Discrete shapes can be described and analyzed using Lie groups, which are mathematical structures having both algebraic and geometrical properties. These structures, borrowed from mathematical physics, are both algebraic groups and smooth manifolds. A key property of a Lie group is that a curved space can be studied, using linear algebra, by local linearization with an exponential map. Here, a discrete shape was a Euclidean-invariant computer representation of an object. Highly variable shapes are known to exist in non-linear spaces where linear analysis tools, such as Pearson's decomposition of principal components, are inadequate. The novel method proposed herein represented a shape as an ensemble of homogenous matrix transforms. The Lie group of homogenous transforms has elements that both represented a local shape and acted as matrix operators on other local shapes. For the manifold, a matrix transform was found to be equivalent to a vector transform in a linear space. This combination of representation and linearization gave a simple implementation for solving a computationally expensive problem. Two medical datasets were analyzed: 2D contours of femoral head-neck cross-sections and 3D surfaces of proximal femurs. The Lie-group method outperformed the established principal-component analysis by capturing higher variability with fewer components. Lie groups are promising tools for medical imaging and data analysis. / Thesis (Ph.D, Computing) -- Queen's University, 2014-01-30 09:49:03.293

The functional properties of nanoparticles highly depend on the surface morphology of the particles, so precise measurements of a particle's morphology enable reliable characterizing of the nanoparticle's properties. Obtaining the measurements requires image analysis of electron microscopic pictures of nanoparticles. Today's labor-intensive image analysis of electron micrographs of nanoparticles is a significant bottleneck for efficient material characterization. The objective of this dissertation is to develop automated morphology analysis methods. Morphology analysis is comprised of three tasks: separate individual particles from an agglomerate of overlapping nano-objects (image segmentation); infer the particle's missing contours (shape inference); and ultimately, classify the particles by shape based on their complete contours (shape classification). Two approaches are proposed in this dissertation: the divide-and-conquer approach and the convex shape analysis approach. The divide-and-conquer approach solves each task separately, taking less than one minute to complete the required analysis, even for the largest-sized micrograph. However, its separating capability of particle overlaps is limited, meaning that it is able to split only touching particles. The convex shape analysis approach solves shape inference and classification simultaneously for better accuracy, but it requires more computation time, ten minutes for the biggest-sized electron micrograph. However, with a little sacrifice of time efficiency, the second approach achieves far superior separation than the divide-and-conquer approach, and it handles the chain-linked structure of particle overlaps well. The capabilities of the two proposed methods cannot be substituted by generic image processing and bio-imaging methods. This is due to the unique features that the electron microscopic pictures of nanoparticles have, including special particle overlap structures, and large number of particles to be processed. The application of the proposed methods to real electron microscopic pictures showed that the two proposed methods were more capable of extracting the morphology information than the state-of-the-art methods. When nanoparticles do not have many overlaps, the divide-and-conquer approach performed adequately. When nanoparticles have many overlaps, forming chain-linked clusters, the convex shape analysis approach performed much better than the state-of-the-art alternatives in bio-imaging. The author believes that the capabilities of the proposed methods expedite the morphology characterization process of nanoparticles. The author further conjectures that the technical generality of the proposed methods could even be a competent alternative to the current methods analyzing general overlapping convex-shaped objects other than nanoparticles.

