Finding the true nondominated points is typically hard for Multi-objective Combinatorial Optimization (MOCO) problems. Furthermore, it is not practical to generate all of them since the number of nondominated points may grow exponentially as the problem size increases. In this thesis, we develop an exact algorithm to find all nondominated points in a specified region. We combine this exact algorithm with a heuristic algorithm that approximates the possible locations of the nondominated points. Interacting with a decision maker (DM), the heuristic algorithm first approximately identifies the region that is of interest to the DM. Then, the exact algorithm is employed to generate all true nondominated points in this region. We conduct experiments on Multi-objective Assignment Problems (MOAP), Multi-objective Knapsack Problems (MOKP) and Multi-objective Shortest Path (MOSP) Problems / and the algorithms work well. Finding the worst possible value for each criterion among the set of efficient solutions has important uses in multi-criteria problems since the proper scaling of each criterion is required by many approaches. Such points are called nadir points. v It is not straightforward to find the nadir points, especially for large problems with more than two criteria. We develop an exact algorithm to find the nadir values for multi-objective integer programming problems. We also find bounds with performance guarantees. We demonstrate that our algorithms work well in our experiments on MOAP, MOKP and MOSP problems. Assuming that the DM&#039 / s preferences are consistent with a quasiconcave value function, we develop an interactive exact algorithm to solve MIP problems. Based on the convex cones derived from pairwise comparisons of the DM, we generate constraints to prevent points in the implied inferior regions. We guarantee finding the most preferred point and our computational experiments on MOAP, MOKP and MOSP problems show that a reasonable number of pairwise comparisons are required.

This thesis describes theory and algorithms for use with Radial Basis Functions (RBFs), emphasising techniques motivated by three particular application areas. In Part I, we apply RBFs to the problem of interpolating to integral data. While the potential of using RBFs for this purpose has been established in an abstract theoretical context, their use has been lacking an easy to check sufficient condition for finding appropriate parent basic functions, and explicit methods for deriving integral basic functions from them. We present both these components here, as well as explicit formulations for line segments in two dimensions and balls in three and five dimensions. We also apply these results to real-world track data. In Part II, we apply Hermite and pointwise RBFs to the problem of surface reconstruction. RBFs are used for this purpose by representing the surface implicitly as the zero level set of a function in 3D space. We develop a multilevel piecewise technique based on scattered spherical subdomains, which requires the creation of algorithms for constructing sphere coverings with desirable properties and for blending smoothly between levels. The surface reconstruction method we develop scales very well to large datasets and is very amenable to parallelisation, while retaining global-approximation-like features such as hole filling. Our serial implementation can build an implicit surface representation which interpolates at over 42 million points in around 45 minutes. In Part III, we apply RBFs to the problem of animation control in the area of motion synthesis---controlling an animated character whose motion is entirely the result of simulated physics. While the simulation is quite well understood, controlling the character by means of forces produced by virtual actuators or muscles remains a very difficult challenge. Here, we investigate the possibility of speeding up the optimisation process underlying most animation control methods by approximating the physics simulator with RBFs.

