Geodesics in Heat

We introduce the heat method for computing the shortest geodesic distance to a specified subset (e.g., point or curve) of a given domain. The heat method is robust, efficient, and simple to implement since it is based on solving a pair of standard linear elliptic problems. The method represents a significant breakthrough in the practical computation of distance on a wide variety of geometric domains, since the resulting linear systems can be prefactored once and subsequently solved in near-linear time. In practice, distance can be updated via the heat method an order of magnitude faster than with state-of-the-art methods while maintaining a comparable level of accuracy. We provide numerical evidence that the method converges to the exact geodesic distance in the limit of refinement; we also explore smoothed approximations of distance suitable for applications where more regularity is required

A triangulation-invariant method for anisotropic geodesic map computation on surface meshes

pre-printThis paper addresses the problem of computing the geodesic distance map from a given set of source vertices to all other vertices on a surface mesh using an anisotropic distance metric. Formulating this problem as an equivalent control theoretic problem with Hamilton-Jacobi-Bellman partial differential equations, we present a framework for computing an anisotropic geodesic map using a curvature-based speed function. An ordered upwind method (OUM)-based solver for these equations is available for unstructured planar meshes. We adopt this OUM-based solver for surface meshes and present a triangulation-invariant method for the solver. Our basic idea is to explore proximity among the vertices on a surface while locally following the characteristic direction at each vertex. We also propose two speed functions based on classical curvature tensors and show that the resulting anisotropic geodesic maps reflect surface geometry well through several experiments, including isocontour generation, offset curve computation, medial axis extraction, and ridge/valley curve extraction. Our approach facilitates surface analysis and processing by defining speed functions in an application-dependent manner

An automatic approach for classification and categorisation of lip morphological traits

Classification of facial traits (e.g., lip shape) is an important area of medical research, for example, in determining associations between lip traits and genetic variants which may lead to a cleft lip. In clinical situations, classification of facial traits is usually performed subjectively directly on the individual or recorded later from a three-dimensional image, which is time consuming and prone to operator errors. The present study proposes, for the first time, an automatic approach for the classification and categorisation of lip area traits. Our approach uses novel three-dimensional geometric features based on surface curvatures measured along geodesic paths between anthropometric landmarks. Different combinations of geodesic features are analysed and compared. The effect of automatically identified categories on the face is visualised using a partial least squares method. The method was applied to the classification and categorisation of six lip shape traits (philtrum, Cupid’s bow, lip contours, lip-chin, and lower lip tone) in a large sample of 4747 faces of normal British Western European descents. The proposed method demonstrates correct automatic classification rate of up to 90%

Accurate Computation of Geodesic Distance Fields for Polygonal Curves on Triangle Meshes Abstract

We present an algorithm for the efficient and accurate computation of geodesic distance fields on triangle meshes. We generalize the algorithm originally proposed by Surazhsky et al. [1]. While the original algorithm is able to compute geodesic distances to isolated points on the mesh only, our generalization can handle arbitrary, possibly open, polygons on the mesh to define the zero set of the distance field. Our extensions integrate naturally into the base algorithm and consequently maintain all its nice properties. For most geometry processing algorithms, the exact geodesic distance information is sampled at the mesh vertices and the resulting piecewise linear interpolant is used as an approximation to the true distance field. The quality of this approximation strongly depends on the structure of the mesh and the location of the medial axis of the distance field. Hence our second contribution is a simple adaptive refinement scheme, which inserts new vertices at critical locations on the mesh such that the final piecewise linear interpolant is guaranteed to be a faithful approximation to the true geodesic distance field.

