Parametrization of General Catmull-Clark Subdivision Surfaces and its Applications

Abstract

A 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 previous approach, the new parametrization is also more efficient for evaluation purpose. This is demonstrated by several applications of the new techniques.Keywords: subdivision, Catmull-Clark surfaces, parametrization. INTRODUCTIONSubdivision surfaces have become popular recently in graphical modeling and animation because of their capability in modeling/representing complex shape of arbitrary topology [6], their relatively high visual quality, and their stability and efficiency in numerical computation. Subdivision surfaces can model/represent complex shape of arbitrary topology because there is no limit on the shape and topology of the control mesh of a subdivision surface. With the parametrization technique for subdivision surfaces becoming available [14] and with the fact that non-uniform B-spline and NURBS surfaces are special cases of subdivision surfaces becoming known [12], we now know that subdivision surfaces cover both parametric forms and discrete forms. Parametric forms are good for design and representation, discrete forms are good for machining and tessellation. Hence, we have a representation scheme that is good for all graphics and CAD/CAM applications.Research work for subdivision surfaces has been done in several important areas, such as surface interpolation [8,[19][20][21][22], surface evaluation [4,[13][14][15][16], surface trimming [9], boolean operations [3], and mesh editing [17]. However, powerful evaluation and analysis techniques for subdivision surfaces have not been fully developed yet. Parametrization methods that have been developed so far are suitable for evaluation purpose only, not for analysis purpose, because these methods either do not have an explicit expression, or are too complicated for each part to be explicit. For instance, in [14], eigen functions are pre-computed numerically and stored in a file. So they can be used for evaluation purpose only. Note that exact evaluation at a point of a subdivision surface is possible only if there is an explicit parametrization of t...

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