A butterfly subdivision scheme for surface interpolation with tension control
ACM Transactions on Graphics (TOG)
Two-scale difference equations I: existence and global regularity of solutions
SIAM Journal on Mathematical Analysis
Two-scale difference equations II. local regularity, infinite products of matrices and fractals
SIAM Journal on Mathematical Analysis
Characterizations of Scaling Functions: Continuous Solutions
SIAM Journal on Matrix Analysis and Applications
Accuracy of lattice translates of several multidimensional refinable functions
Journal of Approximation Theory
Smoothness of Multiple Refinable Functions and Multiple Wavelets
SIAM Journal on Matrix Analysis and Applications
Journal of Approximation Theory
Polynomial generation and quasi-interlpolation in stationary non-uniform subdivision
Computer Aided Geometric Design
Combining 4- and 3-direction subdivision
ACM Transactions on Graphics (TOG)
On C2 triangle/quad subdivision
ACM Transactions on Graphics (TOG)
Matrix-valued subdivision schemes for generating surfaces with extraordinary vertices
Computer Aided Geometric Design
From extension of Loop's approximation scheme to interpolatory subdivisions
Computer Aided Geometric Design
Mathematics and Computers in Simulation
A new interpolation subdivision scheme for triangle/quad mesh
Graphical Models
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Recently the study and construction of quad/triangle subdivision schemes have attracted attention. The quad/triangle subdivision starts with a control net consisting of both quads and triangles and produces finer and finer meshes with quads and triangles. The use of the quad/triangle structure for surface design is motivated by the fact that in CAD modelling, the designers often want to model certain regions with quad meshes and others with triangle meshes to get better visual quality of subdivision surfaces. Though the smoothness analysis tool for regular quad/triangle vertices has been established and C^1 and C^2 quad/triangle schemes (for regular vertices) have been constructed, there is no interpolatory quad/triangle schemes available in the literature. The problem for this is probably that since the template sizes of the local averaging rules of interpolatory schemes for either quad subdivision or triangle subdivision are big, an interpolatory quad/triangle scheme will have large sizes of local averaging rule templates. In this paper we consider matrix-valued interpolatory quad/triangle schemes. In this paper, first we show that both scalar-valued and matrix-valued quad/triangle subdivision scheme can be derived from a nonhomogeneous refinement equation. This observation enables us to treat polynomial reproduction of scalar-valued and matrix-valued quad/triangle schemes in a uniform way. Then, with the result on the polynomial reproduction of matrix-valued quad/triangle schemes provided in our accompanying paper, we obtain in this paper a smoothness estimate for matrix-valued quad/triangle schemes, which extends the smoothness analysis of Levin-Levin from the scalar-valued setting to the matrix-valued setting. Finally, with this smoothness estimate established in this paper, we construct C^1 matrix-valued interpolatory quad/triangle scheme (for regular vertices) with the same sizes of local averaging rule templates as those of Stam-Loop's quad/triangle scheme. We also obtain C^2 matrix-valued interpolatory quad/triangle scheme (for regular vertices) with reasonable sizes of local averaging rule templates.