Flatten Anything: Unsupervised Neural Surface Parameterization

[Rebuttal to Reviewer FN2a]

[W1] Inaccurate claim about the settings of global and local parameterization.

Response: Thanks very much for pointing out our inaccurate claim. Indeed, multi-chart local parameterization is a more suitable choice when dealing with highly-complicated surfaces (such as those ShapeNet-style CAD models). As illustrated in Figure R4-(b) of the uploaded one-page PDF file, our approach cannot well handle such case. We will rectify this claim in our paper and provide the corresponding analyses.

[W2] Enhancing the presentation quality of some figures and table contents.

Response: We are sorry for the figure and table issues. We will carefully fix these problems and enhance the presentation quality.

[Q1&L1] Discussing the applicability of FAM.

Response: Thanks very much for your valuable comments. We will supplement a separate paragraph to detailedly discuss the applicability of our FAM, e.g., failure cases for highly-complicated CAD models and robustness to noisy input models as respectively shown in Figure R4-(b) and Figure R6 of the uploaded one-page PDF file.

[Rebuttal to Reviewer 4cq8]

[W1&Q1] Evaluation of self-intersection.

Response: As suggested, we supplemented quantitative evaluations of self-intersection in Figure R5 of the uploaded one-page PDF file and made comparisons with SLIM for open-surface cases. Generally, it is observed that self-intersection are inevitable both in our FAM architecture and the SLIM approach, but the proportions of self-intersected triangles are very small. In practice, it is not hard to apply post-processing refinement to slightly adjust the distribution of UV coordinates to further relieve or even eliminate self-intersection issues.

[Q2] Handling noisy point clouds.

Response: As suggested, in Figure R6 of the uploaded one-page PDF file, we applied FAM to point clouds added with increasing levels of Gaussian noises (1%, 2%, 4%). It is observed that our approach shows satisfactory robustness to noisy input conditions. In addition, it is worth mentioning that, in Figure 1 of our paper, the displayed models in the third row are real-scanned object/scene, and the displayed models in the last row are direct outputs of existing neural 3D generation tools. Hence, these testing models inevitably contains errors/noises.

[Q3] 1) Advantages of free boundary over fixed regular boundary; 2) Advantages of global parameterization over multi-chart parameterization.

Response: We are sorry for not adequately explaining these issues in our paper. Here we will present more detailed discussions.

1. Indeed, choosing regular boundary facilitates image-based downstream processing, such as compression of geometry images and those conducted in RegGeoNet [42] and Flattening-Net [43] applying deep 2D learning architectures for 3D geometric modeling. However, confining fixed regular boundary typically causes much greater mapping distortion, which is unsuitable for texture mapping, remeshing, and many other shape analysis tasks.

2. Actually, over the years, global surface parameterization has continuously been the mainstream direction of research, since it potentially achieves smooth transitions and uniform distribution of mapping distortion across the entire surface, while multi-chart parameterization typically introduces discontinuities along patch boundaries, causing more obvious visual artifacts for texture mapping (perhaps the most important application scenario in the graphics field) and bringing additional difficulties in many other shape analysis tasks (e.g., remeshing, morphing, compression). For multi-chart parameterization, it is worth emphasizing that only obtaining chart-wise UV maps is not the complete workflow. We need to further perform chart packing to compose the multiple UV domains, without which the actual usefulness can be largely weakened. However, packing is known as a highly non-trivial problem, which is basically skipped in recent neural learning approaches like DiffSR/Nuvo. Still, local parameterization does have its suitable application scenarios for processing highly-complicated surfaces such as typical ShapeNet-style CAD meshes with rich interior structures and severe multi-layer issues, and per-chart distortion can be reduced since the geometry and topology of cropped surface patches become simpler.

[Rebuttal to Reviewer Hnxu]

[W1] Clarifying the working mechanism and necessity of Cut-Net.

Response: The Cut-Net component is indeed necessary in the whole architecture, and we are sorry for not adequately analyzing its working mechanism and effectiveness in the paper.

Actually, the essential difference between Cut-Net and Unwrap-Net lies in that Cut-Net explicitly learns offsets while Unwrap-Net directly performs 3D-to-2D non-linear mapping. Without learning offsets, the inherent smoothness of neural works can impede "tearing" the originally-continuous areas on the 3D surface, thus hindering the creation of cutting seams. As for the reason why no constraints are needed to be imposed over $\mathbf{P}_\mathrm{cut}$, this is exactly the subtlety of the working mechanism of bi-directional cycle mapping. Similarly owing to the inherent smoothness of neural works, Cut-Net is inherently and naturally driven to output continuous point-wise offset values in a way that the resulting opened surface can be easily flattened with small distortion when fed into the subsequent simple 3D-to-2D MLP mapping layers.