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The majority of car crash deaths occur in the front seats because the majority of occupants sit in the front seats. Traditionally, the rear seats were safer than the front seats because a front seated occupant would be closer to rigid structures such as the steering wheel, and they would be closer to the location of the impact. Therefore, government crash test regulations as well as academic and industry testing up to this point have principally focused on the front seats. Since the beginning of efforts to make cars safer, innovations were applied to the front seats first. Only some of these safety innovations have transitioned into the rear seats. Over the years, the front seats have gotten much safer due to advanced seatbelts with pretentioners and load limiters, airbags surrounding the driver, and structural changes to the vehicle frame to prevent intrusion into the occupant compartment. At the same time, occupant safety in the rear seats has also improved, however at only a fraction of the improvement of the front seats. With modern vehicles, the front seats have actually become safer than the rear seats for certain occupants and specific crash types (e.g., adult occupants in frontal crash). The lagging performance of the rear seats represents a problem because thousands of rear-seated occupants are injured or killed each year. With the rise in autonomous driving systems, the amount of occupants sitting in the rear seats, and therefore sustaining injury, could increase dramatically. In this dissertation, rear seats of a range of current vehicles were reconstructed to examine injury risk with the finite element models of two anthropomorphic test devices. These models showed a wide range of injury risks in the reconstructed seats. They were also able to show results similar to sled impact tests with the same vehicles. Knowledge gained from these reconstructions was then used to perform parametric studies on key variables that influence injury risk in the rear seats. From the parametric studies, it was found that the seat back angle, the width of the seatbelt anchors, and the presence of a seatbelt pretensioner had the largest influences on the injury risk. One of the injury mechanisms prevalent in the rear seats is submarining. Submarining likelihood and injury probability is difficult to predict with anthropomorphic test devices; however, human body models can help to improve injury prediction in these cases. To improve the injury prediction capability of human body models, several additions to the models are necessary. This dissertation outlines the investigation of spleen and kidney shapes through statistical shape analysis. This type of analysis allows more customizable human body models which could better capture the injury probability to these organs for a wider range of the population. Finally, subject-specific models of ribs were created to investigate factors affecting the predictive capability of finite element models. The findings and methodology from this body of work have the ability to add critical contributions to the understanding of injury risk and injury mechanisms in the rear seats. / Doctor of Philosophy / The majority of car crash deaths occur in the front seats because the majority of occupants sit in the front seats. Traditionally, the rear seats were safer than the front seats because a front seated occupant would be closer to hard objects such as the steering wheel, and they would be closer to the location of the impact. Therefore, government crash test regulations as well as academic and industry testing up to this point have principally focused on the front seats. Since the beginning of efforts to make cars safer, technology such as seatbelts and airbags were applied to the front seats first. Only some of this technology has been added into the rear seats. Over the years, the front seats have gotten much safer due to all the work focused on the front seats. At the same time, the rear seats have also improved, however at only a fraction of the improvement of the front seats. With modern vehicles, the front seats have actually become safer than the rear seats in some cases. The lagging performance of the rear seats represents a problem because thousands of rear-seated occupants are injured or killed each year. With the rise in self driving cars, the amount of occupants sitting in the rear seats, and therefore sustaining injury, could increase dramatically. In this dissertation, rear seats of a range of current vehicles were reconstructed to examine injury risk with the models of two crash test dummies. These models showed a wide range of injury risks in the reconstructed seats. They were also able to show results similar to physical tests with the same vehicles. Knowledge gained from this work was then used to help look at key variables that influence injury risk in the rear seats. It was found that the angle of the seat back, the width of the seatbelt anchors, and the presence of advanced seatbelts had the largest influences on the injury risk. One of the injury mechanisms prevalent in the rear seats is submarining, where the seatbelt slides up off the hips. Submarining likelihood and injury probability is difficult to predict with crash test dummies; however, human body models can help to improve injury prediction in these cases. To improve the injury prediction capability of human body models, several additions to the models are necessary. This dissertation outlines the investigation of spleen and kidney shapes to allow more customizable human body models which could better capture the injury probability to these organs for a wider range of the population. Finally, subject-specific models of ribs were created to investigate factors affecting the predictive capability of rib models. The findings and methodology from this body of work have the ability to add critical contributions to the understanding of injury risk and injury mechanisms in the rear seats.

