Systematically generalizing planar geometric algorithms to manifold domains is of fundamental importance in computer aided design field.
Based on our theoretical breakthrough, we also develop a set of practical algorithms to generalize triangular B-spline surfaces from planar domains to manifold domains.
This paper presents a general theoretical and computational framework, in which spline surfaces defined over planar domains can be systematically extended to manifold domains with arbitrary topology with or without boundaries.
We discuss the framework for arbitrary target manifolds, while the domain manifold problem was addressed in [J. Comput. Phys. 174(20012001) 759].
We show that the number of extraordinary points of the domain manifold is bounded by the number of segmented components.
In this manifold model, each domain is represented by a Laplacian matrix that captures the closeness to other instances sharing the same label.
We study the affine structure of domain manifolds in depth and prove that the existence of manifold splines is equivalent to the existence of a manifold's affine atlas.
This study proposes a novel multiple-domain manifold (MDM) method to achieve representative features based on singular value decomposition (SVD) and manifold learning.
Firstly, we consider spline functions that are defined on domain manifolds, while the existing constructions are limited to domains that are open subsets of Rd.
The algorithm by Wang [121], referred to as the domain adaptation manifold alignment (DAMA) algorithm, proposes using a manifold alignment [45] process to perform a symmetric transformation of the domain input spaces.
