Research On Implicit Surface Interactive Modelling And Polygonal Mesh Processing

Curves and surfaces modeling is one of most important research fields in computer graphics, computer aided geometric design and computer animation. As curves and surfaces modeling technology developing, its application range becomes wider, such as computer aided design system, medical images three dimensions modeling and reverse engineer, etc. Recent years, as development of computer modeling technology, implicit surfaces are used more and more widely. However, for shape controlling and rendering difficulty, application value of implicit surfaces is limited at certain level.Polygon meshes are usually used as the expression in 3D modeling technology. Data of polygon mesh (the coordinates of the discrete points on 3D objects) can be obtained by the laser fixed measuring equipments. However, original geometric models always have sevel parts of points, and with high noises. Data pre-processing includes point clouds registration, data smoothing and simplification, etc. Implicit surfaces have three obvious advantages: the first is simple expression forms, the second is easy to judge the relative position to a given point, and the third is its high smoothness. Implicit surfaces can change the topology of surfaces easily, so they are very useful for describing continuous and smoothly blend surfaces with complex topology. Based on above problems, we mainly research in implicit surface interactive modeling, and construct polygon mesh expression for implicit surfaces. Furthermore, new mesh processing algorithms such as simplification and fairing for the constructed meshes are also presented. In brief, we made intensive study of implicit surfaces interactive modeling, implicit surfaces polygonization, triangular mesh simplification and triangular mesh fairing, which can be achieved following four main research contributions:(1) Point to control implicit surfaces difficult, we propose a new interactive method for controlling implicit surfaces via optimization. New method can achieve control implicit surfaces in real-time.Two interactive controlling tools are introduced: position controlling handle and normal controlling handle. New implicit surfaces are required to interpolate the adjusted positions and normal vectors at desired vertices on surfaces, which are appointed by user interaction. This problem can be treated as an optimization one. Newton and SQP (Sequential Quadratic Programming) methods are applied to solve it respectively. Particle system is used for real-time rendering of implicit surfaces. This method provides a real-time and interactive method for controlling the shape of implicit surfaces.(2) For implicit surfaces representation and display issues, we proposed two new polygonization methods, which are both based on particle system. Normal and curvature information are used for constructing isomorphic polygonal meshes with implicit surfaces.Method I: Particle system with normal constraint is used for sampling on the given implicit surface to get steady and reasonable sample particles. Two normal added vertices are obtained by extending the particle to an equal distance along the normal vector and the anti-normal vector. Dividing the normal added vertices into tetrahedrons, and then traversing all of the tetrahedrons to find the triangles on implicit surfaces and achieve the final triangular mesh.Method II: Particle system is used for sampling an implicit surface whose fission and death are guided by Gaussian curvature. This strategy leads to curvature adaptive samplings, so there are more small triangles in the high curvature region of constructed mesh. Reconstructed triangular mesh can approximate the implicit surface better. More sample points are obtained by extending a proper distance along each normal vector. These new sample points are used as a sampling on surface of the shell space. Delaunay tetrahedron of these sample points fills in the shell space. Finally, triangles around zero set form a triangulation of the implicit surface. Comparing existing methods, our method is more robust, and can achieve high quality model without post processing.(3) We proposed a new simplification method for triangular mesh obtained by the implicit surfaces based on volume square measure. New method can make as a farther processing for obtained mesh.Triangular meshes can be simplified by minimizing error objective function. Square volume error, shape factor and normal constraint factor of triangles are combined together to define the simplification error, which can be described as a quadratic objective function. New vertices replaced collapced triangles can be determined linearly. Comparing with the QEM method which is the most effective method so far, computation complexity will not be increased. Gaussian curvature factor is computed for each collapced triangle and used to identify strong feature triangle. For non-strong feature triangle, new vertex position is determined by minimizing the error objective function. For strong feature triangle, new vertex is taken the strong feature vertex of the three triangle vertices for preserving the model feature. Each collapced triangle is processing as inner triangle or boundary triangle to keep the boundary feature.(4) A new method for fairing arbitrary triangular mesh is presented. This method can remove noising point on mesh and optimize mesh can be obtained.Mean curvature normal is used to define a quasi-laplacian operator for smoothing inner vertices at a local region. Vertices are moved along the normal direction in a more appropriate velocity which can make mesh smoothing and shape preserving harmonizing well. For the boundary vertices, a new method for estimating the mean curvature normal is presented, so that for an arbitrary triangular mesh, the inner and the boundary vertices can be smoothed by the same smoothing process. Features of the original mesh can be preserved by the weighted mean curvature normal restriction of the neighbors of one vertex effectively.