A New Approach to the Construction of Subdivision Algorithms

TL;DR

This thesis introduces a novel method for constructing subdivision algorithms for generalized quadratic and cubic B-spline surfaces and volumes, emphasizing eigenvector generation, matrix adjustment, and satisfying quality criteria.

Contribution

It presents a new approach using eigenvector-based construction and matrix exponential adjustments for subdivision algorithms, with formal proofs and quality guarantees.

Findings

Algorithms exhibit a subdominant eigenvalue of 1/2

Quadratic case property can be formally proven

Constructed algorithms meet multiple quality criteria

Abstract

In this thesis, a new approach for constructing subdivision algorithms for generalized quadratic and cubic B-spline subdivision for subdivision surfaces and volumes is presented. First, a catalog of quality criteria for these subdivision algorithms is developed, serving as a guideline for the construction process. The construction begins by generating the desired subdominant eigenvectors as the vertices of regular convex 3-polytopes for volumes using circle packings. Subsequently, these polytopes are utilized to construct a Colin-de-Verdiere-matrix for the generalized quadratic and a Colin-de-Verdiere-like matrix for the generalized cubic B-spline subdivision. These matrices are then adjusted using the matrix exponential to obtain subdivision matrices with the desired properties. All subdivision algorithms introduced in this paper empirically exhibit a subdominant eigenvalue of 1/2 with the desired algebraic and geometric multiplicity. For the quadratic case, this property can even be formally proven. Moreover, the corresponding eigenvectors form a convex polytope in the central region for the generalized quadratic B-spline subdivision algorithms, while for the generalized cubic B-spline subdivision algorithms, they represent the refinement of a convex polytope. Additionally, the constructed subdivision algorithms fulfill various other quality criteria, such as affine invariance and convex hull preservation and respecting all symmetries. Furthermore, it is demonstrated that the original Catmull-Clark algorithm is not suitable for generalization to volumetric subdivision and that the established subdivision algorithms [Baj+02] and [JM99] do not exhibit a suitable spectrum for several combinatorial configurations. Additionally, research approaches for the volumetric case are proposed, aiming to generalize from hexahedral to arbitrary structures.