Flatten Anything: Unsupervised Neural Surface Parameterization
Authors: Qijian Zhang
Junhui Hou
Wenping Wang
Ying He
Venue: NeurIPS 2024 poster (2024)
Keywords: Surface Parameterization
UV Unwrapping
Neural Network
Unsupervised Learning
Primary area: machine_vision
TL;DR: An unsupervised neural learning architecture for universal and fully-automated 3D surface parameterization

Abstract: Surface parameterization plays an essential role in numerous computer graphics and geometry processing applications. Traditional parameterization approaches are designed for high-quality meshes laboriously created by specialized 3D modelers, thus unable to meet the processing demand for the current explosion of ordinary 3D data. Moreover, their working mechanisms are typically restricted to certain simple topologies, thus relying on cumbersome manual efforts (e.g., surface cutting, part segmentation) for pre-processing. In this paper, we introduce the Flatten Anything Model (FAM), an unsupervised neural architecture to achieve global free-boundary surface parameterization via learning point-wise mappings between 3D points on the target geometric surface and adaptively-deformed UV coordinates within the 2D parameter domain. To mimic the actual physical procedures, we ingeniously construct geometrically-interpretable sub-networks with specific functionalities of surface cutting, UV deforming, unwrapping, and wrapping, which are assembled into a bi-directional cycle mapping framework. Compared with previous methods, our FAM directly operates on discrete surface points without utilizing connectivity information, thus significantly reducing the strict requirements for mesh quality and even applicable to unstructured point cloud data. More importantly, our FAM is fully-automated without the need for pre-cutting and can deal with highly-complex topologies, since its learning process adaptively finds reasonable cutting seams and UV boundaries. Extensive experiments demonstrate the universality, superiority, and inspiring potential of our proposed neural surface parameterization paradigm. Our code is available at https://github.com/keeganhk/FlattenAnything.

Flatten Anything:
Unsupervised Neural Surface Parameterization
Qijian Zhang', Junhui Hou', Wenping Wang', Ying He
DDppartment of Computer Science, City University of Hong Kong, Hong Kong SAR, China *Department of Computer Science and Engineering, Texas A&M University, Texas, USA Sccooo of Computer Science and Engineering, Nanyang Technological University, Singapore qi 1ewiiannanaaaie cityu. edu. hk, j.. ht..ho@citte tyu. edu. .kk venping@tamu. edu, yheθntu. edu.sp
complex topolegy
real-scanned object
real-scanned scene
real Imoge
(o)
(d)
real Image
(o)
Figure 1: Flatten Anything Model (FAM) for neural surface parameterization: (a) input 3D models; (b) learned UV coordinates; (c) texture mappings: (d) learned cutting seams.
Abstract
Surface parameterization plays an essential role in numerous computer graphics and geometry processing applications. Traditional parameterization approaches are designed for high-quality meshes laboriously created by specialized 3D modelers, thus unable to meet the processing demand for the current explosion of ordinary 3D data. Morcover, their working mechanisms are typically restricted to certain simple topologies, thus relying cumbersome manual effoort (c.g., surface cutting, part segmentation) for pre- pre-processing. In this paper, we introduce the Flatten Anything Model (FAM), unsupervised neural architecture to achieve global free- boundary surface parameterization via learning point-wise wise mappings between 3D points on the target geometric surface and ceaptively-deforme 2D parameter domain. To mimic the actual physical procedures, we ingeniously construct geometrically-interpeeeae surface cutting, UV deforming, unwrapping, and wrapping, which are assembled into bi-directional cycle mapping framework. Compared with previous methods, our FAM directly operates on discrete surface points without utilizing connectivity information, thus significantly reducing the strict requirements for mesh quality
UV coordinates within the
sub-networks with specific functionalities of
*Corresponding author. This work was supported in part by the National Natural Science Foundation of China Excellent Young Scientists Fund 624221 18, and in part by the Hong Kong Research Grants Council under Grants 11219324 and 11219422.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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and even applicable to unstructured point cloud data, More importantly. our FAM is fully-automated without the need for pre-cutting and can deal with hogply-complex topologies, since its learning process adaptively finds reasonable cutting seams and UV boundaries. Extensive experiments demonstrate the universality, superiority, and inspiring potential of our proposed neural surface parameterization paradigm. Our code available at attps://Sa /g/ tiub.. conm///egaatt//llttennyyyya
1 Introduction
Surface parameterization intuitively refers to the process of flattening 3D geometric surface onto a 2D plane. which is typically called the parameter domain. For any 3D spatial point (x, y. lying on the underlying surface, we explicitly map it to a 2D planar coordinate (u, u) while satisfying certain continuity and distortion constraints. Building such point-to-point mappings also known as UV unwrapping, which serves as an indispensable component in modern graphics rendering pipelines for texture mapping and is also widely used in many downstream geometry processing applications, such as remeshing, mesh completion/compressoon detail transfer, surface fitting, and editing, etc.
