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 in Computed Tomography 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.
Contents
- Description
- Reconstruction algorithms
- Fourier Domain Reconstruction Algorithm
- Back Projection Algorithm
- Iterative Reconstruction Algorithm
- Fan Beam Reconstruction
- Tomographic reconstruction software
- Gallery
- References
This article applies in general to tomographic reconstruction for all kinds of tomography, but some of the terms and physical descriptions refer directly to X-ray computed tomography.
Description
The projection of an object, resulting from the tomographic measurement process at a given angle
where
Using the coordinate system of Figure 1, the value of
So the equation above can be rewritten as
where
The Fourier Transform of the projection can be written as
where
In theory, the inverse Radon transformation would yield the original image. The projection-slice theorem tells us that if we had an infinite number of one-dimensional projections of an object taken at an infinite number of angles, we could perfectly reconstruct the original object,
Assuming
Reconstruction algorithms
Practical reconstruction algorithms have been developed to implement the process of reconstruction of a 3-dimensional object from its projections. These algorithms are designed largely based on the mathematics of the Radon transform, statistical knowledge of the data acquisition process and geometry of the data imaging system.
Fourier-Domain Reconstruction Algorithm
Reconstruction can be made using interpolation. Assume
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. Furthermore, it reduces the interpolation error. Yet, the Fourier-Transform algorithm has a disadvantage of producing inherently noisy output.
Back Projection Algorithm
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.
With a sampled discrete system, the inverse Radon Transform is
where
The name back-projection comes from the fact that 1D projection needs to be filtered by 1D Radon kernel (back-projected) in order to obtain a 2D signal. The filter used does not contain DC gain, thus adding DC bias may be desirable. Reconstruction using back-projection allows better resolution than interpolation method described above. However, it induces greater noise because the filter is prone to amplify high-frequency content.
Iterative Reconstruction Algorithm
Iterative algorithm is computationally intensive but it allows to include a priori information about the system
Let
An alternative family of recursive tomographic reconstruction algorithms are the Algebraic Reconstruction Techniques.
Fan-Beam Reconstruction
Use of noncollimated fan beam is common since collimated beam of radiation is difficult to obtain. Fan beams will generate series of line integrals, not parallel to each other, as projections. The fan-beam system will require 360 degrees range of angles which impose mechanical constraint, however, it allows faster signal acquisition time which may be advantageous in certain settings such as in the field of medicine. Back projection follows a similar 2 step procedure that yields reconstruction by computing weighted sum back-projections obtained from filtered projections.
Tomographic reconstruction software
For flexible tomographic reconstruction, open source toolboxes are available, such as TomoPy or the ASTRA toolbox . TomoPy is an open-source Python toolbox to perform tomographic data processing and image reconstruction tasks at the Advanced Photon Source at Argonne National Laboratory. TomoPy toolbox is specifically designed to be easy to use and deploy at a synchrotron facility beamline. It supports reading many common synchrotron data formats from disk through Scientific Data Exchange, and includes several other processing algorithms commonly used for synchrotron data. TomoPy also includes several reconstruction algorithms, which can be run on multi-core workstations and large-scale computing facilities. The ASTRA Toolbox is a MATLAB toolbox of high-performance GPU primitives for 2D and 3D tomography, from 2009-2014 developed by iMinds-Vision Lab, University of Antwerp and since 2014 jointly developed by iMinds-VisionLab, UAntwerpen and CWI, Amsterdam. The toolbox supports parallel, fan, and cone beam, with highly flexible source/detector positioning. A large number of reconstruction algorithms are available through TomoPy and the ASTRA toolkit, including FBP, Gridrec, ART, SIRT, SART, BART, CGLS, PML, MLEM and OSEM. Recently, the ASTRA toolbox has been integrated in the TomoPy framework. By integrating the ASTRA toolbox in the TomoPy framework, the optimized GPU-based reconstruction methods become easily available for synchrotron beamline users, and users of the ASTRA toolbox can more easily read data and use TomoPy’s other functionality for data filtering and artifact correction.
Gallery
Shown in the gallery is the complete process for a simple object tomography and the following tomographic reconstruction based on ART.