Geometry of Dirac Operators

Let $M$ be a compact, oriented, even dimensional Riemannian manifold and let $S$ be a Clifford bundle over $M$ with Dirac operator $D$. Then \[ \textsc{Atiyah Singer: } \quad \text{Ind } \mathsf{D}= \int_M \hat{\mathcal{A}}(TM)\wedge \text{ch}(\mathcal{V}) \] where $\mathcal{V} =\text{Hom}_{\mathbb{C}l(TM)}(\slashed{\mathsf{S}},S)$. We prove the above statement with the means of the heat kernel of the heat semigroup $e^{-tD^2}$. The first outstanding result is the McKean-Singer theorem that describes the index in terms of the supertrace of the heat kernel. The trace of heat kernel is obtained from local geometric information. Moreover, if we use the asymptotic expansion of the kernel we will see that in the computation of the index only one term matters. The Berezin formula tells us that the supertrace is nothing but the coefficient of the Clifford top part, and at the end, Getzler calculus enables us to find the integral of these top parts in terms of characteristic classes.

Mechanical properties and cellular response of novel electrospun nanofibers for ligament tissue engineering: Effects of orientation and geometry

Electrospun nanofibers are a promising material for ligamentous tissue engineering, however weak mechanical properties of fibers to date have limited their clinical usage. The goal of this work was to modify electrospun nanofibers to create a robust structure that mimics the complex hierarchy of native tendons and ligaments. The scaffolds that were fabricated in this study consisted of either random or aligned nanofibers in flat sheets or rolled nanofiber bundles that mimic the size scale of fascicle units in primarily tensile load bearing soft musculoskeletal tissues. Altering nanofiber orientation and geometry significantly affected mechanical properties; most notably aligned nanofiber sheets had the greatest modulus; 125% higher than that of random nanofiber sheets; and 45% higher than aligned nanofiber bundles. Modifying aligned nanofiber sheets to form aligned nanofiber bundles also resulted in approximately 107% higher yield stresses and 140% higher yield strains. The mechanical properties of aligned nanofiber bundles were in the range of the mechanical properties of the native ACL: modulus=158±32MPa, yield stress=57±23MPa and yield strain=0.38±0.08. Adipose derived stem cells cultured on all surfaces remained viable and proliferated extensively over a 7 day culture period and cells elongated on nanofiber bundles. The results of the study suggest that aligned nanofiber bundles may be useful for ligament and tendon tissue engineering based on their mechanical properties and ability to support cell adhesion, proliferation, and elongation.

Discovery of relict subglacial lakes and their geometry and mechanism of drainage

Recent proxy measurements reveal that subglacial lakes beneath modern ice sheets periodically store and release large volumes of water, providing an important but poorly understood influence on contemporary ice dynamics and mass balance. This is because direct observations of how lake drainage initiates and proceeds are lacking. Here we present physical evidence of the mechanism and geometry of lake drainage from the discovery of relict subglacial lakes formed during the last glaciation in Canada. These palaeo-subglacial lakes comprised shallow (<10 m) lenses of water perched behind ridges orientated transverse to ice flow. We show that lakes periodically drained through channels incised into bed substrate (canals). Canals sometimes trend into eskers that represent the depositional imprint of the last high-magnitude lake outburst. The subglacial lakes and channels are preserved on top of glacial lineations, indicating long-term re-organization of the subglacial drainage system and coupling to ice flow.

Automatically Evolving Rotation-invariant Texture Image Descriptors by Genetic Programming

In computer vision, training a model that performs classification effectively is highly dependent on the extracted features, and the number of training instances. Conventionally, feature detection and extraction are performed by a domain-expert who, in many cases, is expensive to employ and hard to find. Therefore, image descriptors have emerged to automate these tasks. However, designing an image descriptor still requires domain-expert intervention. Moreover, the majority of machine learning algorithms require a large number of training examples to perform well. However, labelled data is not always available or easy to acquire, and dealing with a large dataset can dramatically slow down the training process. In this paper, we propose a novel Genetic Programming based method that automatically synthesises a descriptor using only two training instances per class. The proposed method combines arithmetic operators to evolve a model that takes an image and generates a feature vector. The performance of the proposed method is assessed using six datasets for texture classification with different degrees of rotation, and is compared with seven domain-expert designed descriptors. The results show that the proposed method is robust to rotation, and has significantly outperformed, or achieved a comparable performance to, the baseline methods.