Theoretically, for any two geometric surfaces with identical/similar topological structures, there exists a bijective mapping between them. Nevertheless, when the topology of the target 3D surface becomes complicated (e.g., with high genus), one must pre-open the original mesh to sufficiently deveeeopll disk along appropriate cutting seams. Consequently, the current industrial practice for implementing UV unwrapping is typically composed of two stages: (1) manually specifying some necessary cutting seams on the original mesh; (2) applying mature
wowolooy-oriennttteeooo parameterization algorithms (e.g.. LSCM [15], ABF [31, 33)) which have been well integrated into popular 3D modeling software (e.g., Blender, Unity 3Ds Max) to produce per-vertex 2D UV coordinates. In practice, such stage-wise UV unwrapping pipeline still shows the following limitations and inconveniences:
1) Commonly-used surface parameterization algorithms are designed for well-behaved meshes, which typically produced by specialized 3D modelers and technical artists. However, with the advent of user-generated content (UGC) fueled by the rapidly growing 3D sensing, reconstruction [24, 14), and generation [26, 28, 18, 34] techniques, there emerges an urgent need for dealing with ordinary 3D data possibly with unruly anomalies, inferior triangulations, and non-ideal geometries.
2) Despite the existence of few early attempts at heuristic seam generation [32, 8], the process of finding high-quality cutting seams in practical applications still relies on manual efforts and personal experience. Hence, the entire workflow remains subjective and semi-automated, leading to reduced reliability and efficiency, especially when dealing with complex 3D models that users are unfamiliar with. Besides, since cutting is actually achieved via edge selection, one may need to repeatedly adjust the distribution of mesh edges to allow the cutting seams to walk through.
3) The procedures of cutting the original mesh into disk and then flattening the resulting disk onto the parameter domain should have been mutually influenced and jointly optimized; otherwise, the overall surface parameterization results could be sub-optimal.
In recent years, there has emerged a new family of neural parameterization approaches targeted at learning parameterized 3D geometric representations via neural network architectures. The pioneering works of FoldingNet [39] and AtlasNet [11] are among the best two representatives, which can build point-wise mappings via deforming a pre-defined 2D lattice grid to reconstruct the target 3D shape. rectangular structurization of irregular 3D point clouds through geometry image [12, 21] representations. However, these two approaches cannot be regarded as real-sense surface parameterization due to their lack of mapping constraints. DiffSR [2] and Nuvo [36] focus on the local parameterization of the original 3D surface but with explicit and stronger constraints on the learned neural mapping process. Their difference lies in that DiffSR aggregates multi-pated parameterizations to reconstruct the original geometric surface, while Nuvo adaptively assigns surface points to different charts in a probabilistic manner.
RegGeoNet [41] and Flattening-Ner [42] tend to achieve
In this paper, we make the first attempt to investigate neural surface parameterization featured by both global mapping and free boundary. We introduce the Flatten Anything Model (FAM), a universal and aullaea UV unwrapping approach to build point-to-point mappings between 3D points lying on the target geometric surface and 2D UV coordinates within the adaptively-deffoe domain. In general, FAM is designed as an unsupervised learning pipeline (i.c., ground-truth UV
parameter
2

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maps are collected), running in a per-model overfitting manner. To mimic the actual physical process, we ingeniously design a series of paanettrcclllymmmnnngee sub-networks with specific functionalities of surface cutting, UV deforming, 3ID-to-2D unwrapping, and 2D-to-3D wrapping, which are further assembled into a bi-directional cycle mapping framework. By optimizing a combination of different loss functions and auxiliary differential geometric constraints, FAM is able to find reasonable 31) cutting seams and 2D UV boundaries, leading to superior parameterization quality when dealing with different degrees of geometric and topological complexities. Comprehensive experiments demonstrate the advantages of our approach over traditional state-of-the-an approaches. Conclusively, our FAM shows the following several major aspects of characteristics and superiorities:
Compared with traditional mesh parameterization approaches [15, 33,,2), 29]. FAM directly operates on discrete surface points and jointly learns surface cutting, which can be insensitive to mesh triangulations and is able to deal with non-ideal geometries and aibirrarll-complee topologies. Besides, such a pure neural learning architecture can naturally exploit the powerful parallelism of GPUs and is much easier for implementing, tuning, and further extension.
