Tomographic reconstruction
Tomographic reconstruction is a type of multidimensional inverse problem where the challenge is to yield an estimate of a specific system from a finite number of projections.
The mathematical basis for tomographic imaging was laid down by Johann Radon.
A notable example of applications is the reconstruction of computed tomography (CT) where cross-sectional images of patients are obtained in non-invasive manner.
Recent developments have seen the Radon transform and its inverse used for tasks related to realistic object insertion required for testing and evaluating computed tomography use in airport security.
The simplest and easiest way to visualise the method of scanning is the system of parallel projection, as used in the first scanners.
In theory, the inverse Radon transformation would yield the original image.
The discrete Fourier transform (DFT) on each projection yields sampling in the frequency domain.
Combining all the frequency-sampled projections generates a polar raster in the frequency domain.
The polar raster is sparse, so interpolation is used to fill the unknown DFT points, and reconstruction can be done through the inverse discrete Fourier transform.
[4] Reconstruction performance may improve by designing methods to change the sparsity of the polar raster, facilitating the effectiveness of interpolation.
For instance, a concentric square raster in the frequency domain can be obtained by changing the angle between each projection as follow: where
The concentric square raster improves computational efficiency by allowing all the interpolation positions to be on rectangular DFT lattice.
[4] Yet, the Fourier-Transform algorithm has a disadvantage of producing inherently noisy output.
In practice of tomographic image reconstruction, often a stabilized and discretized version of the inverse Radon transform is used, known as the filtered back projection algorithm.
However, it induces greater noise because the filter is prone to amplify high-frequency content.
The iterative algorithm is computationally intensive but it allows the inclusion of a priori information about the system
Fan beams will generate series of line integrals, not parallel to each other, as projections.
The fan-beam system requires a 360-degree range of angles, which imposes mechanical constraints, but it allows faster signal acquisition time, which may be advantageous in certain settings such as in the field of medicine.
An excellent overview can be found in the special issue [5] of IEEE Transaction on Medical Imaging.
Artifact reduction using the U-Net in limited angle tomography is such an example application.
[6] However, incorrect structures may occur in an image reconstructed by such a completely data-driven method,[7] as displayed in the figure.
Therefore, integration of known operators into the architecture design of neural networks appears beneficial, as described in the concept of precision learning.
[8] For example, direct image reconstruction from projection data can be learnt from the framework of filtered back-projection.
[9] Another example is to build neural networks by unrolling iterative reconstruction algorithms.
Tomographic systems have significant variability in their applications and geometries (locations of sources and detectors).
This variability creates the need for very specific, tailored implementations of the processing and reconstruction algorithms.
This is done not only to protect intellectual property, but may also be enforced by a government regulatory agency.
Regardless, there are a number of general purpose tomographic reconstruction software packages that have been developed over the last couple decades, both commercial and open-source.
Most of the commercial software packages that are available for purchase focus on processing data for benchtop cone-beam CT systems.
[20] Shown in the gallery is the complete process for a simple object tomography and the following tomographic reconstruction based on ART.
