## A method for analysis of C 1 -continuity of subdivision surfaces (1998)

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Venue: | SIAM J. Numer. Anal |

Citations: | 24 - 5 self |

### BibTeX

@ARTICLE{Zorin98amethod,

author = {Denis Zorin},

title = {A method for analysis of C 1 -continuity of subdivision surfaces},

journal = {SIAM J. Numer. Anal},

year = {1998},

volume = {37},

pages = {1677--1708}

}

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### Abstract

Abstract. A sufficient condition for C 1-continuity of subdivision surfaces was proposed by Reif [Comput. Aided Geom. Design, 12(1995), pp. 153–174.] and extended to a more general setting in [D. Zorin, Constr. Approx., accepted for publication]. In both cases, the analysis of C 1-continuity is reduced to establishing injectivity and regularity of a characteristic map. In all known proofs of C 1-continuity, explicit representation of the limit surface on an annular region was used to establish regularity, and a variety of relatively complex techniques were used to establish injectivity. We propose a new approach to this problem: we show that for a general class of subdivision schemes, regularity can be inferred from the properties of a sufficiently close linear approximation, and injectivity can be verified by computing the index of a curve. An additional advantage of our approach is that it allows us to prove C 1-continuity for all valences of vertices, rather than for an arbitrarily large but finite number of valences. As an application, we use our method to analyze C 1-continuity of most stationary subdivision schemes known to us, including interpolating butterfly and modified butterfly schemes, as well as the Kobbelt’s interpolating scheme for quadrilateral meshes.

### Citations

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Methods and Applications of Interval Analysis
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- 1979
(Show Context)
Citation Context ...n conjecture for dimension 2, and a counterexample was found by S. Pinchuk in [17].sANALYSIS OF C 1 -CONTINUITY OF SUBDIVISION 1679 Finally, we extensively use interval arithmetics (see, for example, =-=[14]-=-). Overview. In section 2, we describe the notation for subdivision on complexes and state relevant results from [22] and [20]. In section 3 we present the results forming the theoretical foundation o... |

343 | A Butterfly Subdivision Scheme for Surface Interpolation with Tension Control
- DYN, LEVIN, et al.
- 1990
(Show Context)
Citation Context ..., but about families of maps, corresponding to the control points with interval components. Using our method, we analyze interpolating triangular and quadrilateral subdivision schemes — the butterfly =-=[6]-=-, the modified butterfly [24], and the scheme described by Kobbelt [12]. (A similar scheme was proposed earlier by Leber [13].) We also repeat the analysis for two schemes that were analyzed previousl... |

234 |
Analysis of the Behaviour of Recursive Division Surfaces near Extraordinary Points
- DOO, SABIN
- 1978
(Show Context)
Citation Context ...h cases. Our techniques can be also applied in virtually unchanged form to the dual, or corner cutting, subdivision schemes. Two schemes of this type are known to us: the Doo–Sabin subdivision scheme =-=[4]-=- and the Midedge subdivision scheme [9, 16]. For these schemes, C 1 -continuity was already proved for all valences [15, 9, 16]. Using our method, it is possible to perform perturbation analysis of th... |

206 | Interpolating Subdivision for Meshes with Arbitrary Topology
- ZORIN, SCHRÖDER, et al.
- 1996
(Show Context)
Citation Context ..., corresponding to the control points with interval components. Using our method, we analyze interpolating triangular and quadrilateral subdivision schemes — the butterfly [6], the modified butterfly =-=[24]-=-, and the scheme described by Kobbelt [12]. (A similar scheme was proposed earlier by Leber [13].) We also repeat the analysis for two schemes that were analyzed previously by other authors: the Loop ... |

165 |
A four-point interpolatory subdivision scheme for curve design
- Dyn, Gregory, et al.
- 1987
(Show Context)
Citation Context ...Analysis of Kobbelt’s scheme. Kobbelt’s subdivision scheme [12], is an interpolatory scheme defined on quad meshes; in the regular case, the scheme reduces to the tensor product of four-point schemes =-=[5]-=-. There are two challenges in the analysis of this scheme: First, as for the butterfly scheme, the limit surface cannot be expressed in explicit form. In addition, the eigenvalues of the subdivision m... |