Some Inverse Problems in Analysis and Geometry

Thesis (Ph.D.)--University of Washington, 2016-08

Improving the Accuracy of 3-D Reconstruction in Robotic Vision Applications

Thesis (Ph.D.)--University of Washington, 2016-08

On precise three-dimensional environment modeling via UAV-based photogrammetric systems

Abstract : Images acquired from unmanned aerial vehicles (UAVs) can provide data with unprecedented spatial and temporal resolution for three-dimensional (3D) modeling. Solutions developed for this purpose are mainly operating based on photogrammetry concepts, namely UAV-Photogrammetry Systems (UAV-PS). Such systems are used in applications where both geospatial and visual information of the environment is required. These applications include, but are not limited to, natural resource management such as precision agriculture, military and police-related services such as traffic-law enforcement, precision engineering such as infrastructure inspection, and health services such as epidemic emergency management. UAV-photogrammetry systems can be differentiated based on their spatial characteristics in terms of accuracy and resolution. That is some applications, such as precision engineering, require high-resolution and high-accuracy information of the environment (e.g. 3D modeling with less than one centimeter accuracy and resolution). In other applications, lower levels of accuracy might be sufficient, (e.g. wildlife management needing few decimeters of resolution). However, even in those applications, the specific characteristics of UAV-PSs should be well considered in the steps of both system development and application in order to yield satisfying results. In this regard, this thesis presents a comprehensive review of the applications of unmanned aerial imagery, where the objective was to determine the challenges that remote-sensing applications of UAV systems currently face. This review also allowed recognizing the specific characteristics and requirements of UAV-PSs, which are mostly ignored or not thoroughly assessed in recent studies. Accordingly, the focus of the first part of this thesis is on exploring the methodological and experimental aspects of implementing a UAV-PS. The developed system was extensively evaluated for precise modeling of an open-pit gravel mine and performing volumetric-change measurements. This application was selected for two main reasons. Firstly, this case study provided a challenging environment for 3D modeling, in terms of scale changes, terrain relief variations as well as structure and texture diversities. Secondly, open-pit-mine monitoring demands high levels of accuracy, which justifies our efforts to improve the developed UAV-PS to its maximum capacities. The hardware of the system consisted of an electric-powered helicopter, a high-resolution digital camera, and an inertial navigation system. The software of the system included the in-house programs specifically designed for camera calibration, platform calibration, system integration, onboard data acquisition, flight planning and ground control point (GCP) detection. The detailed features of the system are discussed in the thesis, and solutions are proposed in order to enhance the system and its photogrammetric outputs. The accuracy of the results was evaluated under various mapping conditions, including direct georeferencing and indirect georeferencing with different numbers, distributions and types of ground control points. Additionally, the effects of imaging configuration and network stability on modeling accuracy were assessed. The second part of this thesis concentrates on improving the techniques of sparse and dense reconstruction. The proposed solutions are alternatives to traditional aerial photogrammetry techniques, properly adapted to specific characteristics of unmanned, low-altitude imagery. Firstly, a method was developed for robust sparse matching and epipolar-geometry estimation. The main achievement of this method was its capacity to handle a very high percentage of outliers (errors among corresponding points) with remarkable computational efficiency (compared to the state-of-the-art techniques). Secondly, a block bundle adjustment (BBA) strategy was proposed based on the integration of intrinsic camera calibration parameters as pseudo-observations to Gauss-Helmert model. The principal advantage of this strategy was controlling the adverse effect of unstable imaging networks and noisy image observations on the accuracy of self-calibration. The sparse implementation of this strategy was also performed, which allowed its application to data sets containing a lot of tie points. Finally, the concepts of intrinsic curves were revisited for dense stereo matching. The proposed technique could achieve a high level of accuracy and efficiency by searching only through a small fraction of the whole disparity search space as well as internally handling occlusions and matching ambiguities. These photogrammetric solutions were extensively tested using synthetic data, close-range images and the images acquired from the gravel-pit mine. Achieving absolute 3D mapping accuracy of 11±7 mm illustrated the success of this system for high-precision modeling of the environment.

Effet de réduction en sodium sur la texture et la bioaccessibilité des protéines d’un fromage à pâte molle à croûte fleurie

Le sel (NaCl) joue plusieurs rôles importants dans les fromages aux niveaux technologique, microbiologique et organoleptique. Cependant, le sodium, un facteur de risque des maladies cardiovasculaires, est consommé en trop grande quantité par les Canadiens. Comme la réduction du sodium pourrait affecter la texture des fromages à pâte molle à croûte fleurie et la libération des nutriments pendant la digestion, l’impact de la réduction du sodium sur la bioaccessibilité des protéines dans le fromage Brie a été étudié. Ainsi, la composition et les caractéristiques texturales de fromages Brie ayant différentes teneurs en sodium (standard, réduite, substituée au KCl) ont été analysées. Une réduction du sodium de 23 % a été obtenue pour les fromages réduits en sodium. Le temps d’affinage affecte la protéolyse et la texture des fromages. La cinétique de dégradation de la matrice fromagère et la libération des protéines ont été déterminées par une approche de digestion in vitro. La désintégration de la matrice fromagère et la libération des protéines pendant la digestion in vitro ne sont pas différentes entre les fromages expérimentaux. Ces travaux démontrent qu’il est possible de réduire le sodium sans affecter le profil de protéolyse et la texture pendant l’affinage du fromage Brie ainsi que son comportement à la digestion.

