A butterfly subdivision scheme for surface interpolation with tension control
ACM Transactions on Graphics (TOG)
A unified approach to subdivision algorithms near extraordinary vertices
Computer Aided Geometric Design
Interpolating Subdivision for meshes with arbitrary topology
SIGGRAPH '96 Proceedings of the 23rd annual conference on Computer graphics and interactive techniques
The simplest subdivision scheme for smoothing polyhedra
ACM Transactions on Graphics (TOG)
Analysis of Algorithms Generalizing B-Spline Subdivision
SIAM Journal on Numerical Analysis
Subdivision Methods for Geometric Design: A Constructive Approach
Subdivision Methods for Geometric Design: A Constructive Approach
A Method for Analysis of C1-Continuity of Subdivision Surfaces
SIAM Journal on Numerical Analysis
Spectral Analysis of the Transition Operator and Its Applications to Smoothness Analysis of Wavelets
SIAM Journal on Matrix Analysis and Applications
Stationary subdivision and multiresolution surface representations
Stationary subdivision and multiresolution surface representations
Matrix-valued subdivision schemes for generating surfaces with extraordinary vertices
Computer Aided Geometric Design
Refinable bivariate quartic and quintic C2-splines for quadrilateral subdivisions
Journal of Computational and Applied Mathematics
Matrix-valued 4-point spline and 3-point non-spline interpolatory curve subdivision schemes
Computer Aided Geometric Design
Integration of CAD and boundary element analysis through subdivision methods
Computers and Industrial Engineering
Interpolatory quad/triangle subdivision schemes for surface design
Computer Aided Geometric Design
1-ring interpolatory wavelet using function vectors for mobile computing
Proceedings of the 10th International Conference on Virtual Reality Continuum and Its Applications in Industry
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The minimum-supported bivariate C^2-cubic spline on a 6-directional mesh constructed in our previous work [Chui, C.K., Jiang, Q.T., 2003. Surface subdivision schemes generated by refinable bivariate spline function vectors. Appl. Comput. Harmonic Anal. 15, 147-162] can be used to extend Loop's approximation subdivision scheme to introduce some parameter for controlling surface geometric shapes. This extension is achieved by considering matrix-valued subdivisions, resulting in subdivision templates of the same 1-ring template size as Loop's scheme, but with 2-dimensional matrix-valued weights. Another feature accomplished by considering such an extension is that the two components of the refinable vector-valued spline function can be reformulated, by taking certain linear combinations, to convert the approximation scheme to an interpolatory scheme, but at the expense of an increase in template size for the edge vertices. To maintain the 1-ring template size with guarantee of C^2 smoothness for interpolatory surface subdivisions, a non-spline solution is needed, by applying some constructive scheme such as the procedure discussed in our recent work [Chui, C.K., Jiang, Q.T., 2005b. Matrix-valued symmetric templates for interpolatory surface subdivisions I. Regular vertices. Appl. Comput. Harmonic Anal. 19, 303-339]. The main objective of this paper is to develop the corresponding matrix-valued 1-ring templates for the extraordinary vertices of arbitrary valences, for all of the three schemes mentioned above: the extended Loop approximation scheme, its conversion to an interpolatory scheme, and the non-spline 1-ring interpolatory scheme. The discrete Fourier transform (DFT) is applied to analyze the spectral properties of the corresponding subdivision matrices, assuring that the eigenvalues of the subdivision matrices satisfy certain conditions for C^1 smoothness at the extraordinary vertices for all of the three considerations in this paper.