3D mesh cutting for high quality atlas packing

Highlights

• A simple yet effective approach to cut an input surface and generate an atlas.

• A cancellation algorithm which balances the amount of singularities and the distortion of atlas.

• Short cuts which intersect with each other nearly orthogonally to connect all the singularities together.

• Taking our atlas as the input for the subsequent packing algorithm is better than using previous cutting strategies on a benchmark containing 5519 cases.

Abstract

An efficiently packed, low-distortion parameterization with a short boundary can save a great amount of memory and improve both quality and efficiency of rendering. Existing packing methods begin with an input atlas (or parameterization), but the cuts in the input atlas may be not suitable for a high quality result. We propose a simple yet effective approach to cut an input surface and generate an atlas that comprehensively considers the packing efficiency, the mapping distortion and the boundary length. Viewing the desired cuts on the input mesh as the pullback of a low-distortion mapping from a polysquare boundary, we notice that the above three objectives actually imply a small number of cone singularities with angle deficit of $\frac{\pi }{2}k,k\in \mathbb{Z}$, and orthogonally intersected short cuts passing through all singularities. Therefore, we first leverage a cross frame-field to identify a set of singularities and cancel some of them to balance their amount and atlas distortion. Then, the singularities remained are heuristically connected by short cuts which intersect with each other nearly orthogonally. Results show that our method produces a low-distortion and polysquare-like atlas with controllable number of singularities. Comparing with other atlas generation methods only focusing on distortion, taking our atlas as the input for the subsequent packing algorithm (Liu et al., 2019) is better than using previous cutting strategies on a benchmark containing 5519 cases, because the conflicts among those desired objectives for packing are alleviated at an early stage.

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 $\frac{\pi }{2}k,k\in \mathbb{Z}$, 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 $r_{BL}=(B{L}_{rand}-B{L}_{our})/B{L}_{our}$, where $B{L}_{rand}$ means the boundary length under a random order and $B{L}_{our}$ 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.