L’animation faciale est l’un des points clés dans le réalisme des scènes 3D qui mettent en scène des personnages virtuels. Ceci s’explique principalement par les raisons suivantes : le visage et les nombreux muscles qui le composent permettent de générer une multitude d’expressions ; ensuite, notre faculté de perception nous permet de détecter et d’analyser ses mouvements les plus fins. La complexité de ce domaine se retrouve dans les approches existantes par le fait qu’il est très difficile de créer une animation de qualité sans un travail manuel long et fastidieux. Partant de ce constat, cette thèse a pour but de développer des techniques qui contribuent au processus de création d’animations faciales. Trois thèmes sont principalement abordés. Le premier concerne la paramétrisation du visage pour l’animation. La paramétrisation a pour but de définir des moyens de contrôle pour pouvoir déformer et animer le visage. Le second s’oriente sur l’animation, et plus particulièrement sur le transfert d’animation. Le but est de proposer une méthode qui permette d’animer le visage d’un personnage à partir de données variées. Ces données peuvent être issues d’un système de capture de mouvement, ou bien elles peuvent être obtenues à partir de l’animation d’un personnage virtuel qui existe déjà. Enfin, nous nous sommes concentrés sur les détails fins liés à l’animation comme les rides. Bien que ces rides soient fines et discrètes, ces déformations jouent un rôle important dans la perception et l’analyse des émotions. C’est pourquoi nous proposons une technique d’acquisition mono-caméra et une méthode à base de poses références pour synthétiser dynamiquement les détails fins d’animation sur le visage. L’objectif principal des méthodes proposées est d’offrir des solutions afin de faciliter et d’améliorer le processus de création d’animations faciales réalistes utilisées dans le cadre d’applications en temps réel. Nous nous sommes particulièrement concentrés sur la facilité d’utilisation et sur la contrainte du temps réel. De plus, nous offrons la possibilité à l’utilisateur ou au graphiste d’interagir afin de personnaliser sa création et/ou d’améliorer les résultats obtenus / Facial animation is one of the key points of the realism of 3D scenes featuring virtual humans. This is due to several reasons : face and the many muscles that compose it can generate a multitude of expressions ; then, our faculty of perception provides us a great ability to detect and analyze its smallest variations. This complexity is reflected in existing approaches by the fact that it is very difficult to create an animation without a long and a tedious manual work. Based on these observations, this thesis aims to develop techniques that contribute to the process of creating facial animation. Three main themes have been addressed. The first concerns the rigging issue of a virtual 3D face for animation. Rigging aims at defining control parameters in order to deform and animate the face. The second deals with the animation, especially on the animation retargeting issue. The goal is to propose a method to animate a character’s face from various data. These data can be obtained from a motion capture system or from an existing 3D facial animation. Finally, we focus on animation finescale details like wrinkles. Although these are thin and discreet, their deformations play an important part in the perception and analysis of emotions. Therefore we propose a monocular acquisition technique and a reference pose based method to synthetise dynamically animation fine details over the face. The purpose is to propose methods to facilitate and improve the process of creating realistic facial animations for interactive applications. We focused on ease to use in addition to the real-time aspect. Moreover, we offer the possibility to the user or graphist to interact in order to personalize its creation and/or improve the results

The central aim for the research undertaken in this PhD thesis is the development of a model for simulating water droplet movement on a leaf surface and to compare the model behavior with experimental observations. A series of five papers has been presented to explain systematically the way in which this droplet modelling work has been realised. Knowing the path of the droplet on the leaf surface is important for understanding how a droplet of water, pesticide, or nutrient will be absorbed through the leaf surface. An important aspect of the research is the generation of a leaf surface representation that acts as the foundation of the droplet model. Initially a laser scanner is used to capture the surface characteristics for two types of leaves in the form of a large scattered data set. After the identification of the leaf surface boundary, a set of internal points is chosen over which a triangulation of the surface is constructed. We present a novel hybrid approach for leaf surface fitting on this triangulation that combines Clough-Tocher (CT) and radial basis function (RBF) methods to achieve a surface with a continuously turning normal. The accuracy of the hybrid technique is assessed using numerical experimentation. The hybrid CT-RBF method is shown to give good representations of Frangipani and Anthurium leaves. Such leaf models facilitate an understanding of plant development and permit the modelling of the interaction of plants with their environment. The motion of a droplet traversing this virtual leaf surface is affected by various forces including gravity, friction and resistance between the surface and the droplet. The innovation of our model is the use of thin-film theory in the context of droplet movement to determine the thickness of the droplet as it moves on the surface. Experimental verification shows that the droplet model captures reality quite well and produces realistic droplet motion on the leaf surface. Most importantly, we observed that the simulated droplet motion follows the contours of the surface and spreads as a thin film. In the future, the model may be applied to determine the path of a droplet of pesticide along a leaf surface before it falls from or comes to a standstill on the surface. It will also be used to study the paths of many droplets of water or pesticide moving and colliding on the surface.