From extension of Loop's approximation scheme to interpolatory subdivisions

The minimum-supported bivariate C2-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 C2 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 C1 smoothness at the extraordinary vertices for all of the three considerations in this paper.