In this paper, the concept of the basis matrix of B-splines is presented. A general matrix representation, which results in an explicitly recursive matrix formula, for nonuniform Bspline curves of an arbitrary degree is also presented by means of the Toeplitz matrix. New recursive matrix representations for uniform B-spline curves and Bézier curves of an arbitrary degree are obtained as special cases of that for nonuniformB-spline curves. The recursive formula for the basis matrix can be substituted for de Boor-Cox’s formula for B-splines, and it has a better time complexity than de Boor-Cox’s formula when used for computation and conversion of Bspline curves and surfaces between different CAD systems. Finally, some applications of the matrix representations are given in the paper.

In this paper we present a novel approach to construct B-spline wavelets under constraints, taking advantage of the lifting scheme. Constrained B-spline wavelets allowmultiresolution analysis of B-splines which fixes positions, tangents and/or high order derivatives at some user specified parameter values, thus extend the ability of B-spline wavelets: smoothing a curve while preserving user specified "feature points"; representing several segments of a single curve at different resolution levels, leaving no awkward "gaps"; multiresolution editing of B-spline curves under constraints. For a given B-spline order and the number of constraints, both the time and storage complexities of our algorithm are linear in the number of control points. This feature makes our algorithm extremely suitable for large scale datasets.

This paper introduces a new biorthogonal wavelet based on a variant of √2 subdivision by using the lifting scheme. The greatest advantage of this wavelet is its very slow gradual refinement for quadrilateral meshes, which offers the biggest number of resolution levels to control a quadrilateral mesh. Moreover, the resulting wavelet transforms have a linear computational complexity, as they are composed of local and in-place lifting operations only. Feature lines can also be effectively integrated into the wavelet transforms as self-governed boundary curves. The introduced wavelet analysis can be used in a variety of applications such as progressive transmission, data compression, shape approximation and multiresolution rendering. The experiments have shown sufficient stability as well as better performance of the introduced wavelet analysis, as compared to the existing wavelet analyses for quadrilateral meshes of arbitrary topology.

A new efficient biorthogonal wavelet analysis based on the √3 subdivision is proposed in the paper by using the lifting scheme. Since the √3 subdivision is of the slowest topological refinement among the traditional triangular subdivisions, the multiresolution analysis based on the √3 subdivision is more balanced than the existing wavelet analyses on triangular meshes and accordingly offers more levels of detail for processing polygonal models. In order to optimize the multiresolution analysis, the new wavelets, no matter whether they are interior or on boundaries, are orthogonalized with the local scaling functions based on a discrete inner product with subdivision masks. Because the wavelet analysis and synthesis algorithms are actually composed of a series of local lifting operations, they can be performed in linear time. The experiments demonstrate the efficiency and stability of the wavelet analysis for both closed and open triangular meshes with √3 subdivision connectivity. The √3-subdivision-based biorthogonal wavelets can be used in many applications such as progressive transmission, shape approximation, and multiresolution editing and rendering of 3D geometric models.

This paper presents an efficient biorthogonal wavelet construction with the generalized Catmull-Clark subdivision based on the lifting scheme. The subdivision wavelet construction scheme is applicable to all variants of Catmull-Clark subdivision, so it is more universal than the previous wavelet construction for the generalized bicubic B-spline subdivision. Because the analysis and synthesis algorithms of the wavelets are composed of a series of local and in-place lifting operations, they can be performed in linear time. The experiments have demonstrated the stability of the proposed wavelet analysis based on the ordinary Catmull-Clark subdivision. Moreover, the resulting Catmull-Clark subdivision wavelets have better fitting quality than the generalized bicubic B-spline subdivision wavelets at a similar computation cost.

In this paper, a new interpolatory subdivision scheme, called ternary interpolating subdivision, for quadrilateral meshes with arbitrary topology is presented. It can be used to deal with not only extraordinary faces but also extraordinary vertices in polyhedral meshes of arbitrary topologies. It is shown that the ternary interpolating subdivision can generate a C1-continuous interpolatory surface. Some applications with open boundaries and curves to be interpolated are also discussed.

A new identity is proved that represents the k th order Bsplines as linear combinations of the (k+1) th order B-splines. A new method for degree-raising of Bspline curves is presented based on the identity. The new method can be used for all kinds of Bspline curves, that is, both uniform and arbitrarily nonuniform B-spline curves. When used for degree-raising of a segment of a uniform B-spline curve of degree k-1, it can help obtain a segment of curve of degree k that is still a uniform Bspline curve without raising the multiplicity of any knot. The method for degree- raising of Bezier curves can be regarded as the special case of the new method presented. Moreover, the conventional theory for degree-raising, whose shortcoming has been found, is discussed.

We study the existence and computation of spherical rational quartic curves that interpolate Hermite data on a sphere, i.e. two distinct endpoints and tangent vectors at the two points. It is shown that spherical rational quartic curves interpolating such data always exist, and that the family of these curves has n degrees of freedom for any given Hermite data on Sn, n ≥2. A method is presented for generating all spherical rational quartic curves on Sn interpolating given Hermite data.

In this paper, an exponential bound of the distance between a Loop subdivision surface and its control mesh is derived based on the topological structure of the control mesh. The exponential bound is independent of the process of recursive subdivisions and can be evaluated without subdividing the control mesh actually. Using the exponential bound, we can predict the depth of recursion within a user-specified tolerance as well as the error bound after n steps of subdivision. The error-estimating approach can be used in many engineering applications such as surface/surface intersection, mesh generation, NC machining, surface rendering and the like.

Since Doo-Sabin and CatmulI-Clark surfaces were proposed in 1978, eigenstructure, con-vergense and continuity analyses of stationary subdivision have been performed very well, but it has been very difficult to prove the convergence and continuity of non-uniform recursive subdivision surfaces (NURSSes, for short) of arbitrary topology. In fact, so far a problem whether or not there exists the limit surface as well as G1 continuity of a non-uniform Catmull-Clark subdivision has not been solved yet. Here the concept of equivalent knot spacing is introduced. A new technique for eigenanalysis, convergence and continuity analyses of non-uniform CatmulI-Clark surfaces is proposed such that the convergence and G1 continuity of NURSSes at extraordinary points are proved. In addition, slightly improved rules for NURSSes are developed. This offers us one more alternative for modeling free-form surfaces of arbitrary topologies with geometric features such as cusps, sharp edges, creases and darts, while elsewhere maintaining the same order of continuity as B-spline surfaces.