Influence of the cutting edge micro-geometry of PCBN tools on the flank wear in orthogonal quenched and tempered turning M2 steel

Quenched and tempered high-speed steels obtained by powder metallurgy are commonly used in automotive components, such as valve seats of combustion engines. In order to machine these components, tools with high wear resistance and appropriate cutting edge geometry are required. This work aims to investigate the influence of the edge preparation of polycrystalline cubic boron nitride (PCBN) tools on the wear behavior in the orthogonal longitudinal turning of quenched and tempered M2 high-speed steels obtained by powder metallurgy. For this research, PCBN tools with high and low-CBN content have been used. Two different cutting edge geometries with a honed radius were tested: with a ground land (S shape) and without it (E shape). Also, the cutting speed was varied from 100 to 220 m/min. A rigid CNC lathe was used. The results showed that the high-CBN, E-shaped tool presented the longest life for a cutting speed of 100 m/min. High-CBN tools with a ground land and honed edge radius (S shaped) showed edge damage and lower values of the tool’s life. Low-CBN, S-shaped tools showed similar results, but with an inferior performance when compared with tools with high CBN content in both forms of edge preparation.

Combining Leaf Shape and Texture for Costa Rican Plant Species Identification

In the last decade, research in Computer Vision has developed several algorithms to help botanists and non-experts to classify plants based on images of their leaves. LeafSnap is a mobile application that uses a multiscale curvature model of the leaf margin to classify leaf images into species. It has achieved high levels of accuracy on 184 tree species from Northeast US. We extend the research that led to the development of LeafSnap along two lines. First, LeafSnap’s underlying algorithms are applied to a set of 66 tree species from Costa Rica. Then, texture is used as an additional criterion to measure the level of improvement achieved in the automatic identification of Costa Rica tree species. A 25.6% improvement was achieved for a Costa Rican clean image dataset and 42.5% for a Costa Rican noisy image dataset. In both cases, our results show this increment as statistically significant. Further statistical analysis of visual noise impact, best algorithm combinations per species, and best value of , the minimal cardinality of the set of candidate species that the tested algorithms render as best matches is also presented in this research

Automatic Glaucoma Detection Based on Optic Disc Segmentation and Texture Feature Extraction

The use of digital image processing techniques is prominent in medical settings for the automatic diagnosis of diseases. Glaucoma is the second leading cause of blindness in the world and it has no cure. Currently, there are treatments to prevent vision loss, but the disease must be detected in the early stages. Thus, the objective of this work is to develop an automatic detection method of Glaucoma in retinal images. The methodology used in the study were: acquisition of image database, Optic Disc segmentation, texture feature extraction in different color models and classification of images in glaucomatous or not. We obtained results of 93% accuracy

Geometry and topography in James Joyce's Ulysses and Finnegans Wake

Following the development of non-Euclidean geometries from the mid-nineteenth century onwards, Euclid’s system had come to be re-conceived as a language for describing reality rather than a set of transcendental laws. As Henri Poincaré famously put it, ‘[i]f several geometries are possible, is it certain that our geometry [...] is true?’. By examining Joyce’s linguistic play and conceptual engagement with ground-breaking geometric constructs in Ulysses and Finnegans Wake, this thesis explores how his topographical writing of place encapsulates a common crisis between geometric and linguistic modes of representation within the context of modernity. More specifically, it investigates how Joyce presents Euclidean geometry and its topographical applications as languages, rather than ideally objective systems, for describing visual reality; and how, conversely, he employs language figuratively to emulate the systems by which the world is commonly visualised. With reference to his early readings of Giordano Bruno, Henri Poincaré and other critics of the Euclidean tradition, it investigates how Joyce’s obsession with measuring and mapping space throughout his works enters into his more developed reflections on the codification of visual signs in Finnegans Wake. In particular, this thesis sheds new light on Joyce’s developing fascination with the ‘geometry of language’ practised by Bruno, whose massive influence on Joyce is often assumed to exist in Joyce studies yet is rarely explored in any great detail.