Another distinct advantage lies in the smoothness of parameterization. Since mesh parameteriza- tion computes parameters solely for existing vertices, for any point on the mesh that is not a vertex, these approaches resort interpolation within its encompassing triangle to determine the corresponding UV coordinate. Such practice fails to guarantee smoothness, particularly across edges, and this issue is further exacerbated in low-resolution meshes. By contrast, FAM exploits the inherent smooth property [30] of neural networks to learn arbitrary point-to-point mappings, thereby ensuring smoothness for all points on the target surface.
Different from previous multi-patch local parameterization frameworks [11, 2, 36]. FAM focuses global parameterization, more valuable yet much harder problem setting.
Different from previous exxed-boundar structurization frameworks [41, 42], FAM deforms the 2D UV parameter domain adaptively and regularizes the learned neural mapping explicitly, thus significantly reducing discontinuities and distortions.
2
Related Works
2.1 Traditional Parameterization Approaches
Early approaches formulate mesh parameterization as Laplacian problem, where boundary points are anchored to a pre-determined convex 2D curve [7, 91, tackled by sparse linear system solvers. Such linear strategy is appreciated for its simplicity, efficiency, and the guarantee of bijectivity. However, the rigidity of fixing boundary points in the parameter domain usually leads to significant distortions. In response to these issues, free-boundary parameterization algorithms [33, 19] have been proposed, offering more flexible setting by relaxing boundary constraints. Although these approaches bring certain improvements, they often struggle to maintain global bijectivity. More recent state-of-the-ar approaches [29, 35] shift towards minimizing simpler proxy energies by alternating between local and global optimization phases, which not only accelcrate convergence but also enhance the overall parameterization quality. OptCuts [16] explores joint optimization of surface cutting and mapping distortion. Despite the continuous advancements, the working mechanisms of all these approaches fundamentally rely on the connectivity information inherent in mesh structures, casting doubt on their applicability to unstructured point clouds. Moreover, ooieete parameterization approaches typically assume that the inputs are well- behaved meshes possessing relatively regular triangulations. When faced with input meshes of inferior quality characterized by irregular triangulations and anomalies, remeshing becomes an indispensable procedure. Additionally, how to deal with non- ideal geometries and complex topologies (e.g., non-manifolds, multiply-connected components, thin structures) has always been highly challenging.
Unlike surface meshes, unstructured point clouds lack information regarding the connectivity between points, and thus are much more difficult to parameterize. Hence, only a relatively limited amount of prior works [43] have focused on the parameterization of surface point clouds. Early studies investigate various specialized parameterization strategies for disk-type [10, 1, 40], genus-O [44, 171, genus-1 1 [37], well-sampled [13] or low-quality [22] point clouds. Later approaches achieve bixed-boundary spheriica/rectangular parameterization through approximating the Laplace- operators [5] or Teichmüller extremal mappings [23]. More recently, FBCP-PC [4] further pursues free-boundary conformal parameterization, which new point cloud Laplacian approximation scheme is proposed for handling boundary non-convexity.
3

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2.2 Neural Parameterization Approaches
Driven by the remarkable success of deep learning, there is a family of recent works applying neural networks to learn parameterized 3D geometric representations. FoldingNet [39] proposes to deform a uniform 2D grid to reconstruct the target 3D point cloud for unsupervised geometric feature learning. AtlasNet [11 I] applies multi-grid deformation to learn locally parameterized representations. Subse- quently, a series of follow-up researches inherit such a "folding-style" parameterization paradigm as pioneered by [39, 11] and investigate different aspects of modifications. GTIF [3] introduces graph topology inference and filtering mechanisms, empowering the decoder to preserve more representative geometric features in the latent space. EleStruc [6] proposes to perform shape reconstruction from learnable 3D elementary structures, rather than a pre-defined 2D lattice. Similarly, TearingNet [27] adaptively breaks the edges of an initial primitive graph for emulating the topology of the target 3D point cloud, which can effectively deal with higher-geme or multi-object inputs. In fact, the ultimate goal of the above line of approaches is to learn expressive shape codewords by means of reforrattoon-rivee 3D surface reconstruction. The characteristics of surface parameterization, i.c., the mapping process between 3D surfaces and 2D parameter domains, are barely considered.
