Elsevier

Computer-Aided Design

Volume 38, Issue 4, April 2006, Pages 392-403
Computer-Aided Design

G1 surface modelling using fourth order geometric flows

https://doi.org/10.1016/j.cad.2005.11.002Get rights and content

Abstract

We use three fourth order geometric partial differential equations to efficiently solve several surface modelling problems, including surface blending, N-sided hole filling and free-form surface fitting with the G1 boundary continuity. The non-linear equations we use include the surface diffusion flow, the quasi surface diffusion flow and the Willmore flow. These non-linear equations are discretized using a mixed finite element method based on a combination of the Loop's basis and the linear basis. The proposed approach is simple, efficient and gives very desirable results.

Introduction

We use three geometric partial differential equations (GPDEs) to solve several surface modelling problems. The GPDEs we use include the surface diffusion flow, the quasi surface diffusion flow and the Willmore flow. These equations are non-linear and geometrically intrinsic, i.e. they do not depend upon any particular parametrization. The problems we solve include surface blending, N-sided hole filling and free-form surface fitting with the G1 boundary continuity.

For the problems of surface blending and N-sided hole filling, we are given triangular surface meshes of the surrounding area. Triangular surface patches need to be constructed to fill the openings enclosed by the surrounding surface mesh and interpolate its boundary with the G1 continuity (see Fig. 1, Fig. 7). For the free-form surface fitting problem, we are given a set of points. We construct a surface which interpolates the points with the specified G1 continuity. The free-form surface fitting problem is the most general, including surface blending and N-sided hole filling as its special cases.

There are basically two classes of approaches for solving a GPDE on any domain. One class is based on the generalized finite divided differences, the other is based on the finite element method (see [3], [13], [18], [25]). The approach we adopt in this paper is based on a mixed finite element method. The generalized finite divided difference method has been used in [47]. It is well known that the finite divided difference method is simpler and easier to implement, but lacks the convergence analysis. The finite element method is not as simple as the finite divided difference method, but is based on a well developed mathematical foundation. In this paper, we use a mixed finite element method consisting of using the Loop's subdivision basis for representing the surfaces and the linear basis for representing the curvatures. As the Loop's subdivision surface is C2 (except for the extraordinary vertices whose valences are not six), its curvature is continuous, we use the C0 linear element to approximate the curvature.

Our two-step strategy for solving these surface modelling problems is as follows: firstly we construct an initial triangular surface mesh (‘filler’) using any automatic, semi-automatic free-form modelling techniques (see [1], [23], [32]) or more advanced techniques based on the level set method (see [12], [40]). One may also interactively edit this ‘filler’ to meet some weak assumptions for an initial solution shape. This ‘filler’ may be bumpy or noisy, and it does not have to satisfy the smoothness boundary conditions, though it may roughly characterize the shape of the surface to be constructed. Secondly, we deform the initial mesh by solving a suitable geometric flow. Unlike most of the previous free-form modelling techniques, our approach treats the smooth boundary continuity constraints without any prior estimation of normals or derivative jets along the boundary. In this paper, we devote our attention to solving GPDEs with the G1 boundary continuity constraints, rather than the construction of the initial filler.

Earlier research on using PDEs to handle the surface modelling problems can be traced back to Bloor et al.'s work at the end of the 1980s [6], [7]. They use the biharmonic equation on a rectangular domain to solve the blending and the hole filling problems. The biharmonic equation on a rectangular domain is linear, so can be solved easily. Recently, this equation and its generalizations have been frequently used in interactive surface design (see [17], [39]) and interactive sculpting (see [16]). Lowe et al. present a PDE method in [27] with certain functional constraints, such as geometric constraints, physical criteria and engineering requirements, which can be incorporated into the shape design process. However, the biharmonic equation is not geometrically intrinsic and its solution (the geometry of the surface) depends on the concrete parametrization. Furthermore, these methods are inappropriate for modelling surfaces with arbitrary shaped boundaries. In contrast, the equations used in this paper are geometrically intrinsic.

The mean curvature flow (MCF) and its variations have been intensively used to smooth or fair noisy surfaces (see [3], [11], [14], [29] for references). MCF has been shown to be the most important and efficient flow for fairing or denoising surface meshes. However, for the surface modelling anti design problems, MCF cannot achieve the G1 smooth joining of the different patches. Recently, fourth order geometric flows have been used to solve the surface blending problem and the free-form surface fitting problem (see [10], [35], [36], [47]). In [35], fair meshes with the G1 conditions are created in the special case where the meshes are assumed to have the subdivision connectivity, using the surface diffusion flow. In this work, the local surface parametrization is used to estimate the surface curvatures. The same equation is used in [36] for smoothing meshes while satisfying the G1 boundary conditions. The outer fairness (the smoothness in the classical sense) and the inner fairness (the regularity of the vertex distribution) criteria are used in their fairing process. The finite element method is used by Clarenz et al. [10] to solve the Willmore flow equation, based on a new variational formulation of the flow, for the surface restoration.