164 |
A unified approach to subdivision algorithms near extraordinary vertices
- Reif
- 1995
(Show Context)
Citation Context ... that make it possible to perform the C 1 -continuity tests automatically. The principal result allowing one to analyze C 1 -continuity of most subdivision schemes is the sufficient condition of Reif =-=[18]-=-. This condition reduces the analysis of stationary subdivision to the analysis of a single map, called the characteristic map, for each valence of vertices in the mesh. The analysis of C 1 -continuit... |

139 | Interpolatory subdivision on open quadrilateral nets with arbitrary topology
- KOBBELT
- 1996
(Show Context)
Citation Context ... interval components. Using our method, we analyze interpolating triangular and quadrilateral subdivision schemes — the butterfly [6], the modified butterfly [24], and the scheme described by Kobbelt =-=[12]-=-. (A similar scheme was proposed earlier by Leber [13].) We also repeat the analysis for two schemes that were analyzed previously by other authors: the Loop [19] scheme and the Catmull–Clark scheme [... |

80 | The simplest subdivision scheme for smoothing polyhedra
- PETERS, REIF
- 1997
(Show Context)
Citation Context ...lied in virtually unchanged form to the dual, or corner cutting, subdivision schemes. Two schemes of this type are known to us: the Doo–Sabin subdivision scheme [4] and the Midedge subdivision scheme =-=[9, 16]-=-. For these schemes, C 1 -continuity was already proved for all valences [15, 9, 16]. Using our method, it is possible to perform perturbation analysis of the type that we have described above. We wil... |

64 | Analysis and Application of Subdivision Surface
- Schweitzer
- 1996
(Show Context)
Citation Context ...theoretical analysis of the surface has to be performed. For subdivision on arbitrary meshes, even the analysis of the basic property of the surfaces, C 1 -continuity, poses a considerable challenge; =-=[19, 9, 15, 16, 20]-=-. In this paper we describe a set of theoretical results and algorithms that make it possible to perform the C 1 -continuity tests automatically. The principal result allowing one to analyze C 1 -cont... |

63 |
Conditions for tangent plane continuity over recursively generated B-spline surfaces
- Ball, Storry
- 1988
(Show Context)
Citation Context ...rix subdivision. Initial discrete fourier transform (DFT) analysis that we use to find the control points for the characteristic map for invariant schemes follows the well-established pattern used in =-=[1, 9, 19, 24, 15]-=-. 1 In general, it is not true that a map is injective if it is regular on its domain even if the domain is the plane and the map is polynomial. This statement is known as the Jacobian conjecture for ... |

58 |
Stationary subdivision and multiresolution surface representation
- Zorin
- 1998
(Show Context)
Citation Context ... 1679 Finally, we extensively use interval arithmetics (see, for example, [14]). Overview. In section 2, we describe the notation for subdivision on complexes and state relevant results from [22] and =-=[20]-=-. In section 3 we present the results forming the theoretical foundation of our method. In section 3.1, we discuss the basic properties of the matrix subdivision schemes, and we derive estimates for t... |

48 |
Theory and Application of the z-Transform Method
- Jury
- 1964
(Show Context)
Citation Context ...mial. The filter is stable, if all roots of the polynomial a(z) are inside the unit circle. A variety of tests exist for this condition; for our purposes, the algebraic Marden–Jury test is convenient =-=[11]-=-. With appropriate rescaling of the variable it can be used to prove that all roots of a polynomial are inside the circle of any given radius r. As the test requires only a simple algebraic calculatio... |

43 | Analysis of algorithms generalizing B-spline subdivision
- Peters
- 1998
(Show Context)
Citation Context ...theoretical analysis of the surface has to be performed. For subdivision on arbitrary meshes, even the analysis of the basic property of the surfaces, C 1 -continuity, poses a considerable challenge; =-=[19, 9, 15, 16, 20]-=-. In this paper we describe a set of theoretical results and algorithms that make it possible to perform the C 1 -continuity tests automatically. The principal result allowing one to analyze C 1 -cont... |

30 |
Using parameters to increase smoothness of curves and surfaces generated by subdivision, Computer Aided Geometric Design 7
- Dyn, Levin, et al.
- 1990
(Show Context)
Citation Context ...sign, we can conclude that the map is regular. Our derivations are similar to the derivations in Chapters 2 and 3 of Cavaretta, Dahmen, and Micchelli [2], and those found in Dyn, Levin, and Micchelli =-=[7]-=-. We have to consider convergence not only of the scheme, but also of the corresponding scheme for differences, which, in general, is a matrix subdivision scheme. For this reason, some of the theorems... |