We supplemented its ablation study in Figure R3 of the uploaded one-page PDF file. For models that are simpler to open ("human-face" and "mobius-strip"), removing Cut-Net does not lead to obvious performance degradation. However, for the other complex models, the removal of Cut-Net causes highly-distorted surface flattening, demonstrating the necessity of the Cut-Net component.

[W2] Evaluating multi-chart local parameterization.

Response: Actually, multi-chart local parameterization approaches are applicable not only to open-surface models but also to more complex topologies. Originally, we did not make comparisons with these approaches in experiments because they represent another different surface parameterization paradigm. Over the years, global surface parameterization has continuously been the mainstream direction of research, since it potentially achieves smooth transitions and uniform distribution of mapping distortion across the entire surface, while multi-chart parameterization typically introduces discontinuities along patch boundaries, causing more obvious visual artifacts for texture mapping (perhaps the most important application scenario in the graphics field) and bringing additional difficulties in many other shape analysis tasks (e.g., remeshing, morphing, compression).

For multi-chart parameterization, it is worth emphasizing that only obtaining chart-wise UV maps is not the complete workflow. We need to further perform chart packing to compose the multiple UV domains, without which the actual usefulness can be largely weakened. However, packing is known as a highly non-trivial problem, which is basically skipped in recent neural learning approaches like DiffSR/Nuvo. Still, local parameterization does have its suitable application scenarios for processing highly-complicated surfaces such as typical ShapeNet-style CAD meshes with rich interior structures and severe multi-layer issues, and per-chart distortion can be reduced since the geometry and topology of cropped surface patches become simpler.

Still, we agree that supplementing the results of multi-chart parameterization can help better understand the characteristics and advantages of our approach. Hence, here we conducted experiments to present the results of the latest work of Nuvo in Figure R1 of the uploaded one-page PDF file. We can observe that its chart assignment capability is still not stable. It often occurs that some spatially-disconnected surface patches are assigned to the same chart. Moreover, the critical procedure of chart packing is actually ignored in Nuvo. Directly merging the rectangular UV domains is not a valid packing. A real-sense packing should adaptively adjust the positions, poses, and sizes of each UV domain via combinations of translation, rotation, and scaling.

[W3] Experimenting with complex-topology models with manual surface cutting.

Response: Actually, in Figure 7 of our appendix, we have already performed such comparison setting. The reviewer can refer to our appendix contents for the corresponding analyses.

In this case, we manually specified the potentially-optimal cutting seams for the SLIM approach, achieving the conformality metric of 0.089, while our approach also produces satisfactory surface cuts and achieves slightly worsen conformality performance of 0.117. This comparison suggests that our FAM potentially performs comparably to optimal results obtained with manual efforts.

As for applying local parameterization to different complex models, please refer to our response to your preceding comment [W2], where we experimented with Nuvo and provided results in Figure R1 of the uploaded one-page PDF file.

Dear Reviewer Hnxu,

Thank you very much for evaluating this work. During the previous rebuttal phase, we made sincere efforts to directly address each of your concerns and questions. Please let us know if there is anything else we can clarify further. We would be delighted to seize this opportunity to discuss it with you.

[Rebuttal to Reviewer eo5W]

[W1&Q1] Discussing the advantages of global parameterization over multi-chart parameterization (e.g., Nuvo).

Response: Actually, over the years, global surface parameterization has continuously been the mainstream direction of research, since it potentially achieves smooth transitions and uniform distribution of mapping distortion across the entire surface, while multi-chart parameterization typically introduces discontinuities along patch boundaries, causing more obvious visual artifacts for texture mapping (perhaps the most important application scenario in the graphics field) and bringing additional difficulties in many other shape analysis tasks (e.g., remeshing, morphing, compression).

For multi-chart parameterization, it is worth emphasizing that only obtaining chart-wise UV maps is not the complete workflow. We need to further perform chart packing to compose the multiple UV domains, without which the actual usefulness can be largely weakened. However, packing is known as a highly non-trivial problem, which is basically skipped in recent neural learning approaches like DiffSR/Nuvo. Still, local parameterization does have its suitable application scenarios for processing highly-complicated surfaces such as typical ShapeNet-style CAD meshes with rich interior structures and severe multi-layer issues, and per-chart distortion can be reduced since the geometry and topology of cropped surface patches become simpler.