Level set methods have also been used in surface fairing and surface reconstruction (see [2], [5], [9], [31], [43], [48]). In these methods, surfaces are formulated as level surfaces of 3D functions, which are usually defined from the signed distance over Cartesian grids of a volume. An evolution PDE on the volume governs the behavior of the level surface. To reduce the computation complexity, Bertalmio et al. [5] solve the PDE in a narrow band for deforming vectorial functions on surfaces. The level set method is also used in hole filling of surfaces. The pioneering work by Davis et al. [12] addresses the problem of filling holes with complicated topology via the volumetric data diffusion. Inspired by this work, Verdera et al. [40] propose an alternative by introducing a system of geometric PDEs derived from image inpainting algorithms.

Another category of surface fairing research is based on optimization techniques. In this category, one constructs an optimization problem that minimizes certain objective functions [20], [22], [30], [34], [41], such as the thin plate energy, the membrane energy [24], the total curvature [25], [42], or the sum of distances [28]. Using local interpolation, fitting or replacing differential operators with divided difference operators, the optimization problems are discretized to arrive at finite dimensional linear or non-linear systems. Approximate solutions are then obtained by solving the constructed systems. In general, such an approach is quite computationally intensive.

We use the surface diffusion flow, the quasi surface diffusion flow and the Willmore flow for solving each of the surface modelling problems. We propose a novel stratagem for the treatment of the boundary conditions, a beautiful combination of the Loop's subdivision space and the linear element space. The proposed approach is simple and easy to implement. It is general, i.e. solves several surface modelling problems in the same manner, and gives very desirable results for a range of the complicated free-form surface models. Furthermore, it avoids the estimation of normals, tangents or curvatures on the boundaries.

The rest of the paper is organized as follows: Section 2 describes the non-linear GPDEs used in this paper. Section 3 gives the variational forms of GPDEs. In Section 4, we give the details of the discretization of the variational forms in both the spatial and the temporal directions. Comparative examples to illustrate the different effects achievable from the solutions of these GPDEs are given in Section 5. Some mathematical details are provided in appendices.

Section snippets

Partial differential equation models

Let us consider the non-linear GPDE models used in this paper. Several involved differential geometry operators, such as the Laplace–Beltrami operator, gradient and mean curvature, are presented in Appendix A.

Variational form

In Eqs. (2.1), (2.2), (2.3), the surface point p is an unknown to be determined. Solving these fourth order equations for the unknown p using the finite element method is not straightforward. Instead, we formulate each of the fourth order equations into a coupled system of two second order equations by treating the mean curvature H(p) as an unknown variable. As a result, (2.1), (2.2) are reformulated as{pt=ΔM[H(p)]n(p),H(p)=12n(p)TΔMp,and{pt=[ΔMH(p)+2H(p)(H(p)2K(p))]n(p),H(p)=12n(p)TΔ

Solution of GPDEs

Consider a triangular surface mesh Ω with vertices p1,p2,…,pn. For each vertex pi, we associate it with a C2 smooth basis function ϕi. Then the surface M is represented asp=j=1nϕj(q)pjR3,qΩ,where p1,p2,…,pn are regarded as the control vertices of M. Now we classify these control vertices into several categories. The first category consists of interior vertices, denoted as p1,p2,,pn0. The positions of these vertices are to be determined (unknown). For the problems of hole filling and

Examples and remarks

In this section, we give several examples to illustrate the different effects of QSDF, SDF and WF and how these GPDEs are used to solve the different modelling problems in a uniform fashion.

Acknowledgements

The work was supported by in part by NSFC grant 10371130, National Key Basic Research Project of China (2004CB318000).

Guoliang Xu is a Professor of Academy of Mathematics and System Sciences, Chinese Academy of Science. His academic interests include computational geometry, computer aided geometric design, computational mathematics and computer graphics. He has been involved in several government funded projects that have investigated these areas of research and currently concentrating on using geometric partial different equations in surface design and modeling.

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    Guoliang Xu is a Professor of Academy of Mathematics and System Sciences, Chinese Academy of Science. His academic interests include computational geometry, computer aided geometric design, computational mathematics and computer graphics. He has been involved in several government funded projects that have investigated these areas of research and currently concentrating on using geometric partial different equations in surface design and modeling.

    Pan Qing is a Ph.D. student under Guoliang Xu at Academy of Mathematics and System Sciences. Her research interests are in the areas of computer aided geometric design and computational mathematics.

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