Multi-sheet surface rebinning methods for reconstruction from asymmetrically truncated cone beam projections: I. Approximation and optimality

The mechanical motion of the gantry in conventional cone beam CT scanners restricts the speed of data acquisition in applications with near real time requirements. A possible resolution of this problem is to replace the moving source detector assembly with static parts that are electronically activated. An example of such a system is the Rapiscan Systems RTT80 real time tomography scanner, with a static ring of sources and axially offset static cylinder of detectors. A consequence of such a design is asymmetrical axial truncation of the cone beam projections resulting, in the sense of integral geometry, in severely incomplete data. In particular we collect data only in a fraction of the Tam-Danielsson window, hence the standard cone beam reconstruction techniques do not apply. In this work we propose a family of multi-sheet surface rebinning methods for reconstruction from such truncated projections. The proposed methods combine analytical and numerical ideas utilizing linearity of the ray transform to reconstruct data on multi-sheet surfaces, from which the volumetric image is obtained through deconvolution. In this first paper in the series, we discuss the rebinning to multi-sheet surfaces. In particular we concentrate on the underlying transforms on multi-sheet surfaces and their approximation with data collected by offset multi-source scanning geometries like the RTT. The optimal multi-sheet surface and the corresponding rebinning function are found as a solution of a variational problem. In the case of the quadratic objective, the variational problem for the optimal rebinning pair can be solved by a globally convergent iteration. Examples of optimal rebinning pairs are computed for different trajectories. We formulate the axial deconvolution problem for the recovery of the volumetric image from the reconstructions on multi-sheet surfaces. Efficient and stable solution of the deconvolution problem is the subject of the second paper in this series (Betcke and Lionheart 2013 Inverse Problems 29 115004). © 2013 IOP Publishing Ltd

Accelerated Statistical Image Reconstruction Algorithms and Simplified Cost Functions for X-ray Computed Tomography.

Statistical image reconstruction methods are poised to replace traditional methods like filtered back-projection (FBP) in commercial X-ray computed tomography (CT) scanners. Statistical methods offer many advantages over FBP, including incorporating physical effects and physical constraints, modeling of complex imaging geometries, and imaging at lower X-ray doses. But, the use of statistical methods is limited due to many practical problems. This thesis proposes methods to improve four aspects of statistical methods: reconstruction time, beam hardening, non-negativity constraints, and organ motion. To reduce the reconstruction time, several novel iterative algorithms are proposed that are adapted to multi-core computing, including a hybrid ordered subsets (OS) / iterative coordinate descent (ICD) approach. This approach leads to a reduction in reconstruction time, and it also makes the ICD algorithm robust to the initial guess image. Statistical methods have accounted for beam hardening by using more information than needed by traditional FBP-based methods like the Joseph-Spital (JS) method. This thesis proposes a statistical method that uses exactly the same beam hardening information as the JS method while suppressing beam hardening artifacts. Directly imposing the non-negativity constraints can increase the computation time of algorithms such as the preconditioned conjugate gradient (PCG) method. This thesis proposes a modification of the penalized-likelihood cost function for monoenergetic transmission tomography, and a corresponding PCG algorithm, that reduce reconstruction time when enforcing nonnegativity. Organ motion during a scan causes image artifacts, and in some cases these artifacts are more apparent when standard statistical methods are used. A preliminary simulation study of a new approach to remove motion artifacts is presented. The distinguishing feature of this approach is that it does not require any new information from the scanner. The target applications of this research effort are 3-D volume reconstructions for axial cone-beam and helical cone-beam scanning geometries of multislice CT (MSCT) scanners.Ph.D.Electrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/60749/1/someshs_1.pd

Simulação e reconstrução em tomografia cardio-torácica

Doutoramento em Engenharia ElectrotécnicaEsta dissertação aborda o tema da Tomografia Axial Computorizada com especial enfoque na reconstrução da imagem, ou seja, nos métodos matemáticos que, dado um conjunto de amostras da intensidade de radiação X transmitida, recolhidas segundo uma geometria de irradiação específica, permitem obter o mapa da distribuição tri-dimensional (3D) do coeficiente de atenuação linear no interior do objecto. Em termos mais formais, dado que, do ponto de vista matemático, o processo de aquisição dos dados corresponde à amostragem da transformada de Radon da função que descreve a distribuição do coeficiente de atenuação no volume sob estudo, estamos interessados nos métodos matemáticos que permitem calcular a transformada inversa de Radon a partir de conjuntos suficientemente densos de amostras. O ambiente de simulação utilizado, desenvolvido de forma a permitir a aquisição de amostras da transformada de Radon de um modelo digital dinâmico do tórax humano (4D MCAT) segundo qualquer uma das geometrias de amostragem actualmente em uso nos equipamentos modernos, permitiu a implementação e avaliação de novos métodos de reconstrução. Nomeadamente, são propostos novos métodos de reconstrução rápida baseados na teoria de Fourier e no cálculo da transformada de Fourier a partir de amostras não equiespaçadas, quer para geometria de aquisição paralela quer divergente, e é abordado o tema da reconstrução dinâmica sendo exploradas várias técnicas de reconstrução que permitem reduzir os efeitos do movimento nos estudos cardio-torácicos. Os estudos realizados permitem demonstrar que o desempenho dos métodos de Fourier propostos é equivalente, no que respeita à qualidade de imagem, ao desempenho do método de Retroprojecção Filtrada, habitualmente utilizado nos equipamentos comerciais mas a custos computacionais significativamente mais baixos.This thesis addresses the theme of Computed Tomography, with special emphasis on tomographic reconstruction techniques, this means, the mathematical methods which, given a set of samples of transmitted radiation intensity, acquired with a specific sampling geometry, provide a threedimensional (3D) map of linear attenuation coefficient of the given object. Since, from a mathematical point of view, the acquisition process corresponds to the sampling of the Radon transform of the function describing the spatial distribution of attenuation coefficient in the volume under study, we are interested in methods and algorithms leading to diverse numerical implementations of the inverse Radon transform from sufficiently dense sets of samples. The developed simulation environment, capable to perform virtual acquisition of Radon transform’s samples of a digital anatomical model, in our case the dynamical digital model of human thorax (4D MCAT), through any of the sampling geometries currently used in commercial equipments, enabled the implementation and evaluation of new reconstruction methods. We propose new methods for fast reconstruction based on Fourier theory and on the calculation of Fourier transform for non-equispaced samples, both for parallel and for divergent acquisition geometries, and we deal with the topic of dynamic reconstruction, by exploring some reconstruction techniques capable of minimizing movement artifacts in cardio-thoracic images. The studies performed demonstrated that the performance of the proposed Fourier methods regarding image quality is similar to the Filtered Backprojection method, traditionally used in commercial equipments but at significantly lower computational costs