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Mesh Parameterization
Published in Kai Hormann, N. Sukumar, Generalized Barycentric Coordinates in Computer Graphics and Computational Mechanics, 2017
In this chapter introduces mesh parameterization, an application of generalized barycentric coordinates. By “unfolding” a surface onto a 2D space, mesh parameterization has many possible applications, such as texture mapping. In this chapter we present some basic notions of topology that characterize the class of surfaces that admit a parameterization. Then we focus on the specific case of a topological disk with its boundary mapped to a convex polygon. In this setting, Tutte’s barycentric mapping theorem not only gives sufficient conditions, but also a practical algorithm to compute a parameterization. We outline the main argument of the simple and elegant proof of Tutte’s theorem by Gortler, Gotsman, and Thurston [172]. It is remarkable that their proof solely uses basic topological notions together with a counting argument. Finally, we mention the importance of the weights in the quality of the result, and demonstrate how mean value coordinates can be used to reduce the distortions.
Editable texture map generation and optimization technique for 3D visualization presentation
Published in Computer-Aided Design and Applications, 2018
Tsung-Chien Wu, Jiing-Yih Lai, Watchama Phothong, Douglas W. Wang, Chao-Yaug Liao, Ju-Yi Lee
The mesh parameterization technique in computational geometry provides several practical applications. Sheffer et al. [16] and Hormann et al. [9] introduced and summarized several typical methods of mesh parameterization and its applications, e.g. texture mapping, normal mapping, detail transfer, morphing, mesh completion, editing, database, remeshing, and surface fitting. The available techniques for mesh parameterization can be divided into types relating to distortion minimization, fixed or free boundary, or numerical complexity. For distortion minimization, an objective function can be formulated in terms of angles, areas or distances, and it is minimized to yield the optimized mapping of the model from the 3D domain to the parametric domain (called UV domain hereafter). The fixed boundary can be obtained by a simple formulation, allowing for an easy solution, but, the distortion in parameterization is quite large. In contrast, the free boundary has less distortion in parameterization, but obtaining the solution is time-consuming in computation because the boundary is considered as part of the solution. Numerical complexity is divided into linear and nonlinear methods. The nonlinear method is complex and requires more computational time, but it yields less distortion in the result.
Generation and quality improvement of 3D models from silhouettes of 2D images
Published in Journal of the Chinese Institute of Engineers, 2018
Watchama Phothong, Tsung-Chien Wu, Chun-Yeh Yu, Jiing-Yih Lai, Douglas W. Wang, Chao-Yaug Liao
In texture mapping, an integrated algorithm based on the conformal mesh parameterization and a technique for direct texture mapping is developed. The conformal mesh parameterization is employed to convert 3D meshes onto a 2D (UV) domain and keep the shape of the meshes on the UV domain. The direct texture mapping algorithm is primarily composed of three phases: grouping of 3D triangles, extraction of object image pixels, and placement of texture map pixels. Figure 18(a) shows the texture map for the ‘sport shoe’ and Figure 18(b) shows the 3D color model, where four different views of the model with color texture are displayed. The current result shows the completeness of the entire process from multiple 2D images of the object to a complete 3D model with color texture. A detailed discussion of the proposed texture mapping method can be shown in Wu et al. (2016).