On extraordinary rules of quad-based interpolatory subdivision schemes
Introduction
Subdivision schemes are computationally efficient algorithms for representing smooth surfaces by applying a few steps of a refinement operator to a given polygonal mesh, roughly describing the desired limit shape. Each application of the refinement operator aims at splitting the edges and faces of the current mesh to obtain a finer and smoother version. Compared with parametric surface representations, subdivision schemes are no longer restricted to work with tensor-product meshes, but can advantageously operate on polygonal meshes of arbitrary manifold topology in order to generate surfaces of arbitrary topology. Polygonal meshes consisting entirely of quadrilateral faces are called quadrilateral meshes. Using the term valence to refer to the number of edges incident to a vertex, we have that, in case of quadrilateral meshes, all vertices of valence 4 are regular or ordinary, whereas vertices of valence other than 4 are extraordinary. In case of quadrilateral meshes, the refinement operator responsible for producing the finer mesh from the coarser one is specified by a set of topological and geometrical rules that can vary according to the properties that the limit surface is required to satisfy. More precisely, if the limit surface is required to pass through all the vertices of the given initial mesh, then the refinement operator relies upon topological rules that retain the vertices of the coarser mesh and insert new vertices in correspondence to the midpoint of its edges and the centroid of its faces. Such vertices are respectively called edge-points and face-points, and are hereinafter denoted by E and F (see Fig. 1, Fig. 2). A refined mesh is then obtained by constructing new edges and faces in the following way: first, we create all new edges of the refined mesh by connecting each face-point to the edge-points of the edges surrounding the face, and each mesh vertex to the edge-points of the edges incident on it; then, all new faces are simply obtained by the loop of four new edges. The role of the geometrical rules is instead to specify the positions of the new points. Edge-point and face-point rules consist in computing affine combinations of the vertices lying in the neighborhood of each edge or face of the coarser mesh. The choice of the weights to be used in the affine combination has been always considered a difficult problem. In fact, in order to maximize the global smoothness of the limit surface it is necessary to apply special edge- and face-point rules in the neighborhood of all extraordinary vertices. Such rules, besides being expected to be dependent on the valence of the extraordinary vertex, should involve the least possible number of control points in its vicinity in order to increase the locality of the scheme and consequently reduce the computational cost of each refinement step. The major challenge in designing subdivision schemes thus consists in finding a suitable trade-off between the locality of the subdivision rules and the visual quality of the resulting limit surface.
Focusing on the class of quad-based interpolatory subdivision schemes generalizing the tensor-product version of the Dubuc–Deslauriers 4-point scheme, we can find proposals featured by edge-point rules that either involve vertices from the coarser mesh or only a subset of of them. The existing schemes falling into the first group (see Kobbelt, 1996, Li and Ma, 2007, Li and Zheng, 2012), besides more computationally expensive, are -smooth with unbounded curvature at extraordinary points. As a matter of fact, dealing with refinement rules of larger size not only increases the computational costs for generating the limit surface, but remarkably complicates the tuning of the weights appearing in the affine combination such that bounded curvature at extraordinary points is hardly satisfied. In light of this, we believe strategic to restrict our attention to the subclass of interpolatory subdivision schemes for closed quadrilateral meshes that compute
- (i)
new edge-points near extraordinary vertices of valence N by means of an affine combination of vertices from the coarser mesh;
- (ii)
new face-points near extraordinary vertices of valence N by means of an affine combination of vertices from the coarser mesh.
The remainder of this article consists of six sections. In Section 2 we describe the edge-point and face-point rules characterizing the class of interpolatory subdivision schemes discussed in this work. For such schemes, in Section 3 we construct the local subdivision matrix providing a compact representation of a single refinement step in the vicinity of an extraordinary vertex of valence N, and in Section 4 we derive the explicit formulation of the associated characteristic polynomial and all its roots. In Section 5, we first recall some known results from the literature and then we derive which conditions have to be satisfied by the weights of the extraordinary rules to guarantee smoothness of the limit surface and boundedness of curvature at extraordinary points. Finally, in Section 6 we exploit the derived conditions to easily check these features in the limit surfaces obtained by the application of special extraordinary rules recently proposed in the literature. We also show that the obtained constraints can be used to design new extraordinary rules able to produce limit surfaces of the same quality as the existing proposals, but at a reduced computational cost. Conclusions are drawn in Section 7.
Section snippets
Edge-point and face-point rules
We consider an interpolatory subdivision scheme on quadrilateral meshes generalizing the tensor-product of the 4-point Dubuc–Deslauriers scheme (Deslauriers and Dubuc, 1989, Dubuc, 1986). This means that, when the mesh is regular, that is each vertex has valence , the rule for computing the edge-point is nothing but the 4-point scheme applied to the vertices (see Fig. 1), i.e. while the rule for computing the face-point is exactly the
The local subdivision matrix
By ordering the points counterclockwise along every ring of each sector proceeding outwards from the extraordinary vertex and labeling compatibly within the sectors, the subdivision rules in (2.1)–(2.2) and (2.4), (2.5), (2.6) allow one to construct a local subdivision matrix of the form where , and
The characteristic polynomial
Let , denote the eigenvalues of the matrix in (3.1). Furthermore, whenever μ is an eigenvalue of , we call ν the Fourier index of μ and we write .
The results in Section 3 allow us to write the complete spectrum of the local subdivision matrix as Now let denote the identity matrix. To work out the explicit expressions of the eigenvalues of each block we have to compute the roots of the characteristic polynomial
Constraints on the weights of edge- and face-point extraordinary rules
We start by providing a brief summary of known results concerning the analysis of bivariate subdivision schemes at extraordinary points, since they are needed to understand the content of the following subsections. To get more detailed explanations and see the related proofs we refer the reader to Peters and Reif (2008), Reif (1995), Zorin, 1997, Zorin, 2000.
Numerical examples: special weights settings
In this section, we consider interpolatory subdivision schemes from the literature which fall into the general class studied in this paper. Such schemes are featured by
- (i)
the regular rules in (2.1)–(2.2), obtained from the tensor-product of the Dubuc–Deslauriers interpolatory 4-point scheme (Deslauriers and Dubuc, 1989, Dubuc, 1986);
- (ii)
the extraordinary rules in (2.4), (2.5), (2.6) and (2.3), (2.4), (2.5) for the cases and , respectively.
Conclusions
In this work we studied which constraints are required to be respected by the weights of the extraordinary stencils shown in Fig. 2 to obtain a limit surface that is -continuous and with bounded curvature at the extraordinary points. The necessary conditions defined by analyzing the eigenvalues of the subdivision matrix are summarized in Table 3 for all extraordinary points of valence . Moreover, a standard procedure that checks sufficient conditions for the regularity of the
Acknowledgements
This work was partially supported by Italian GNCS-INdAM within the research project entitled “Mathematical and computational models for the representation of complicated geometries” and MIUR-PRIN 2012 (grant No. 2012MTE38N). The authors are grateful to Laura Valerio for cooperating in producing the pictures contained in this article and to the anonymous referees for their careful reading of the manuscript.
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