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.
The purpose of each method is to solve for vector
are unknown vectors of the original image, while
is inside a region in the measurement corresponding to
To simplify the concept of interior reconstruction, the matrices
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.
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.
is inside a region in the measurement corresponding to
In CT image reconstruction, it has To simplify the concept of interior reconstruction, the matrices
are applied to image reconstruction instead of a complex operator.
The result depends on how good the extrapolation function
A frequent choice is at the boundary of the two regions.
[1][2][3][4] The extrapolation method is often combined with a priori knowledge,[5][6] and an extrapolation method which reduces calculation time is shown below.
can be calculated as follows: The reconstructed image can be calculated as follows: It is assumed that at the boundary of the interior region;
solves the problem, and is known as the adaptive extrapolation method.
The response in the outside region can be calculated as follows: Consider the generalized inverse
satisfying Define so that Hence, The above equation can be solved as considering that
If the goal function can be some kind of normal, this is known as the minimal norm method.
[16][17][18] 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.
[19] 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.