The Star Unfolding from a Geodesic Curve

An unfolding of a polyhedron P is obtained by `cutting' the surface of P in such a way that it can be flattened into the plane into a single polygon. For most practical and theoretic applications, it is desirable for an algorithm to produce an unfolding which is simple, that is, non-overlapping. Currently, two methods for unfolding which guarantee non-overlap for convex polyhedra are known, the source unfolding, and the star}unfolding. Both methods involve computing shortest paths from a single source point on the polyhedron's surface. In this thesis, we attempt to prove non-overlap of a variant called the geodesic star unfolding. This unfolding, much like the star unfolding, is computed by cutting shortest paths from each vertex to λ, a geodesic curve on the surface of a convex polyhedron P, and also cutting λ itself. Non-overlap of this case was conjectured by Demaine and Lubiw (2011). We are unsuccessful in completely proving non-overlap, though we present a number of partial results, and discuss some areas for future study. We first develop a new proof for non-overlap of the star unfolding from a point. The original proof of non-overlap was given by Aronov and O'Rourke (2009). This new proof uses a partitioning of the unfolding around the ridge tree. Each edge of the ridge tree serves as a base edge to a pair of congruent triangles; in this way, the whole unfolding is decomposed into these pairs which are called kites. We prove non-overlap by showing that pairwise, no two kites in the unfolding overlap each other, by a method which bounds the surface angle of the source images to either side of any path through the ridge tree. In addition to its simplicity compared to the previous proof, this new method easily generalizes to prove non-overlap for some cases of the star unfolding from geodesic curves. Specifically, we show non-overlap for two classes of geodesic curves, geodesic loops, and fully-extended S-shaped geodesics, by showing that the surface angle of the source images in those two cases are bounded. We also investigate a class of curves called fully-extended C-shaped geodesics for which the proof cannot hold directly. We show some specific cases where we are able to create a supplementary proof to show non-overlap, though non-overlap for the class as a whole remains unproven

Design and Optimization of a Robot for Abrasive Waterjet Polishing of Hydraulic Turbine Blades

