We present an efficient method for reconstructing complex geometry using an elliptic Partial Differential Equation (PDE) formulation. The integral part of this work is the use of three-dimensional curves within the physical space which act as boundary conditions to solve the PDE. The chosen PDE is solved explicitly for a given general set of curves representing the original shape and thus making the method very efficient. In order to improve the quality of results for shape representation we utilize an automatic parameterization scheme on the chosen curves. With this formulation we discuss our methodology for shape representation using a series of practical examples.

<p>This thesis describes initial language extensions to the Modelica language to define a more general language called PDEModelica, with built-in support for modeling with partial differential equations (PDEs). Modelica® is a standardized modeling language for objectoriented, equation-based modeling. It also supports component-based modeling where existing components with modified parameters can be combined into new models. The aim of the language presented in this thesis is to maintain the advantages of Modelica and also add partial differential equation support.</p><p>Partial differential equations can be defined using a coefficient-based approach, where a predefined PDE is modified by changing its coefficient values. Language operators to directly express PDEs in the language are also discussed. Furthermore, domain geometry description is handled and language extensions to describe geometries are presented. Boundary conditions, required for a complete PDE problem definition, are also handled.</p><p>A prototype implementation is described as well. The prototype includes a translator written in the relational meta-language, RML, and interfaces to external software such as mesh generators and PDE solvers, which are needed to solve PDE problems. Finally, a few examples modeled with PDEModelica and solved using the prototype are presented.</p> / Report code: LiU-Tek-Lic-2002:63.

Yes / This paper extends the PDE method of surface generation. The governing partial differential equation is generalised to sixth order to increase its flexibility. The PDE is solved analytically, even in the case of general boundary conditions, making the method fast. The boundary conditions, which control the surface shape, are specified interactively, allowing intuitive manipulation of generic shapes. A compact user interface is presented which makes use of direct manipulation and other techniques for 3D interaction.

One of the challenging problems in geometric modeling and computer graphics is the construction of realistic human facial geometry. Such geometry are essential for a wide range of applications, such as 3D face recognition, virtual reality applications, facial expression simulation and computer based plastic surgery application. This paper addresses a method for the construction of 3D geometry of human faces based on the use of Elliptic Partial Differential Equations (PDE). Here the geometry corresponding to a human face is treated as a set of surface patches, whereby each surface patch is represented using four boundary curves in the 3-space that formulate the appropriate boundary conditions for the chosen PDE. These boundary curves are extracted automatically using 3D data of human faces obtained using a 3D scanner. The solution of the PDE generates a continuous single surface patch describing the geometry of the original scanned data. In this study, through a number of experimental verifications we have shown the efficiency of the PDE based method for 3D facial surface reconstruction using scan data. In addition to this, we also show that our approach provides an efficient way of facial representation using a small set of parameters that could be utilized for efficient facial data storage and verification purposes.

In this paper we study a methodology for the numerical simulation of stable structures of fluid membranes and vesicles in biological organisms. In particular, we discuss the effects of spontaneous curvature on vesicle cell membranes under the bending energy for given volume and surface area. The geometric modeling of the vesicle shapes are undertaken by means of surfaces generated as Partial Differential Equations (PDEs). We combine PDE based geometric modeling with numerical optimization in order to study the stable shapes adopted by the vesicle membranes. Thus, through the PDE method we generate a generic template of a vesicle membrane which is then efficiently parameterized. The parameterization is taken as a basis to set up a numerical optimization procedure which enables us to predict a series of vesicle shapes subject to given surface area and volume.

Yes / An important issue in many of today's biometric applications is the development of efficient and accurate techniques for representing related 3D data. Such data is often available through the process of digitization of complex geometric objects which are of importance to biometric applications. For example, in the area of 3D face recognition a digital point cloud of data corresponding to a given face is usually provided by a 3D digital scanner. For efficient data storage and for identification/authentication in a timely fashion such data requires to be represented using a few parameters or variables which are meaningful. Here we show how mathematical techniques based on Partial Differential Equations (PDEs) can be utilized to represent complex 3D data where the data can be parameterized in an efficient way. For example, in the case of a 3D face we show how it can be represented using PDEs whereby a handful of key facial parameters can be identified for efficient storage and verification.

