Shuhua Lai

Loop subdivision surface based progressive interpolation

A new method for constructing interpolating Loop subdivision surfaces is presented. The new method is an extension of the progressive interpolation technique for B-splines. Given a triangular mesh M, the idea is to iteratively upgrade the vertices of M to generate a new control mesh

Similarity based interpolation using Catmull–Clark subdivision surfaces

\overline{M}

Parametrization of General Catmull-Clark Subdivision Surfaces and its Applications

such that limit surface of

Progressive interpolation using loop subdivision surfaces

\overline{M}

Chapter 5: Smooth Surface Reconstruction Using Doo-Sabin Subdivision Surfaces

would interpolate M. It can be shown that the iterative process is convergent for Loop subdivision surfaces. Hence, the method is well-defined. The new method has the advantages of both a local method and a global method, i.e., it can handle meshes of any size and any topology while generating smooth interpolating subdivision surfaces that faithfully resemble the shape of the given meshes. The meshes considered here can be open or closed.

Locally adjustable interpolation for meshes of arbitrary topology

A new method for constructing a Catmull–Clark subdivision surface (CCSS) that interpolates the vertices of a given mesh with arbitrary topology is presented. The new method handles both open and closed meshes. Normals or derivatives specified at any vertices of the mesh (which can actually be anywhere) can also be interpolated. The construction process is based on the assumption that, in addition to interpolating the vertices of the given mesh, the interpolating surface is also similar to the limit surface of the given mesh. Therefore, construction of the interpolating surface can use information from the given mesh as well as its limit surface. This approach, called similarity based interpolation, gives us more control on the smoothness of the interpolating surface and, consequently, avoids the need of shape fairing in the construction of the interpolating surface. The computation of the interpolating surface’s control mesh follows a new approach, which does not require the resulting global linear system to be solvable. An approximate solution provided by any fast iterative linear system solver is sufficient. Nevertheless, interpolation of the given mesh is guaranteed. This is an important improvement over previous methods because with these features, the new method can handle meshes with large number of vertices efficiently. Although the new method is presented for CCSSs, the concept of similarity based interpolation can be used for other subdivision surfaces as well.

Near-Optimum adaptive tessellation of general catmull-clark subdivision surfaces

AbstractA new parametrization technique and its applications for general Catmull-Clark subdivision surfaces are presented. The new technique extends J. Stam’s work by redefining all the eigen basis functions in the parametric representation for general Catmull-Clark subdivision surfaces and giving each of them an explicit form. The entire eigenstructure of the subdivision matrix and its inverse are computed exactly and explicitly with no need to precompute anything. Therefore, the new representation can be used not only for evaluation purpose, but for analysis purpose as well. The new approach is based on an Ω-partition of the parameter space and a detoured subdivision path. This results in a block diagonal matrix with constant size diagonal blocks (7×7) for the corresponding subdivision process. Consequently, eigen decomposition of the matrix is always possible and is simpler and more efficient. Furthermore, since the number of eigen basis functions required in the new approach is only one half of the pr...

Voxelization of free-form solids represented by catmull-clark subdivision surfaces

A new method for constructing interpolating Loop subdivision surfaces is presented. The new method is an extension of the progressive interpolation technique for B-splines. Given a triangular mesh M, the idea is to iteratively upgrade the vertices of M to generate a new control mesh M such that limit surface of M interpolates M. It can be shown that the iterative process is convergent for Loop subdivision surfaces. Hence, the method is well-defined. The new method has the advantages of both a local method and a global method, i.e., it can handle meshes of any size and any topology while generating smooth interpolating subdivision surfaces that faithfully resemble the shape of the given meshes.

Mesh clustering by approximating centroidal Voronoi tessellation

A new technique for the reconstruction of a smooth surface from a set of 3D data points is presented. The reconstructed surface is represented by an everywhere C1-continuous subdivision surface which interpolates all the given data points. The new technique consists of two major steps. First, an efficient surface reconstruction method is applied to produce a polyhedral approximation to the given data set M. A Doo-Sabin subdivision surface that smoothly passes through all the points in the given data set M is then constructed. The Doo-Sabin subdivision surface is constructed by iteratively modifying the vertices of the polyhedral approximation until a new control mesh Mmacr, whose Doo-Sabin subdivision surface interpolates M, is reached. This iterative process converges for meshes of any size and any topology. Therefore the surface reconstruction processes well-defined. The new technique has the advantages of both a local method and a global method, and the surface reconstruction process can reproduce special features such as edges and corners faithfully.

Fast Spherical Mapping for Genus-0 Meshes

A new method for constructing a smooth surface that interpolates the vertices of an arbitrary mesh is presented. The mesh can be open or closed. Normals specified at vertices of the mesh can also be interpolated. The interpolating surface is obtained by locally adjusting the limit surface of the given mesh (viewed as the control mesh of a Catmull-Clark subdivision surface) so that the modified surface would interpolate all the vertices of the given mesh. The local adjustment process is achieved through locally blending the limit surface with a surface defined by non-uniform transformations of the limit surface. This local blending process can also be used to smooth out the shape of the interpolating surface. Hence, a surface fairing process is not needed in the new method. Because the interpolation process does not require solving a system of linear equations, the method can handle meshes with large number of vertices. Test results show that the new method leads to good interpolation results even for complicated data sets. The new method is demonstrated with the Catmull-Clark subdivision scheme. But with some minor modification, one should be albe to apply this method to other parametrizable subdivision schemes as well.