A butterfly subdivision scheme for surface interpolation with tension control

Abstract

A new interpolatory subdivision zscheme for surface design is presented. The new scheme is designed for a general triangulation of control points and has a tension parameter that provides design flexibility.The resulting limit surface is C' for a specified range of the tension parameter, with a few exceptions. Application of the butterfly scheme and the role of the tension parameter are demonstrated by several examples.

Non-linear subdivision using local spherical coordinates

“…While performed on regular triangular meshes, it can be seen as a tensor product of the four-point scheme. It has been first presented in [14] and then extended to irregular meshes in [27], and generates C 1 surfaces. Another example is the "Loop" scheme [25], which is not interpolating, but gives a smoother limit surface.…”

Subdivision interpolating implicit surfaces

“…Dyn interpolating subdivision surface model based on a butterfly scheme for a regular topology [19]. This scheme was later improved by Zorin to accommodate topologically irregular triangular meshes [20].…”

“…By iteratively moving each added vertex along the gradient direction, we arrive at a point that lies on the implicit surface and take this point as the correspondence point or projection point of the added vertex. As the tessellated surface is quite similar to the interpolating subdivision surface [19,20], we call the resulting surface as subdivision interpolating implicit surface. This method can produce the same fine scalable triangles as the interpolating subdivision surface (see Fig.…”

Spherical Wavelets: Texture Processing

“…This results in the hat function referred to earlier. In the Butterfly scheme [7] a new value is found as:…”