Row Action Algebraic Reconstruction Techniques Implementation in Graphic Cards using SCILAB (Benchmark in Medical Imaging, Seismic Imaging and Walnut Imaging)

In this article we present a SCILAB implementation of algebraic iterative reconstruction methods for discretisation of inverse problems in imaging. These so-called row action methods rely on semi-convergence for achieving the necessary regularisation of the problem. We implement this method using SCILAB and provide a few simplified test problems: medical tomography, seismic tomography and walnut tomography.Numerical results show the capability of this method for the original and perturbed right-hand side vector.


Introduction
Iterative reconstruction method for computing solutions to discretization of inverse problems have been used for decades in medical imaging, geophysics, material science and many other disciplines that involve 2D and 3D imaging [1]. This article presents algebraic reconstruction techniques to solve large linear systems of the form ≅ , ∈ ℝ × used in tomography and many other inverse problems. We assume that the elements of matrix A are nonnegative and contains non zero rows or columns.
There are no restriction on the dimensions.We can write the system of linear equations with m equations with n unknown variables: = , = 1, 2, … , where and are column vectors. Observe that them equation are hyperplanesin ℝ , ann-dimensional Euclidean space.

Algebraic Reconstruction Techniques
This section may be divided by subheadings. It should provide a concise and precise description of the experimental results, their interpretation as well as the experimental conclusions that can bedrawn.
Algebraic Reconstruction Technique (ART) is one of the method used in commercial machine.The earliest row action ART iterative method proposed by S. Kaczmarz in 1937 in his paper "Angenäherte auflösung von systemen linearer gleichungen". [2] The core of ART method is the row-action method that treat the equation one at a time during the iterations. ART method find orthogonal projection +1 from to the hyperplane = for = 1, 2, … , . For each iteration, +1 will convergent to the solution if the vectors 1 , 2 , …, span ℝ . Therefore, in each iteration, we calculate the iterative vector:

Test Problems
The rowaction ART is benchmarked using 4 cases, of which two cases are x-ray CT scan tests, a case from seismic tomography, and lastly a case of CT-scan of walnut.

Medical Imaging
Parallel beams tomography arise from medical sciences and used in medical imaging, X-ray CT scan geometries. We use parallel rays in some degrees to calculate the position and the interior of an object. This techniques also known as first generation, translate-rotate pencil beam geometry. Fan beams tomography also arise from medical sciences and used in medical imaging, X-ray CT scan geometries. In this techniques, we use point beam with fan-like forms rays through an object. These techniques also known as second generation, translate-rotate fan beam geometry.The object used in the experiment is modified Shepp Logan phantom with the discretization [3].

Figure 2.Illustration of Shepp Logan Phantom
Characteristics of matrix used in experiment is given in the Table 1. Both parallel beams and fan beams tomography coefficient matrix is based on AIR Tools -A MATLAB package of algebraic iterative reconstruction methods [1]. By multiplication ofthe desired coefficient matrix with modified Shepp Logan phantom, we obtain the RHS vector.

Seismic Tomography
Seismic tomography is a technique for imaging Earth subsurface characteristics. We placed the sources on the right side of the area and the receivers on the left side and the surface of the area.

Figure 3.Illustration of Seismic Tomography
Seismic tomography is a technique for imaging Earth sub surface characteristics. The object used in the experiment is tectonic phantom with the discretization as in [3].  Table 2 shows the characteristic of matrix used in the experiment.

Walnut Tomography
Tomographic X-ray data of a walnut taken from [4]. We use X-ray sinogram of a single 2D slice of the walnut of two different resolutions, 82 × 82and 164 × 164. Projections measured by 120 fan beams tomography of a walnut.  Algebraic reconstruction techniques (ART) produces an iterative solution with given coefficient matrix and right hand side vector. We reproduce approximate solution with increased iteration numbers by using original coefficient matrix and right hand side vector. Both tomography techniques produces different results below.
Additionally, we test the problems by giving some perturbation to the right hand side vector in the form of random error vector. With an additional vector, we also reproduce approximate solution with increased iteration numbers.
= + ‖ ‖ ‖ ‖ where eis a random error vector and is some small number (in this experiment we use = 0.05).