17 | Subdivision schemes
- Dyn
- 1992
(Show Context)
Citation Context ...roach: it is simpler to apply and more general. Our estimates of the errors of linear approximations rely on the work of Cavaretta, Dahmen, and Micchelli [2], and on the work of Cohen, Dyn, and Levin =-=[3]-=- on matrix subdivision. Initial discrete fourier transform (DFT) analysis that we use to find the control points for the characteristic map for invariant schemes follows the well-established pattern u... |

15 |
A counterexample to the strong real Jacobian conjecture
- Pinchuk
- 1994
(Show Context)
Citation Context ...is regular on its domain even if the domain is the plane and the map is polynomial. This statement is known as the Jacobian conjecture for dimension 2, and a counterexample was found by S. Pinchuk in =-=[17]-=-.sANALYSIS OF C 1 -CONTINUITY OF SUBDIVISION 1679 Finally, we extensively use interval arithmetics (see, for example, [14]). Overview. In section 2, we describe the notation for subdivision on complex... |

12 |
Edge and vertex insertion for a class of subdivision surfaces
- Habib, Warren
- 1996
(Show Context)
Citation Context ...theoretical analysis of the surface has to be performed. For subdivision on arbitrary meshes, even the analysis of the basic property of the surfaces, C 1 -continuity, poses a considerable challenge; =-=[19, 9, 15, 16, 20]-=-. In this paper we describe a set of theoretical results and algorithms that make it possible to perform the C 1 -continuity tests automatically. The principal result allowing one to analyze C 1 -cont... |

9 |
Smoothness of subdivision on irregular meshes, Constr. Approx
- Zorin
- 2004
(Show Context)
Citation Context ...BDIVISION 1679 Finally, we extensively use interval arithmetics (see, for example, [14]). Overview. In section 2, we describe the notation for subdivision on complexes and state relevant results from =-=[22]-=- and [20]. In section 3 we present the results forming the theoretical foundation of our method. In section 3.1, we discuss the basic properties of the matrix subdivision schemes, and we derive estima... |

7 |
Piecewise smooth surface reconsruction
- Hoppe, DeRose, et al.
- 1994
(Show Context)
Citation Context ...2] ... [k − 1,N]. With this ordering of vertices, the subdivision matrix 3 An example of a scheme which is not invariant with respect to some isomorphism is the piecewisesmooth scheme of Hoppe et al. =-=[10]-=-shas the form (2.5) ANALYSIS OF C 1 -CONTINUITY OF SUBDIVISION 1685 ⎛ a00 b ⎜ S = ⎜ ⎝ T 0 ··· bT N−1 . . . . . . .. . c0 A00 ··· A0 N−1 cN−1 AN−10 ··· AN−1 N−1 where Aj j ′ are k × k matrices with ent... |

1 |
A geometric approarturbation theory of matrices and matrix pencils
- Edelman, Elmroth, et al.
- 1997
(Show Context)
Citation Context ...ible to use a combination of symbolic and numerical methods to obtain all necessary information. We believe that a satisfactory solution of this problem requires methods similar to those developed in =-=[8]-=-. While it might not be possible to determine the Jordan normal form exactly, one can find all possible Jordan normal forms of matrices that are obtained from the original subdivision matrix by a smal... |

1 |
Interpolierende subdivisionsalgorithmen
- Leber
- 1994
(Show Context)
Citation Context ...erpolating triangular and quadrilateral subdivision schemes — the butterfly [6], the modified butterfly [24], and the scheme described by Kobbelt [12]. (A similar scheme was proposed earlier by Leber =-=[13]-=-.) We also repeat the analysis for two schemes that were analyzed previously by other authors: the Loop [19] scheme and the Catmull–Clark scheme [15]. For the latter schemes we extend the analysis to ... |

1 |
Subdivision Surfaces with Boundary, in preparation
- Zorin, Duchamp
(Show Context)
Citation Context ... closed meshes: in fact, we have successfully used them to establish C 1 -continuity on the boundary of several variations of common subdivision schemes. These results are discussed in a future paper =-=[23]-=-. One important, although typically not the most difficult, aspect of the problem is not addressed by our method. As it could be seen from our analysis of the butterfly and Kobbelt subdivision schemes... |