Point-set manifold processing for computational mechanics: thin shells, reduced order modeling, cell motility and molecular conformations

In many applications, one would like to perform calculations on smooth manifolds of dimension d embedded in a high-dimensional space of dimension D. Often, a continuous description of such manifold is not known, and instead it is sampled by a set of scattered points in high dimensions. This poses a serious challenge. In this thesis, we approximate the point-set manifold as an overlapping set of smooth parametric descriptions, whose geometric structure is revealed by statistical learning methods, and then parametrized by meshfree methods. This approach avoids any global parameterization, and hence is applicable to manifolds of any genus and complex geometry. It combines four ingredients: (1) partitioning of the point set into subregions of trivial topology, (2) the automatic detection of the local geometric structure of the manifold by nonlinear dimensionality reduction techniques, (3) the local parameterization of the manifold using smooth meshfree (here local maximum-entropy) approximants, and (4) patching together the local representations by means of a partition of unity. In this thesis we show the generality, flexibility, and accuracy of the method in four different problems. First, we exercise it in the context of Kirchhoff-Love thin shells, (d=2, D=3). We test our methodology against classical linear and non linear benchmarks in thin-shell analysis, and highlight its ability to handle point-set surfaces of complex topology and geometry. We then tackle problems of much higher dimensionality. We perform reduced order modeling in the context of finite deformation elastodynamics, considering a nonlinear reduced configuration space, in contrast with classical linear approaches based on Principal Component Analysis (d=2, D=10000's). We further quantitatively unveil the geometric structure of the motility strategy of a family of micro-organisms called Euglenids from experimental videos (d=1, D~30000's). Finally, in the context of enhanced sampling in molecular dynamics, we automatically construct collective variables for the molecular conformational dynamics (d=1...6, D~30,1000's)

Improving the Accuracy of CT-derived Attenuation Correction in Respiratory-Gated PET/CT Imaging

The effect of respiratory motion on attenuation correction in Fludeoxyglucose (18F) positron emission tomography (FDG-PET) was investigated. Improvements to the accuracy of computed tomography (CT) derived attenuation correction were obtained through the alignment of the attenuation map to each emission image in a respiratory gated PET scan. Attenuation misalignment leads to artefacts in the reconstructed PET image and several methods were devised for evaluating the attenuation inaccuracies caused by this. These methods of evaluation were extended to finding the frame in the respiratory gated PET which best matched the CT. This frame was then used as a reference frame in mono-modality compensation for misalignment. Attenuation correction was found to affect the quantification of tumour volumes; thus a regional analysis was used to evaluate the impact of mismatch and the benefits of compensating for misalignment. Deformable image registration was used to compensate for misalignment, however, there were inaccuracies caused by the poor signal-to-noise ratio (SNR) in PET images. Two models were developed that were robust to a poor SNR allowing for the estimation of deformation from very noisy images. Firstly, a cross population model was developed by statistically analysing the respiratory motion in 10 4DCT scans. Secondly, a 1D model of respiration was developed based on the physiological function of respiration. The 1D approach correctly modelled the expansion and contraction of the lungs and the differences in the compressibility of lungs and surrounding tissues. Several additional models were considered but were ruled out based on their poor goodness of fit to 4DCT scans. Approaches to evaluating the developed models were also used to assist with optimising for the most accurate attenuation correction. It was found that the multimodality registration of the CT image to the PET image was the most accurate approach to compensating for attenuation correction mismatch. Mono-modality image registration was found to be the least accurate approach, however, incorporating a motion model improved the accuracy of image registration. The significance of these findings is twofold. Firstly, it was found that motion models are required to improve the accuracy in compensating for attenuation correction mismatch and secondly, a validation method was found for comparing approaches to compensating for attenuation mismatch

Improving the Accuracy of CT-derived Attenuation Correction in Respiratory-Gated PET/CT Imaging

The effect of respiratory motion on attenuation correction in Fludeoxyglucose (18F) positron emission tomography (FDG-PET) was investigated. Improvements to the accuracy of computed tomography (CT) derived attenuation correction were obtained through the alignment of the attenuation map to each emission image in a respiratory gated PET scan. Attenuation misalignment leads to artefacts in the reconstructed PET image and several methods were devised for evaluating the attenuation inaccuracies caused by this. These methods of evaluation were extended to finding the frame in the respiratory gated PET which best matched the CT. This frame was then used as a reference frame in mono-modality compensation for misalignment. Attenuation correction was found to affect the quantification of tumour volumes; thus a regional analysis was used to evaluate the impact of mismatch and the benefits of compensating for misalignment. Deformable image registration was used to compensate for misalignment, however, there were inaccuracies caused by the poor signal-to-noise ratio (SNR) in PET images. Two models were developed that were robust to a poor SNR allowing for the estimation of deformation from very noisy images. Firstly, a cross population model was developed by statistically analysing the respiratory motion in 10 4DCT scans. Secondly, a 1D model of respiration was developed based on the physiological function of respiration. The 1D approach correctly modelled the expansion and contraction of the lungs and the differences in the compressibility of lungs and surrounding tissues. Several additional models were considered but were ruled out based on their poor goodness of fit to 4DCT scans. Approaches to evaluating the developed models were also used to assist with optimising for the most accurate attenuation correction. It was found that the multimodality registration of the CT image to the PET image was the most accurate approach to compensating for attenuation correction mismatch. Mono-modality image registration was found to be the least accurate approach, however, incorporating a motion model improved the accuracy of image registration. The significance of these findings is twofold. Firstly, it was found that motion models are required to improve the accuracy in compensating for attenuation correction mismatch and secondly, a validation method was found for comparing approaches to compensating for attenuation mismatch