3D mesh cutting for high quality atlas packing☆
Introduction
In computer graphics, parametrizing a mesh into 2D atlas, i.e. texture mapping, is ubiquitously used to store surface signals for rendering purposes (Hormann et al., 2008; Sheffer et al., 2006). For the sake of efficient storage, fast and high-quality rendering, it is important to generate a compactly packed atlas with low distortion and short boundaries for a given mesh. However, a good trade-off among all requirements mathematically involves a complicated multi-objective, nonlinear and non-convex optimization problem to solve.
Directly modeling and solving this problem is computationally expensive and sometimes prohibitive. Instead, a practical way is to decompose the multi-objective optimization into multi-stages with different objectives being processed at different stages separately in relatively cheap cost. For atlas packing, the first stage typically cuts and parametrizes the mesh into a 2D atlas, while the second stage continues to refine and pack the atlas. The major objective in the first stage (e.g. Li et al., 2018; Zhu et al., 2020) is to keep distortion low and cuts short, and the second stage mainly focuses on achieving high packing efficiency and maintaining short boundaries (e.g. Limper et al., 2018; Liu et al., 2019) without introducing much additional distortion.
Obviously, the input of the second stage has a strong influence to the final packing quality (see Fig. 1), because typical packing methods (Limper et al., 2018; Liu et al., 2019) usually have to create an amount of new cuts to slice the input atlas for higher packing efficiency but they naively keep the original boundaries untouched. Improper boundaries introduced by the input atlas may impose strong yet unnecessary restrictions for packing (see Fig. 2). Therefore, it is quite natural to ask the first stage to generate an atlas in favor of the second stage. The key here is to find suitable cuts on the input mesh, since the parameterization is mainly driven by the boundaries. As demonstrated in recent works (Limper et al., 2018; Liu et al., 2019), polysquare-like input atlas is advantageous for high-quality packing, so ideal cuts are expected to directly yield a parameterized atlas with low distortion, polysquare layouts and short boundaries.
Checking the pullback of a low-distortion mapping from a polysquare, we find two important properties of ideal cuts which favors low-distortion parametrization with high packing efficiency: the cone singularities through which the cuts must pass should have angle deficit of , and cuts should intersect with each other nearly orthogonally. Therefore, we leverage cross-frame generation methods developed for quadrangulation (Knöppel et al., 2013) to identify the cone singularities, but relax stringent constraints for quadrangulation such as feature and boundary alignment to adapt to texture mapping. In other words, the singularities in the cross-frame field are squeezed out purely based on the intrinsic geometry of surface for concentrating its Gaussian curvature. Furthermore, to balance the distortion of parameterization and the amount of singularities, we clean out some unnecessary ones by an iterative optimization process. Once done, singularities will be progressively connected via short cuts in an orthogonal manner. We notice that the ordering of picking singularities in this process has a large influence on resulting cut length. As the evidence shown in the inset, it demonstrates the distribution of the relative boundary length , where means the boundary length under a random order and is the one generated by our cut method. Thus, to make the total cut length as small as possible, we solve this NP-hard problem by a heuristic search which always looks one step further instead of simply adopting a trivial greedy strategy. After parameterization, the cuts lead to a low-distortion, polysquare-like atlas, which can be seamlessly used for subsequent process on packing as an input. Our method is simple and efficient, and comprehensive evaluations on a large dataset show that the atlas generated by our method generally leads to better packing quality than that from previous approaches.
Section snippets
Related work
Since our work builds upon prior research on cross-frame field generation, cut optimization, parameterization, and atlas packing, we briefly review the most relevant approaches.
Cross-frame field generation Cross frame field, or 4-RoSy vector field, has been extensively studied since it was firstly introduced in Palacios and Zhang (2007). A thorough review on this topic is performed in Vaxman et al. (2016). By representing a cross field using rotational angles and integer period jumps, a
Method
The problem investigated in this paper can be stated as follows: given a surface tessellated by a triangle mesh with arbitrary topology, how to acquire a high-quality atlas packing? As shown in Fig. 3, starting from a surface mesh, we firstly obtain a set of candidate cone singularities by computing a smooth cross frame field. Secondly, we cancel some singularities in the candidate set under control of parameterization distortion (Section 3.1). Thirdly, we compute a cut to flatten the surface
Experiments
In this section, we will demonstrate the efficacy of our cutting method through quantitative comparisons and ablation studies using a large dataset containing thousands of models. All the experiments were performed on a desktop with AMD RyzenTM 9 3900X CPU and 64 GB of RAM. The computational time for some selected examples is listed in Table 1.
Dataset We build our testing dataset by processing models of the dataset released by Liu et al. (2019) containing 5588 models. Each model has been
Conclusions
Inspired by the pullback of a low distorted mapping from a polysquare, we propose a method to generate short and orthogonal cuts passing through a few cone singularities with angle deficit of to flatten the input surface into a polysquare-like atlas with low distortion for compact packing. Taking the output of our cutting method as the input atlas, the state-of-the-art packing methods can get considerable benefits from our re-parameterization and produce better results, which has been
CRediT authorship contribution statement
ShiyiWang: Conceptualization, Methodology, Software, Data curation; Jiong Chen: Writing-Original draft preparation; Xifeng Gao: Validation, Investigation; Hujun Bao: Supervision; Jin Huang: Supervision, Writing-Reviewing and Editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was supported by National Key R&D Program of China (No. 2020AAA0108901) and Zhejiang Provincial Science and Technology Program in China under Grant 2021C01108.
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Editor: Thomas Takacs.