Constrained Texture Mapping And Foldover-free Condition

Texture mapping has been widely used in image processing and graphics to enhance the realism of CG scenes. However to perfectly match the feature points of a 3D model with the corresponding pixels in texture images, the parameterisation which maps a 3D mesh to the texture space must satisfy the positional constraints. Despite numerous research efforts, the construction of a mathematically robust foldover-free parameterisation subject to internal constraints is still a remaining issue. In this paper, we address this challenge by developing a two-step parameterisation method. First, we produce an initial parameterisation with a method traditionally used to solve structural engineering problems, called the bar-network. We then derive a mathematical foldover-free condition, which is incorporated into a Radial Basis Function based scheme. This method is therefore able to guarantee that the resulting parameterization meets the hard constraints without foldovers

Constrained parameterization with applications to graphics and image processing.

Surface parameterization is to establish a transformation that maps the points on a surface to a specified parametric domain. It has been widely applied to computer graphics and image processing fields. The challenging issue is that the usual positional constraints always result in triangle flipping in parameterizations (also called foldovers). Additionally, distortion is inevitable in parameterizations. Thus the rigid constraint is always taken into account. In general, the constraints are application-dependent. This thesis thus focuses on the various constraints depended on applications and investigates the foldover-free constrained parameterization approaches individually. Such constraints usually include, simple positional constraints, tradeoff of positional constraints and rigid constraint, and rigid constraint. From the perspective of applications, we aim at the foldover-free parameterization methods with positional constraints, the as-rigid-as-possible parameterization with positional constraints, and the well-shaped well-spaced pre-processing procedure for low-distortion parameterizations in this thesis. The first contribution of this thesis is the development of a RBF-based re-parameterization algorithm for the application of the foldover-free constrained texture mapping. The basic idea is to split the usual parameterization procedure into two steps, 2D parameterization with the constraints of convex boundaries and 2D re-parameterization with the interior positional constraints. Moreover, we further extend the 2D re-parameterization approach with the interior positional constraints to high dimensional datasets, such as, volume data and polyhedrons. The second contribution is the development of a vector field based deformation algorithm for 2D mesh deformation and image warping. Many presented deformation approaches are used to employ the basis functions (including our proposed RBF-based re-parameterization algorithm here). The main problem is that such algorithms have infinite support, that is, any local deformation always leads to small changes over the whole domain. Our presented vector field based algorithm can effectively carry on the local deformation while reducing distortion as much as possible. The third contribution is the development of a pre-processing for surface parameterization. Except the developable surfaces, the current parameterization approaches inevitably incur large distortion. To reduce distortion, we proposed a pre-processing procedure in this thesis, including mesh partition and mesh smoothing. As a result, the resulting meshes are partitioned into a set of small patches with rectangle-like boundaries. Moreover, they are well-shaped and well-spaced. This pre-processing procedure can evidently improve the quality of meshes for low-distortion parameterizations