While most statistical applications involve real numbers, some demand complex numbers. Statistical shape analysis is one such area. The complex Bingham distribution is utilized in the shape analysis of landmark data in two dimensions. Previous analysis of data arising from this distribution involved classical statistical techniques. In this report, a full Bayesian inference was carried out on the posterior distribution of the parameter matrix when data arise from a complex Bingham distribution. We utilized a Markov chain Monte Carlo algorithm to sample the posterior distribution of the parameters. A Metropolis-Hastings algorithm sampled the posterior conditional distribution of the eigenvalues while a successive conditional Monte Carlo sampler was used to sample the eigenvectors. The method was successfully verifi ed on simulated data, using both at and informative priors. / text

Realistic assessments of liver injury risk for the entire occupant population require incorporating inter-subject variations into numerical human models. The main objective of this study was to quantify the variations in shape and material properties of the human liver. Statistical shape analysis was applied to analyze the geometrical variation using a surface set of 15 adult human livers recorded in an occupant posture. Principal component analysis was then utilized to obtain the modes of variation, the mean model, and a set of 95% statistical boundary shape models. Specimen-specific finite element (FE) models were employed to quantify material and failure properties of human liver parenchyma. The mean material model parameters were then determined, and a stochastic optimization approach was utilized to determine the standard deviations of the material model parameters. The distributions of the material parameters were used to develop probabilistic FE models of the liver implemented in THUMS human FE model to simulate oblique impact tests under three impact speeds. In addition, the influence of organ preservation on the biomechanical responses of animal livers was investigated using indentation and tensile tests. Results showed that the first five modes of the human liver shape models accounted for more than 70% of the overall anatomical variations. The Ogden material model with two parameters showed a good fit to experimental tensile data before failure. Significant changes of the biomechanical responses of liver parenchyma were found after cooling or freezing storage. The force-deflection responses of THUMS model with probabilistic liver material models were within the test corridors obtained from cadaveric tests. Significant differences were observed in the maximum and minimum principal Green-Lagrangian strain values recorded in the THUMS liver model with the default and updated average material properties. The results from this study could help in the development of more biofidelic human models, which may provide a better understanding of injury mechanisms of the liver during automobile collisions. / Ph. D.

Cette thèse a pour objet l'analyse statistique de formes, dans le contexte de l'imagerie médicale.Dans le champ de l'imagerie médicale, l'analyse de formes est utilisée pour décrire la variabilité morphologique de divers organes et tissus. Nous nous focalisons dans cette thèse sur la construction d'un modèle génératif et discriminatif, compact et non-linéaire, adapté à la représentation de formes.Ce modèle est évalué dans le contexte de l'étude d'une population de patients atteints de la maladie d'Alzheimer et d'une population de sujets contrôles sains. Notre intérêt principal ici est l'utilisationdu modèle discriminatif pour découvrir les différences morphologiques les plus discriminatives entre une classe de formes donnée et des formes n'appartenant pas à cette classe. L'innovation théorique apportée par notre modèle réside en deux points principaux : premièrement, nous proposons un outil pour extraire la différence discriminative dans le cadre Support Vector Data Description (SVDD) ; deuxièmement, toutes les reconstructions générées sont anatomiquementcorrectes. Ce dernier point est dû au caractère non-linéaire et compact du modèle, lié à l'hypothèse que les données (les formes) se trouvent sur une variété non-linéaire de dimension faible. Une application de notre modèle à des données médicales réelles montre des résultats cohérents avec les connaissances médicales. / This thesis addresses statistical shape analysis, in the context of medical imaging. In the field of medical imaging, shape analysis is used to describe the morphological variability of various organs and tissues. Our focus in this thesis is on the construction of a generative and discriminative, compact and non-linear model, suitable to the representation of shapes. This model is evaluated in the context of the study of a population of Alzheimer's disease patients and a population of healthy controls. Our principal interest here is using the discriminative model to discover morphological differences that are the most characteristic and discriminate best between a given shape class and forms not belonging in that class. The theoretical innovation of our work lies in two principal points first, we propose a tool to extract discriminative difference in the context of the Support Vector Data description (SVDD) framework ; second, all generated reconstructions are anatomicallycorrect. This latter point is due to the non-linear and compact character of the model, related to the hypothesis that the data (the shapes) lie on a low-dimensional, non-linear manifold. The application of our model on real medical data shows results coherent with well-known findings in related research.

