Research On Surface Fitting And Deformation For Meshes

With the fast development of 3D scanning and related techniques, 3D models are now widely used as an emerging type of media in 3D games, computer animation, industrial model design, computer simulation, digital cultural heritage protection and so on. Techniques for processing 3D geometry data have also become a hot research topic in computer graphics, which is called Digital Geometry Processing (DGP). The research contents of DGP cover modeling, processing and application of 3D digital geometric models. No matter in industry or academic world, much attention has been paid on DGP, and techniques have been greatly developed. However, as the related research continues to develop and the applied range is being enlarged, there are still many challenges in DGP.In this thesis we focus on 3D representation based on surfaces. 3D surfaces can be classified into two categories: continuous form surfaces and discrete form surfaces. Continuous form surfaces include spline surfaces, implicit surfaces, subdivision surfaces and so on. And discrete form surfaces include mesh and point cloud. Based on mesh representation, this thesis presents several novel techniques for spline surface fitting and deformation for 3D models. The main contributions are as follows:(1) We propose a new method for constructing rational bi-cubic spline surfaces on irregular meshes.For spline surface fitting on arbitrary meshes, we extend B-spline method to irregular meshes through the decomposition and classification of uniform bi-cubic B-spline basis function. Given a quad mesh of control points, a basis function is constructed for each control point. Then the surface is defined by the weighted combination of all the control points using their associated basis functions. This surface is a piecewise bi-cubic rational parametric polynomial surface. It is an extension to uniform B-spline surfaces in the sense that its definition is an analogy of the B-spline surface, and it produces a uniform bi-cubic B-spline surface if the control mesh is a regular quad mesh. (2) We present to use the cross product of normals as the object function term and propose a new method to resize models with constraints.To reuse existing mesh models, resizing is often necessary to satisfy engineering requirements. We present a new resizing method in this thesis. The resizing is driven by scaling each edge of the mesh. And an objective function component, which is expressed as the cross product of normal vectors of triangles before and after scaling, is devised to optimize the resizing model. Its geometric meaning is to minimizing the variation of normal vector of every triangle. In the sense of Willmore energy, the cross product item can approximately minimize the variation of the Willmore energy before and after resizing. For some important featured regions of the model, whose shape should be accurately preserved, a constrained resizing method is also presented. Lagrange multiplier method is used to preserve the shape of these regions. However, existing resizing methods have not considered the accurate preserving of these important featured regions.(3) We present a feature sensitive deformation method.Features are crucial for accurate representing geometry, as well as understanding and analysis of models. Therefore, the feature details should be well-preserved during deformation. Based on feature sensitive metric, we pay more attention on the feature regions of the mesh models. Firstly, taking unit normal vectors into account, we derive a FS Laplacian operator, which is more sensitive to featured regions of mesh models than existing operators. Secondly, we use the 1-ring tetrahedron in the dual mesh as the basis deformation cell. To preserve the shape of the tetrahedron, we introduce linear tetrahedron constraints minimizing both the distortion of the base triangle and the change of the corresponding height. These ensure that geometric details are accurately preserved during deformation. The time complexity of our new method is similar to that of existing linear Laplacian methods.