Modelling and Animation using Partial Differential Equations. Geometric modelling and computer animation of virtual characters using elliptic partial differential equations.

This work addresses various applications pertaining to the design, modelling and animation of parametric surfaces using elliptic Partial Differential Equations (PDE) which are produced via the PDE method. Compared with traditional surface generation techniques, the PDE method is an effective technique that can represent complex three-dimensional (3D) geometries in terms of a relatively small set of parameters. A PDE-based surface can be produced from a set of pre-configured curves that are used as the boundary conditions to solve a number of PDE. An important advantage of using this method is that most of the information required to define a surface is contained at its boundary. Thus, complex surfaces can be computed using only a small set of design parameters. In order to exploit the advantages of this methodology various applications were developed that vary from the interactive design of aircraft configurations to the animation of facial expressions in a computer-human interaction system that utilizes an artificial intelligence (AI) bot for real time conversation. Additional applications of generating cyclic motions for PDE based human character integrated in a Computer-Aided Design (CAD) package as well as developing techniques to describe a given mesh geometry by a set of boundary conditions, required to evaluate the PDE method, are presented. Each methodology presents a novel approach for interacting with parametric surfaces obtained by the PDE method. This is due to the several advantages this surface generation technique has to offer. Additionally, each application developed in this thesis focuses on a specific target that delivers efficiently various operations in the design, modelling and animation of such surfaces.The project files will not be available online

A survey of partial differential equations in geometric design

YesComputer aided geometric design is an area where the improvement of surface generation techniques is an everlasting demand since faster and more accurate geometric models are required. Traditional methods for generating surfaces were initially mainly based upon interpolation algorithms. Recently, partial differential equations (PDE) were introduced as a valuable tool for geometric modelling since they offer a number of features from which these areas can benefit. This work summarises the uses given to PDE surfaces as a surface generation technique togethe

Modelling of oedemous limbs and venous ulcers using partial differential equations

BACKGROUND: Oedema, commonly known as tissue swelling, occurs mainly on the leg and the arm. The condition may be associated with a range of causes such as venous diseases, trauma, infection, joint disease and orthopaedic surgery. Oedema is caused by both lymphatic and chronic venous insufficiency, which leads to pooling of blood and fluid in the extremities. This results in swelling, mild redness and scaling of the skin, all of which can culminate in ulceration. METHODS: We present a method to model a wide variety of geometries of limbs affected by oedema and venous ulcers. The shape modelling is based on the PDE method where a set of boundary curves are extracted from 3D scan data and are utilised as boundary conditions to solve a PDE, which provides the geometry of an affected limb. For this work we utilise a mixture of fourth order and sixth order PDEs, the solutions of which enable us to obtain a good representative shape of the limb and associated ulcers in question. RESULTS: A series of examples are discussed demonstrating the capability of the method to produce good representative shapes of limbs by utilising a series of curves extracted from the scan data. In particular we show how the method could be used to model the shape of an arm and a leg with an associated ulcer. CONCLUSION: We show how PDE based shape modelling techniques can be utilised to generate a variety of limb shapes and associated ulcers by means of a series of curves extracted from scan data. We also discuss how the method could be used to manipulate a generic shape of a limb and an associated wound so that the model could be fine-tuned for a particular patient

Modelling oedemous limbs and venous ulcers using partial differential equations

Background Oedema, commonly known as tissue swelling, occurs mainly on the leg and the arm. The condition may be associated with a range of causes such as venous diseases, trauma, infection, joint disease and orthopaedic surgery. Oedema is caused by both lymphatic and chronic venous insufficiency, which leads to pooling of blood and fluid in the extremities. This results in swelling, mild redness and scaling of the skin, all of which can culminate in ulceration. Methods We present a method to model a wide variety of geometries of limbs affected by oedema and venous ulcers. The shape modelling is based on the PDE method where a set of boundary curves are extracted from 3D scan data and are utilised as boundary conditions to solve a PDE, which provides the geometry of an affected limb. For this work we utilise a mixture of fourth order and sixth order PDEs, the solutions of which enable us to obtain a good representative shape of the limb and associated ulcers in question. Results A series of examples are discussed demonstrating the capability of the method to produce good representative shapes of limbs by utilising a series of curves extracted from the scan data. In particular we show how the method could be used to model the shape of an arm and a leg with an associated ulcer. Conclusion We show how PDE based shape modelling techniques can be utilised to generate a variety of limb shapes and associated ulcers by means of a series of curves extracted from scan data. We also discuss how the method could be used to manipulate a generic shape of a limb and an associated wound so that the model could be fine-tuned for a particular patient

Shape Morphing Using PDE Surfaces

NoA methodology for shape morphing using partial differential equation (PDE) surfaces is presented in this work. The use of the PDE formulation shows how shape morphing can be based on a boundary-value approach by which intermediate shapes can be created. Furthermore, the mathematical properties of the method give rise to several alternatives in which morphing one shape into another can be achieved. Three of these alternatives are presented here. The first one is based on the gradual variation of the weighted sum of the boundary conditions for each surface, the second one consists of varying the Fourier mode for which the PDE is solved whilst the third results from a combination of the first two. Examples showing the efficiency of these methodologies are presented. Thus, it is shown that the PDE based approach for morphing, when combined with a parametric variation of the boundary conditions, is capable of obtaining smooth intermediate surfaces automatically

Fast generation of 3D deformable moving surfaces

Dynamic surface modeling is an important subject of geometric modeling due to their extensive applications in engineering design, entertainment and medical visualization. Many deformable objects in the real world are dynamic objects as their shapes change over time. Traditional geometric modeling methods are mainly concerned with static problems, therefore unsuitable for the representation of dynamic objects. Apart from the definition of a dynamic modeling problem, another key issue is how to solve the problem. Because of the complexity of the representations, currently the finite element method or finite difference method is usually used. Their major shortcoming is the excessive computational cost, hence not ideal for applications requiring real-time performance. We propose a representation of dynamic surface modeling with a set of fourth order dynamic partial differential equations (PDEs). To solve these dynamic PDEs accurately and efficiently, we also develop an effective resolution method. This method is further extended to achieve local deformation and produce n-sided patches. It is demonstrated that this new method is almost as fast and accurate as the analytical closed form resolution method and much more efficient and accurate than the numerical methods

PDE Face: A Novel 3D Face Model

YesWe introduce a novel approach to face models, which exploits the use of Partial Differential Equations (PDE) to generate the 3D face. This addresses some common problems of existing face models. The PDE face benefits from seamless merging of surface patches by using only a relatively small number of parameters based on boundary curves. The PDE face also provides users with a great degree of freedom to individualise the 3D face by adjusting a set of facial boundary curves. Furthermore, we introduce a uv-mesh texture mapping method. By associating the texels of the texture map with the vertices of the uv mesh in the PDE face, the new texture mapping method eliminates the 3D-to-2D association routine in texture mapping. Any specific PDE face can be textured without the need for the facial expression in the texture map to match exactly that of the 3D face model

PDE-based Facial Animation: Making the Complex Simple

YesDirect parameterisation is among the most widely used facial animation techniques but requires complicated ways to animate face models which have complex topology. This paper develops a simple solution by introducing a PDE-based facial animation scheme. Using a PDE face model means we only need to animate a group of boundary curves without using any other conventional surface interpolation algorithms. We describe the basis of the method and show results from a practical implementation.EPSR