Development of a New 3D Reconstruction Algorithm for Computed Tomography (CT)

Abstract

Model-based computed tomography (CT) image reconstruction is dominated by iterative algorithms. Although long reconstruction times remain as a barrier in practical applications, techniques to speed up its convergence are object of investigation, obtaining impressive results. In this thesis, a direct algorithm is proposed for model-based image reconstruction. The model-based approximation relies on the construction of a model matrix that poses a linear system which solution is the reconstructed image. The proposed algorithm consists in the QR decomposition of this matrix and the resolution of the system by a backward substitution process. The cost of this image reconstruction technique is a matrix vector multiplication and a backward substitution process, since the model construction and the QR decomposition are performed only once, because of each image reconstruction corresponds to the resolution of the same CT system for a different right hand side.Several problems regarding the implementation of this algorithm arise, such as the exact calculation of a volume intersection, definition of fill-in reduction strategies optimized for CT model matrices, or CT symmetry exploit to reduce the size of the system. These problems have been detailed and solutions to overcome them have been proposed, and as a result, a proof of concept implementation has been obtained.Reconstructed images have been analyzed and compared against the filtered backprojection (FBP) and maximum likelihood expectation maximization (MLEM) reconstruction algorithms, and results show several benefits of the proposed algorithm. Although high resolutions could not have been achieved yet, obtained results also demonstrate the prospective of this algorithm, as great performance and scalability improvements would be achieved with the success in the development of better fill-in strategies or additional symmetries in CT geometry. ResumenEn la reconstrucción de imagen de tomografía axial computerizada (TAC), en su modalidad model-based, prevalecen los algoritmos iterativos. Aunque los altos tiempos de reconstrucción aún son una barrera para aplicaciones prácticas, diferentes técnicas para la aceleración de su convergencia están siendo objeto de investigación, obteniendo resultados impresionantes. En esta tesis, se propone un algoritmo directo para la reconstrucción de imagen model-based. La aproximación model-based se basa en la construcción de una matriz modelo que plantea un sistema lineal cuya solución es la imagen reconstruida. El algoritmo propuesto consiste en la descomposición QR de esta matriz y la resolución del sistema por un proceso de sustitución regresiva. El coste de esta técnica de reconstrucción de imagen es un producto matriz vector y una sustitución regresiva, ya que la construcción del modelo y la descomposición QR se realizan una sola vez, debido a que cada reconstrucción de imagen supone la resolución del mismo sistema TAC para un término independiente diferente.Durante la implementación de este algoritmo aparecen varios proble...

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

“…where A is the system matrix, of dimensions M×N, M rows and N columns; b, of dimensions 1×M, contains the detector element measurements; and x, of dimensions N×1, is the a priori unknown image [25].…”

“…We need to define a set of projections that produces at least N linearly independent rows in A. If there are sufficient projections in the matrix A, there will be enough information to uniquely solve the linear system (1), and that implies [25]- [27]: d×P>N (2) where d is the number of detectors considered in the detector panel, P is the number of projections and N is the number of voxels in the field of view. Notice that M = d×P is the number of detector element measurements.…”

“…The most CPU time-consuming part of the QRfactorization method is calculated and stored only once for a given CT system, and from then on, each image reconstruction only involves a matrix vector product and a backward substitution process. The authors have obtained promising results [24], [25] for the studies of the QR-factorization stability including the errors introduced by the reconstruction process. Additionally, a heuristic has been described in [25] to exploit the particular CT sparse matrix structure allowing a reduction in the number of projections, without compromising image quality.…”

“…The authors have obtained promising results [24], [25] for the studies of the QR-factorization stability including the errors introduced by the reconstruction process. Additionally, a heuristic has been described in [25] to exploit the particular CT sparse matrix structure allowing a reduction in the number of projections, without compromising image quality.…”

“…The reconstructed image obtained with QR-factorization is unique and equivalent to the solution obtained with the least squares method as long as the corresponding system matrix is injective [25]- [27].…”