RÉSUMÉ Dans l’industrie de fabrication de turbine hydraulique, toutes les surfaces de turbines qui sont en contact avec de l’eau devraient être polies afin d’obtenir la qualité et l’efficacité maximales. Pour cela, il est nécessaire d’utiliser une méthode de polissage qui peut avoir accès à toutes les surfaces des turbines incluant leurs bords, leurs zones restreintes et leurs courbures serrées. En raison des propriétés particulières qu’offre la technique de polissage par jet d’eau abrasif, celle-ci peut être utilisée pour accomplir cette tâche. Par conséquent, dans cette recherche, les propriétés de cette méthode non-conventionnelle sont examinées dans un premier temps et les principaux paramètres affectant ses performances sont alors déterminés. Ensuite, les conditions nécessaires de manipulations de la buse de pulvérisation vis-à-vis des surfaces courbes sont étudiées et les propriétés d’un bras robotisé pour manipuler celle-ci sont obtenues afin de réaliser cette tâche d’une manière appropriée. Par après, plusieurs mécanismes robotiques tels que des mécanismes sériels, parallèles à membrures, parallèles à câbles, et des robots hybrides sont considérés et leurs capacités à être utilisé dans ce processus sont analysées. Il est alors démontré qu’une l’architecture hybride est le meilleur candidat à retenir pour le design d’un robot de polissage par jet d’eau abrasif. Ensuite, l’architecture conceptuelle d’un robot hybride à 5 DDL est proposée. La structure du robot est constituée d’un mécanisme parallèle à câbles à 3 DDL et d’un poignet sériel à 2 DDL. Afin d’améliorer les propriétés cinématiques du mécanisme à câbles tout en minimisant le nombre d’actionneurs nécessaires, il est proposé d’utiliser des différentiels pour guider ce robot manipulateur. Aussi, la rigidité et la compacité du mécanisme sont améliorées en utilisant une liaison prismatique. Par la suite, les systèmes à câbles différentiels sont examinés et les différences entre leurs propriétés cinématiques et celles de systèmes actionnés indépendamment pour chaque câble sont décrites. Il est démontré que la force résultante de tous les câbles d’un différentiel à câbles doit être prise en compte dans son analyse cinématique. En effet, dans un système différentiel planaire, la direction de la force résultante n’est pas fixée vers un point particulier. Mais plutôt, elle se déplace dans le plan de ce système différentiel. Cette propriété peut être bénéfique pour les propriétés cinématiques des robots à câbles. En comparant deux types d’espace de travail de plusieurs robots planaires actionnés par des mécanismes différentiels par rapport à leurs équivalents pleinement actionnés, il est alors montré qu’en utilisant ces mécanismes, les espaces de travail des robots planaires à câbles peuvent être améliorés. Cependant, cette même propriété qui augmente la plage de variation de la direction de la force résultante dans un câble différentiel, diminue aussi son amplitude. Ainsi, le design optimal d’un différentiel à câble résulte d’un compromis entre ces deux propriétés.----------ABSTRACT In hydraulic turbine manufacturing, all surfaces of the turbines which are in contact with the water flow should be polished to obtain the desired quality and maximal efficiency. For this, it is needed to use an effective polishing method which can have access to all surfaces of the turbines including edges, narrow areas and tight bends. Because of the particular properties of the abrasive waterjet polishing technique, it can be used to accomplish this task. Therefore, in this research, the properties of this non-conventional method are first investigated and the main parameters affecting its performance are then determined. Next, the manipulation requirements of the jet nozzle over free-form surfaces are studied and the properties of a robotic arm to appropriately perform this task are obtained. Afterwards, several robotic mechanisms, e.g., serial, linkage-driven parallel, cabledriven parallel, and hybrid robots are considered and their abilities to be used in this process are investigated. It is then shown that a hybrid architecture is the best candidate for the design of an abrasive waterjet polishing robot. Next, the conceptual design of a 5-DOF hybrid robot is proposed. The structure of this robot is made of a 3-DOF cable-driven parallel mechanism and a 2-DOF serial wrist. To improve the kinematic properties of the cable-driven mechanism while the number of required actuators is kept at a minimum, it is proposed to use cable differentials to drive this manipulator. Also, the rigidity and compactness of the mechanism is improved through the use of a prismatic joint in its structure. Afterwards, differentially driven cable systems are investigated and the differences between their kinematic properties and these of independently actuated cables are described. It is shown that the resultant force of all cables of a cable differential should be taken into account in its kinematic analysis. Indeed, in a planar differential, the direction of the resultant force is not fixed toward a particular point. Instead, it moves within the plane of that differential. This property can be beneficial in the kinematic properties of differentially driven cable robots. By comparing two types of workspaces of several planar robots actuated by differentials with their fully actuated counterparts, it is then shown that using these mechanisms, these workspaces of planar cable robots can be improved. However, the same property that increases the range of variation of the resultant force direction in a cable differential, decreases its magnitude. Thus, the optimal design of a cable differential is a trade-off between these two properties. Next, a synthesis method is presented to find all possible arrangements of the cable differentials to generalize the idea of using such mechanisms in the design of planar cable robots. Additionally, the application of differentials in spatial robots is also investigated and it is shown that they have properties similar to the planar types

Groupwise non-rigid registration for automatic construction of appearance models of the human craniofacial complex for analysis, synthesis and simulation

Finally, a novel application of 3D appearance modelling is proposed: a faster than real-time algorithm for statistically constrained quasi-mechanical simulation. Experiments demonstrate superior realism, achieved in the proposed method by employing statistical appearance models to drive the simulation, in comparison with the comparable state-of-the-art quasi-mechanical approaches.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Groupwise non-rigid registration for automatic construction of appearance models of the human craniofacial complex for analysis, synthesis and simulation

Finally, a novel application of 3D appearance modelling is proposed: a faster than real-time algorithm for statistically constrained quasi-mechanical simulation. Experiments demonstrate superior realism, achieved in the proposed method by employing statistical appearance models to drive the simulation, in comparison with the comparable state-of-the-art quasi-mechanical approaches