Yes

This thesis describes initial language extensions to the Modelica language to define a more general language called PDEModelica, with built-in support for modeling with partial differential equations (PDEs). Modelica® is a standardized modeling language for objectoriented, equation-based modeling. It also supports component-based modeling where existing components with modified parameters can be combined into new models. The aim of the language presented in this thesis is to maintain the advantages of Modelica and also add partial differential equation support. Partial differential equations can be defined using a coefficient-based approach, where a predefined PDE is modified by changing its coefficient values. Language operators to directly express PDEs in the language are also discussed. Furthermore, domain geometry description is handled and language extensions to describe geometries are presented. Boundary conditions, required for a complete PDE problem definition, are also handled. A prototype implementation is described as well. The prototype includes a translator written in the relational meta-language, RML, and interfaces to external software such as mesh generators and PDE solvers, which are needed to solve PDE problems. Finally, a few examples modeled with PDEModelica and solved using the prototype are presented. / <p>Report code: LiU-Tek-Lic-2002:63.</p>

Diffusion is a phenomenon in which particles move from regions of higher density to regions of lower density. Many physical systems, in fields as diverse as plant biology and finance, are known to involve diffusion phenomena. Typically, diffusion systems are modeled by partial differential equations (PDEs), which include certain parameters. These parameters characterize a given diffusion system. Therefore, for both modeling and simulation of a diffusion system, one has to either know or determine these parameters. Moreover, as PDEs are infinite order dynamic systems, for computational purposes one has to approximate them by a finite order model. In this thesis, we investigate these two issues of model reduction and parameter estimation by considering certain specific cases of heat diffusion systems. We first address model reduction by considering two specific cases of heat diffusion systems. The first case is a one-dimensional heat diffusion across a homogeneous wall, and the second case is a two-dimensional heat diffusion across a homogeneous rectangular plate. In the one-dimensional case we construct finite order approximations by using some well known PDE solvers and evaluate their effectiveness in approximating the true system. We also construct certain other alternative approximations for the one-dimensional diffusion system by exploiting the different modal structures inherently present in it. For the two-dimensional heat diffusion system, we construct finite order approximations first using the standard finite difference approximation (FD) scheme, and then refine the FD approximation by using its asymptotic limit. As for parameter estimation, we consider the same one-dimensional heat diffusion system, as in model reduction. We estimate the parameters involved, first using the standard batch estimation technique. The convergence of the estimates are investigated both numerically and theoretically. We also estimate the parameters of the one-dimensional heat diffusion system recursively, initially by adopting the standard recursive prediction error method (RPEM), and later by using two different recursive algorithms devised in the frequency domain. The convergence of the frequency domain recursive estimates is also investigated.

In the thesis we investigate two problems on Partial Differential Equations (PDEs) in differential geometry and fluid mechanics. First, we prove the weak L^p continuity of the Gauss-Codazzi-Ricci (GCR) equations, which serve as a compatibility condition for the isometric immersions of Riemannian and semi-Riemannian manifolds. Our arguments, based on the generalised compensated compactness theorems established via functional and micro-local analytic methods, are intrinsic and global. Second, we prove the vanishing viscosity limit of an incompressible fluid in three-dimensional smooth, curved domains, with the kinematic and Navier boundary conditions. It is shown that the strong solution of the Navier-Stokes equation in H^r+1 (r &GT; 5/2) converges to the strong solution of the Euler equation with the kinematic boundary condition in H^r, as the viscosity tends to zero. For the proof, we derive energy estimates using the special geometric structure of the Navier boundary conditions; in particular, the second fundamental form of the fluid boundary and the vorticity thereon play a crucial role. In these projects we emphasise the linkages between the techniques in differential geometry and mathematical hydrodynamics.