A generalized surface subdivision scheme of arbitrary order with a tension parameter

Convergence and Normal Continuity Analysis of Nonstationary Subdivision Schemes Near Extraordinary Vertices and Faces

“…In fact, if v k < 1 the scheme yields trigonometric splines, if v k = 1 polynomial splines and if v k > 1 hyperbolic splines. In [22], the authors prove that the limit surface obtained by applying the generalized spline schemes of order d to a regular mesh is C d−2 -continuous, while in the neighborhood of extraordinary elements the C 1 -continuity of the limit surface is shown only by numerical evidence. Here we use Theorem 4.1 and Theorem 4.2 to prove convergence and normal continuity of the limit surfaces.…”

“…The difficulties concerning the analysis of a level-dependent subdivision scheme in the neighborhood of an extraordinary vertex/face, are due to the fact that the well-established approach based on the spectral analysis of the subdivision matrix and on the study of the characteristic map is not applicable. Thus, we use and generalize the notion of asymptotical equivalence between stationary and non-stationary subdivision schemes (known only for schemes defined on regular meshes), and show that normal continuity of a non-stationary scheme in the vicinity of an extraordinary element can be obtained by assuming that the matrix sequence identifying it converges towards the matrix S (identifying a C 1 -regular, standard, stationary scheme) faster than λ k 1 , where λ 1 denotes the real, double subdominant eigenvalue of S. The sufficient conditions we propose are used for the analysis of the family of approximating non-stationary subdivision schemes presented in [22]. The members of the latter family are a generalization of exponential spline surfaces to quadrilateral meshes of arbitrary topology whose normal continuity is conjectured and shown only by numerical evidence in [22,Section 5].…”

“…Thus, we use and generalize the notion of asymptotical equivalence between stationary and non-stationary subdivision schemes (known only for schemes defined on regular meshes), and show that normal continuity of a non-stationary scheme in the vicinity of an extraordinary element can be obtained by assuming that the matrix sequence identifying it converges towards the matrix S (identifying a C 1 -regular, standard, stationary scheme) faster than λ k 1 , where λ 1 denotes the real, double subdominant eigenvalue of S. The sufficient conditions we propose are used for the analysis of the family of approximating non-stationary subdivision schemes presented in [22]. The members of the latter family are a generalization of exponential spline surfaces to quadrilateral meshes of arbitrary topology whose normal continuity is conjectured and shown only by numerical evidence in [22,Section 5]. Due to the lack of existing theoretical results for the analysis of level-dependent subdivision schemes, we believe that our contribution could mark a first step forward towards a deeper understanding of non-stationary subdivision with a consequent increase of its use in different fields of application.…”

Construction and analysis of unified 4-point interpolating nonstationary subdivision surfaces

“…Reif [19] has established a generalized technique to the SSs near extraordinary vertices. Fang et al [20] introduced the unified stationary SS of arbitrary order with image controlling variable, but it does not hold up the surfaces like sphere and hyperboloid. Recently, Ghaffar et al [21] have introduced tensor products of nine-tic B-spline.…”

Construction and Analysis of Binary Subdivision Schemes for Curves and Surfaces Originated from Chaikin Points

“…The detailed information about refinement rules and Laurent polynomial can be found in [1][2][3]. Surface modeling via subdivision is very important topic in computer graphics and computer aided geometric design and it has been studied by several authors; see surveys [4][5][6][7][8][9][10] and references therein.…”