Originally, our experiments did not include Nuvo (which has not yet been formally published) as its official code is not publicly available. Here, we have supplemented the results of Nuvo using the suggested third-party implementation, as presented in Figure R1 of the uploaded one-page PDF file. We can observe that its chart assignment capability is still not stable. It often occurs that some spatially-disconnected surface patches are assigned to the same chart. Moreover, the critical procedure of chart packing is actually ignored in Nuvo. Directly merging the rectangular UV domains is not a valid packing. A real-sense packing should adaptively adjust the positions, poses, and sizes of each UV domain via combinations of translation, rotation, and scaling.

[W2] Discussing the highly-related work of OptCuts ([Li et al., TOG 2018]).

Response: Thanks very much for reminding us of this highly-related work. Overall, Optcuts is a traditional optimization-based surface parameterization algorithm operating on mesh models. Its prominent feature lies in the joint optimization of surface cutting and mapping distortion. Comparatively, our neural parameterization paradigm still shows advantages in terms of flexibility (not limited to well-behaved meshes; applicable to point clouds), convenience (exploiting GPU parallelism; much easier for implementing and tuning), and parameterization smoothness (not limited to mesh vertices).

In Figure R2 of the uploaded one-page PDF file, we supplemented the results of OptCuts on some of our used testing models. It is observed that, although OptCuts is able to jointly obtain reasonable surface cuts and UV unwrapping results, its performances are generally sub-optimal compared with ours (referring to "human-hand" in Figure 3 of our paper, "lion" and "torus-double" in Figure 4 of our paper, as well as "hsf-cylinder" in Figure R4-(a) of the uploaded one-page PDF file).

[W3&Q3] Clarifying the working mechanism and necessity of Cut-Net.

Response: The Cut-Net component is indeed necessary in the whole architecture, and we are sorry for not adequately analyzing its working mechanism and effectiveness in the paper.

Actually, the essential difference between Cut-Net and Unwrap-Net lies in that Cut-Net explicitly learns offsets while Unwrap-Net directly performs 3D-to-2D non-linear mapping. Without learning offsets, the inherent smoothness of neural works can impede "tearing" the originally-continuous areas on the 3D surface, thus hindering the creation of cutting seams. As for the reason why no constraints are needed to be imposed over $\mathbf{P}_\mathrm{cut}$, this is exactly the subtlety of the working mechanism of bi-directional cycle mapping. Similarly owing to the inherent smoothness of neural works, Cut-Net is inherently and naturally driven to output continuous point-wise offset values in a way that the resulting opened surface can be easily flattened with small distortion when fed into the subsequent simple 3D-to-2D MLP mapping layers.

We supplemented its ablation study in Figure R3 of the uploaded one-page PDF file. For models that are simpler to open ("human-face" and "mobius-strip"), removing Cut-Net does not lead to obvious performance degradation. However, for the other complex models, the removal of Cut-Net causes highly-distorted surface flattening, demonstrating the necessity of the Cut-Net component.

[Q2] Stress test and potential failure cases.

Response: We supplemented the same stress test on a Hilbert-space-filling-shaped cylinder model. As shown in Figure R4-(a) of the uploaded one-page PDF file, we can also obtain the basically optimal solution.

As for potential failure cases, since FAM has no hard requirement for data quality or complexity, our neural model accepts arbitrary input surfaces to output UV maps, unlike many other traditional parameterization algorithms whose program running directly raises errors and fails for non-well-behaved mesh inputs. Still, it does not mean that FAM works well in all cases. Typically, as shown in Figure R4-(b) of the uploaded one-page PDF file, processing the highly-complicated ShapeNet-style CAD model with rich interior structures and many multi-layer issues shows inferior UV unwrapping quality. Although our learned cutting seams are generally reasonable and texture mapping looks relatively regular from outside, it is hard to deal with the complicated interior structures.

Thanks for the authors detailed response to my questions. I think this work may have a good impact on the field of geometry processing. As before, I suggest to accept this paper.

Dear Reviewer eo5W,

We are delighted to have successfully addressed your questions. Thank you very much for acknowledging and positively recommending our work.

Thanks to the authors for their detailed response, it has explained my questions well.