Axis-Aligned Parameterizatior And Its Applications

In recent years,there are various applications of computer graphics in scientific research,industrial design,and our daily life entertainment.This field develops rapidly and is playing a more and more important role.The computation of parameterization mappings is a fundamental task in computer graphics,since it is at a central place of representation and processing of surfaces.In general,the parameterization mappings are required to be inversion-free and low distortion.Other requirements from different applications leads to different kinds of parameterization mappings.In this dissertation,a special kind of parameterization mappings is studied,where the parametric domain has a structure with axis-aligned boundaries.Compared to general parametric domain with free boundary,the regularity of the boundaries this kind of parametric domain leads to the advantages in some points.The solution of three application-driven problems:atlas packing problem,integer-grid mappings generating problem and discrete Chebyshev nets generating problem,are given in this dissertation based on the properties of this special kind of parameterization mappings.Texture atlas should be with high packing efficiency,to improve the utilization rate of the space.Based on the fact that the axis-aligned structure can easily be decomposed into rectangles,a novel method to refine an input atlas with bounded packing efficiency is presented in this dissertation to a generate space-efficient atlas.The basic idea is to convert the general polygon packing problem to an easier rectangle packing problem via the rectangle decomposition of axis-aligned structure.Given a parameterized mesh with no flipped triangles,this method proposes a new angle-driven deformation strategy to transform the general parametric domain into axis-aligned structures,then decomposes the charts into rectangles,and obtain a rectangle packing result balancing the trade-off between the packing efficiency and the length of cut.A following deformation for the rectangle packing result is performed to reduce the distortion while bounding the packing efficiency and retaining bijection.This method is proven to be effective in the experiments.The result atlases have much higher and bounded packing efficiency,while the distortion only increases a little.Compared to the state-of-the-art method,this method is much faster and achieves higher and bounded packing efficiency.Integer-grid mappings are an important category of parameterization mappings,which are used in quadrilateral meshing,paneling,animation,and simulation.But it is a challenging task to generate such a mapping without inversion.This dissertation presents a solution of generating integer-grid mappings using the axis-aligned structure:axis-aligned integer-grid parameterization mappings.Besides the requirements of gen-eral axis-aligned parameterizations mappings,the parametric domain needs to satisfy the global seamless constraints,the boundary edges must lie on the integer iso-lines,and coordinates of singularities must be integers.So the integer constraints in the integer-grid mappings are converted to the integer coordinate constraints of the axis-aligned structure to lower the difficulty.Given an input triangle mesh coupled with a global seamless parameterization,this method first constructs a parameterization-consistent cut that connects the singularities,and unfold the input global seamless parameterization on the parametric plane.Then the parametric domain is deformed into the axis-aligned structure and finally turned into the desired structure without violating the constraints of inversion-free and global seamless.The practical robustness of this method is demon-strated in the experiments with a wide variety of complex models.And the efficacy and superiority on two geometry processing tasks,including quadrilateral meshing and pixelation of meshes,are shown in this dissertation.Quad meshes with identical edge lengths,i.e.discrete Chebyshev nets,are widely used in architecture,engineering,and artistic design.Therefore,generating such a quad mesh that is approximate to a given 3D model is a valuable problem.Based on the property that the axis-aligned structure can be easily cut from the net-shape mate-rial,a method to generate discrete Chebyshev nets for various 3D models and fabri-cate them with physical net-shape material is presented in this dissertation.The first step is to generate discrete Chebyshev nets via generating frame fields that are discrete integrable,orientation preserving,approximate unit-length,and angle-bounded on the input meshes.And the second step is to partition the discrete Chebyshev net into a set of patches that can be unfolded into axis-aligned structures without overlap,and then to assign each patch a specification so that the home users can use it to cut,deform and as-semble the patches to fabricate the discrete Chebyshev nets with common tools such as scissors and pliers.This method generated digital discrete Chebyshev nets from various triangle meshes in the experiments,and some of them were fabricated with welded wire mesh.Eight participants were invited to the process of fabrication and all of them suc-ceeded in fabricating,thus the usability of the method and the specifications is proven.