Parallel Beams Tomography Result
In this subsection, we present the result for the parallel beams tomography for the original and perturbed RHS. By some observations, the more ART iterations performed, the approximation data represented by scaled images looksbetter than small ART iterations. Table 3 shows that the approximate solution is closer to exact solution, shown by smaller value of ‖ * − ‖. On the other hand, the perturbed right hand side also give approximate solution that is closer to exact solution, shown by smaller value of ‖ * − ‖ , even with slower rate of convergence. The graphical view of the error rates forevery iterationsare shown in Figure 10.
(a) (b) Figure 10.Parallel beams tomography error rates for (a) original RHS and (b) perturbed RHS As shown by the table and images, both cases converge to the exact solution as iteration increases, even though it converges to different values of norm because of perturbation that we have given above.

Fan Beams Tomography Result
In this subsection, we present the result for the fan beams tomography for the original and perturbed RHS. As in the previous subsection, Figures 11 -14 show the approximation data represented by the scaled images using SCILAB's plotlib 0.46 toolbox [https://atoms.scilab.org/toolboxes/plotlib/0.46] for selected iterations (1, 10, 100 and 1000 iterations). By some observations, the more ART iterations performed, the approximation data represented by scaled images looksbetter than small ART iterations. Table 4 shows that the approximate solution is closer to exact solution, shown by smaller value of ‖ * − ‖. On the other hand, the perturbed right hand side also give approximate solution that is closer to exact solution, shown by smaller value of ‖ * − ‖ , even with slower rate of convergence due to the perturbed condition.We can see that the fan beams tomography gives the similar result with parallel beams tomography. The graphical view of the error rates for every iterations give similar conclusion with parallel beams tomography as shown in Figure 15.

Seismic Tomography Result
In this subsection, we apply ART iterations for the seismic tomography data. Figures 16  -19 show the approximation data represented by the scaled images using SCILAB's plotlib 0.46 toolbox [https://atoms.scilab.org/toolboxes/plotlib/0.46] for selected iterations (1, 10, 100 and 1000 iterations). The seismic tomography also gives us a similar conclusion: increased ART iterations converges better while perturbed images look worse than a smaller number of iterations. The error rates progression also agrees with both parallel beams and fan beams tomography as shown in Table 5 and Figure 20.

Walnut Tomography Result
In this subsection, we apply the ART iterations for the walnut tomography. The real walnut tomography data was capturedon 82 × 82 and 164 × 164resolutions. Figures 21 -28 show the approximation data represented by the scaled images using SCILAB's plotlib 0.46 toolbox [https://atoms.scilab.org/toolboxes/plotlib/0.46] for selected iterations (1, 10, 100 and 1000 iterations).     The left side images are ART iterations for walnut tomography with 82 × 82 resolution for the original problem, on the other hand, the right side images are ART iterations for the perturbed problem. As shown by images and small calculation, both problems converge and closer to the exact solution as iteration increased.

Conclusions
Both parallel beams tomography and fan beams tomography are widely used in CT scan geometries. This experiment shows that the parallelisation of algebraic reconstruction techniques to solve system of linear equations is successful. By adding more ART iterations, approximate solutions is closer to exact solution shown by smaller value of norm ‖ * − ‖, which is also matched with approximation data that represented by scaled image.
With an additional error vector in hand right side vector, which we called perturbed systems, the approximate solution is harder to converge, again, shown by the value of norm ‖ * − ‖ even though time elapsed for solving the equation is not significantly different.
Since the perturbed vector is based on the magnitude of the right hand side, the walnut tomography -which has smallest maximum right hand side value -has the best approximation result for perturbed right hand side.