Les différentes étapes et processus de la fabrication additive (FA) induisent des erreurs de sources multiples et complexes qui soulèvent des problèmes majeurs au niveau de la qualité géométrique du produit fabriqué. Par conséquent, une modélisation effective des écarts géométriques est essentielle pour la FA. Le paradigme Skin Model Shapes (SMS) offre un cadre intégral pour la modélisation des écarts géométriques des produits manufacturés et constitue ainsi une solution efficace pour la modélisation des écarts géométriques en FA.Dans cette thèse, compte tenu de la spécificité de fabrication par couche en FA, un nouveau cadre de modélisation à base de SMS est proposé pour caractériser les écarts géométriques en FA en combinant une approche dans le plan et une approche hors plan. La modélisation des écarts dans le plan vise à capturer la variabilité de la forme 2D de chaque couche. Une méthode de transformation des formes est proposée et qui consiste à représenter les effets de variations sous la forme de transformations affines appliquées à la forme nominale. Un modèle paramétrique des écarts est alors établi dans un système de coordonnées polaires, quelle que soit la complexité de la forme. Ce modèle est par la suite enrichi par un apprentissage statistique permettant la collecte simultanée de données des écarts de formes multiples et l'amélioration des performances de la méthode.La modélisation des écarts hors plan est réalisée par la déformation de la couche dans la direction de fabrication. La modélisation des écarts hors plan est effectuée à l'aide d'une méthode orientée données. Sur la base des données des écarts obtenues à partir de simulations par éléments finis, deux méthodes d'analyse modale: la transformée en cosinus discrète (DCT) et l'analyse statistique des formes (SSA) sont exploitées. De plus, les effets des paramètres des pièces et des procédés sur les modes identifiés sont caractérisés par le biais d'un modèle à base de processus Gaussien.Les méthodes présentées sont finalement utilisées pour obtenir des SMSs haute-fidélité pour la fabrication additive en déformant les contours de la couche nominale avec les écarts prédits et en reconstruisant le modèle de surface non idéale complet à partir de ces contours déformés. Une toolbox est développée dans l'environnement MATLAB pour démontrer l'efficacité des méthodes proposées. / The intricate error sources within different stages of the Additive Manufacturing (AM) process have brought about major issues regarding the dimensional and geometrical accuracy of the manufactured product. Therefore, effective modeling of the geometric deviations is critical for AM. The Skin Model Shapes (SMS) paradigm offers a comprehensive framework aiming at addressing the deviation modeling problem at different stages of product lifecycle, and is thus a promising solution for deviation modeling in AM. In this thesis, considering the layer-wise characteristic of AM, a new SMS framework is proposed which characterizes the deviations in AM with in-plane and out-of-plane perspectives. The modeling of in-plane deviation aims at capturing the variability of the 2D shape of each layer. A shape transformation perspective is proposed which maps the variational effects of deviation sources into affine transformations of the nominal shape. With this assumption, a parametric deviation model is established based on the Polar Coordinate System which manages to capture deviation patterns regardless of the shape complexity. This model is further enhanced with a statistical learning capability to simultaneously learn from deviation data of multiple shapes and improve the performance on all shapes.Out-of-plane deviation is defined as the deformation of layer in the build direction. A layer-level investigation of out-of-plane deviation is conducted with a data-driven method. Based on the deviation data collected from a number of Finite Element simulations, two modal analysis methods, Discrete Cosine Transform (DCT) and Statistical Shape Analysis (SSA), are adopted to identify the most significant deviation modes in the layer-wise data. The effect of part and process parameters on the identified modes is further characterized with a Gaussian Process (GP) model. The discussed methods are finally used to obtain high-fidelity SMSs of AM products by deforming the nominal layer contours with predicted deviations and rebuilding the complete non-ideal surface model from the deformed contours. A toolbox is developed in the MATLAB environment to demonstrate the effectiveness of the proposed methods.