QR-Factorization Algorithm for Computed Tomography (CT): Comparison With FDK and Conjugate Gradient (CG) Algorithms

Abstract

Even though QR-factorization of the system matrix for tomographic devices has been already used for medical imaging, to date, no satisfactory solution has been found for solving large linear systems, such as those used in Computed Tomography (CT) (in the order of 10 6 equations). In computed tomography, the Feldkamp, Davis and Kress back projection algorithm (FDK) and iterative methods like conjugate gradient (CG) are the standard methods used for image reconstruction. As the image reconstruction problem can be modelled by a large linear system of equations, QR-factorization of the system matrix could be used to solve this system. Current advances in computer science enable the use of direct methods for solving such a large linear system. The QR-factorization is a numerically stable direct method for solving linear systems of equations, which is beginning to emerge as an alternative to traditional methods, bringing together the best from traditional methods. QR-factorization was chosen because the core of the algorithm, from the computational cost point of view, is pre-calculated and stored only once for a given CT system, and from then on, each image reconstruction only involves a backward substitution process and the product of a vector by a matrix. Image quality assessment was performed comparing contrast to noise ratio (CNR) and noise power spectrum (NPS); performances regarding sharpness were evaluated by the reconstruction of small structures using data measured from a small animal 3D CT. Comparisons of QR-factorization with FDK and conjugate gradient (CG) methods show that QR-factorization is able to reconstruct more detailed images for a fixed voxel size.

CT image reconstruction with SuiteSparseQR factorization package

“…This package gives us a lot of basic sparse matrices algebraic algorithms and also provides the possibility of using BLAS threads as well as Intel Threading Building Block (TBB) that can parallelize the factorization code and allows us to exploit our hardware resources to the maximum. In works like (Rodríguez-Alvarez et al, 2018) they perform the QR factorization to reconstruct small CT images but as they mention, the method is not parallelized or optimized to reduce the fill-in of the matrix, which the implementation in this packages does. In our previous work (Chillarón et al, 2018), we did a preliminary study in which we applied the multifrontal sparse QR (SPQR) method to reconstruct phantom images, in order to determine its feasibility.…”

Computed tomography medical image reconstruction on affordable equipment by using Out-Of-Core techniques

“…Unlike direct methods, iterative methods often generate patchy or blocky artifacts in the reconstructed images due to overregularization [16], [20]. Therefore, direct algebraic methods such as the QR factorization [21], [22] have been explored recently. Although they usually require a greater number of views than the iterative ones (as was shown in a previous work [23]), they are much more accurate when the rank of the weights matrix is complete.…”

“…Therefore, direct algebraic methods such as the QR factorization [21], [22] have been explored recently. Although they usually require a greater number of views than the iterative ones (as was shown in a previous work [23]), they are much more accurate when the rank of the weights matrix is complete.…”

Parallel CT Reconstruction for Multiple Slices Studies with SuiteSparseQR Factorization Package

“…Using iterative methods, we can reduce the number of views or projections that we use to a really small number if we make a good selection of the projection angles [9,10,6]. However, with the direct methods [11,7], we have reached the conclusion that it is necessary that the matrix of the CT system has full rank, which will determine the number of projections required according to the image resolution that you want to achieve. Regardless, both approaches need a lower number of X-ray projections than other methods.…”