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In image reconstruction, 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 called the region of interest (ROI). Interior reconstruction can be applied to either dental or cardiac CT images, but the concept is not limited to CT. Interior reconstruction is applied using one of various methods.
- 1 Methods
- 2 Comparison of the above interior reconstruction methods
- 3 See also
- 4 Notes
The purpose of each method is to solve for vector in the following problem:
Let be the region of interest (ROI) and be the region outside of . Assume , , , are known matrices; and are unknown vectors of the original image, while and are vector measurements of the responses, being known and unknown. Further, is inside of the region , () and , in the region , (), is outside of region . Also, is inside of a region in the measurement corresponding to . This region is denoted as , (), while is outside of the region . This region corresponds to , and is denoted as , ().
For CT image reconstruction purposes, .
In order to simplify the concept of the interior reconstruction, the matrices , , , are applied to image reconstruction instead of using complicated operators.
The first interior reconstruction method under consideration below is the extrapolation method. That is a local tomography method which eliminates the truncation artifacts but introduces another kind of artifact, i.e. "bowl effect". Then we examine an improvement called the adaptive extrapolation method. The iterative extrapolation method below also improves the reconstruction results. In some special cases, the exact reconstruction can be found for the interior reconstruction. The local inverse method explained below modifies the local tomography method and perhaps can improve the reconstruction result of the local tomography. Further, the iterative reconstruction method can be applied to interior reconstruction. Among the above methods, extrapolation is often applied.
Assume , , , are known matrices; and are unknown vectors; is a known vector; is an unknown vector. We desire to know the vector . Here it is assumed that and are the original image, while and are measurements of the responses. Vector is at the inside of the region , () where is also called the Region of interest (ROI). Vector is at the outside of the region . The outside region is referred as , () and is at the inside of a region in the measurement corresponding to . This region is denoted as , (). Also, the region of vector , which is at the outside of the region , corresponds to and is denoted as , (). In CT image reconstruction, it has
The question is: What is the solution of this problem in the region of ?
In order to simplify the concept of the interior reconstruction, the matrices , , , are applied to image reconstruction instead of using a complicated operator.
The response in the outside region can be a guess ; for example, assume it is
In the above formula a simple solution of is obtained; it is written as . This is called the extrapolation method. The result is dependent on how good the guess function or extrapolation function is. A frequent choice is
at the boundary of the two regions. The examples of the extrapolation method can be seen in the cited references    . The extrapolation method is often combined with a priori knowledge.  
(There is also a fast extrapolation method to reduce calculation time, shown below.)
Adaptive extrapolation method
Assume a rough solution, and , is obtained from the extrapolation method described above. The response in the outside region can be calculated as follows:
The reconstructed image can be calculated as following,
It is assumed that
at the boundary of the interior region. Here is a solution of this problem. This method is referred to as the adaptive extrapolation method. is the adaptive extrapolation function. The adaptive extrapolation method can be seen in the cited references     .
Iterative extrapolation method
It is assumed that a rough solution, and , is obtained from the extrapolation method described above.
The reconstruction can be obtained as following,
Here is an extrapolation function. It is assumed that
is one solution of this problem. The iterative extrapolation method can be seen in the cited reference.
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 satisfying
The above equation can be solved as
is the generalized inverse of , i.e.
The solution can be simplified as
The matrix is referred to as the local inverse of a matrix corresponding to . This method is referred to 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 a minimal norm method. We have
and is known.
where , and are weighting constants of the minimization and is some kind of norm. The often used norms are , , , or total variation (TV) norm, or some combination of the above norms. An example of this method is also referred to as the projection onto convex sets (POCS) method.  
In special situations, the interior reconstruction can be obtained as an analytical solution. The solution of is exact in such cases.
The extrapolated data often convolutes to a kernel function. After data is extrapolated, the size of the data is increased N times, where N = 2 ~ 3. If this 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). There exists an algorithm; it analytically calculates the contribution from the part of the extrapolated data. The calculation time can be omitted compared to the original convolution calculation. Hence with this algorithm the calculation of a convolution using the extrapolated data is not increased noticeably. This is referred to as the fast extrapolation.
Comparison of the above interior reconstruction methods
- The extrapolation method is suitable to the situation where
- i.e. a small truncation artifacts situation.
- The adaptive extrapolation method is suitable to the situation where
- i.e. a normal truncation artifacts situation. This method also offers a rough solution for the exterior region.
- The iterative extrapolation method is suitable to the situation where
- i.e. a normal truncation artifacts situation. This method gets better interior reconstruction compared to the adaptive reconstruction, but at the price of totally missing the result in the exterior region.
- The local tomography is suitable to the situation where
- i.e. a largest truncation artifacts situation. There are no truncation artifacts for this method. However there is a fixed error in the reconstruction. The error is not dependent on the value of .
- The local inverse method is the same as local tomography. It is suitable to the situation where
- i.e. a largest truncation artifacts situation. There are no truncation artifacts for this method. However there is a fixed error in the reconstruction. The error is not dependent on the value of . The error is perhaps smaller than with local tomography.
- The iterative reconstruction method achieves a good result, but at the price of huge calculations.
- The analytic method achieves the exact result, but it is only functional for some special situations.
- The fast extrapolation method can get the same results as the other extrapolation methods. It can be applied to the above interior reconstruction methods to reduce the calculation.
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- Minimum polynomial extrapolation
- Multigrid method
- Prediction interval
- Regression analysis
- Richardson extrapolation
- Static analysis
- Trend estimation
- Extrapolation domain analysis
- Dead reckoning
- Image reconstruction
- Truncation artifact
- Exterior reconstruction
- Local tomography
- Local inverse
- Generalized inverse
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