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 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.

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.