104, i.e.
The solution can be simplified as
The matrix 
formula_108 
is known as the local inverse of matrix 
formula_109, corresponding to formula_6. This is known as the local inverse method.
Iterative reconstruction method.
Here a goal function is defined, and this method iteratively achieves the goal. If the goal function can be some kind of normal, this is known as the minimal norm method.
subject to
and 
formula_113 is known,
where formula_114, formula_115 and formula_116 are weighting constants of the minimization and formula_117 is some kind of norm. Often-used norms are formula_118, formula_119, formula_120, formula_121 total variation (TV) norm or a combination of the above norms. An example of this method is the projection onto convex sets (POCS) method.
Analytical solution.
In special situations, the interior reconstruction can be obtained as an analytical solution; the solution of formula_1 is exact in such cases.
Fast extrapolation.
Extrapolated data often convolutes to a kernel function. After data is extrapolated its size is increased "N" times, where "N" = 2 ~ 3. If the data needs to be convoluted to a known kernel function, the numerical calculations will increase log("N")·"N" times, even with the fast Fourier transform (FFT). An algorithm exists, analytically calculating the contribution from part of the extrapolated data. The calculation time can be omitted, compared to the original convolution calculation; with this algorithm, the calculation of a convolution using the extrapolated data is not noticeably increased. This is known as fast extrapolation.
Comparison of methods.
The extrapolation method is suitable in a situation where 
The adaptive extrapolation method is suitable for a situation where 
The iterative extrapolation method is suitable for a situation in which 
Local tomography is suitable for a situation in which 
The local inverse method, identical to local tomography, suitable in a situation in which 
The iterative reconstruction method obtains a good result with large calculations. Although the analytic method achieves an exact result, it is only functional in some situations. The fast extrapolation method can get the same results as the other extrapolation methods, and can be applied to the above interior reconstruction methods to reduce the calculation.