In contrast to the extensive research in the fields of deep learning- basedd 3D geometric reconstruction and feature learning, there only exist few studies that particularly focus on neural surface param- eterization. DiftSR [2] adopts the basic multi multi-patch reconstruction framework [11] and explicitly regularizes multiple differential surface properties. NSM [25] explores neural encoding of surface maps by overfitting a neural network to an existing UV parameterization pre-computed via standard mesh parameterization algorithms [38, 29). DA-Wand [20] constructs a paraneterizatton-oorineed mesh segmentation framework. Around a specified triangle, it learns to select a local sub-region, which is supposed to be sufficiently developable to produce low-distortion parameterization. Inherit- ing the geometry image (GI) [12] representation paradigm, RegGeoNet [41] and Flattening-Net [42] propose to learn deep regular representations of unstructured 3D point clouds. However, these two approaches lack explicit constraints on the parameterization distortions, and their local-to-glon -oglobal assembling procedures are hard to control. More recently, Nuvo [36] proposes a neural UV mapping framework that operates on oriented 3D points sampled from arbitrary 3D representations, liberating from the stringent quality demands of mesh triangulation. This approach assigns the original surface to multiple charts and ignores the packing procedure, thus essentially differing from our targeted global parameterization setting.
3 Proposed Method
Our FAM operates on discrete spatial points lying on the target 3D geometric surface for achieving global free-boundary parameterization in an unsupervised learning manner. Denote by P E R x3 a set of unstructured points without any connectivity information, we aim at point-wisely parameterizing (i.e., row-wisely mapping) P onto the planar domain to produce UV coordinates Q E IR Nx2
Technical Motivation. The most straightforward way of learning such a surface flattening process is to directly impose certain appropriate regularizers over the resulting 2D UV coordinates Q. However, due to the missing of ground-truths, it is impossible explicitly supervise the generation of Q, and it empirically turns out that designing and minimizing such regularizers cannol produce satisfactory results. Alternatively, another feasible scheme to choose an opposite mapping direction wrap the adaptively-defomme and potentially-optimal 2D UV coordinates Q E I onto the target surface by generating new set of unstructured points P E Vx3 which should approximate the input P by minimizing certain point set similarity metries. Under ideal situations where P and P are losslessly matched, we can equivalently deduce the desired UV coordinate of each point in P (i.e., for certain 3D point P; E P perfectly matched with generated 3D point P. E P. which is row-wisely mapped from 93 E Q, then we know the UV coordinate of p, should be âz). However, due to the practical difficulty of reconstructing point sets with high-precision point-wise matching, the generated PP is only able to coarsely recover the target surface, thereby Q cannot be treated as the resulting 2D UV coordinates of the input P. Still, despite the inaccuracy of such inverse 2D-to-3D wrapping process, provides valuable cues indicating how the target 3D surface is roughly transformed from the 2D parameter domain, which naturally motivates us to combine the two opposite mapping processes into a bi-directional joint learning architecture. To build associations between the unwrapping and wrapping processes and promote mapping bijectivity, we further introduce cycle mapping mechanisms

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Lyyle
Deform Nen---et
Wrap-No ---Net
Cut-Net
Unwrap-Net
[o]
surface-sampled Al-generated real-scanned
Unwrap- Net
Cut-Net
Wrap-Aie
[b)
P
Peycle
N
Figure 2: Illustration of bi-directional cycle mapping, composed (a) 2D-33D-22D cycle mapping branch, and (b) 3D 2D --2)>31 cycle mapping branch. Modules with the same color share network parameters. (c) shows the learned cutting seams. (d) shows the checker-image texture mapping.
and impose consistency constraints. Thus, our FAM is designed bi-directional cycle mapping workflow, in which all sub-networks are parameter-sharing and jointly-otimizze
3.1 Bi-directional Cycle Mapping
As illustrated in Figure 2, there are two parallel learning branches comprising a series of geometrically- meaningful sub-networks built upon point-wisety shared multi-layer perceptrons (MLPs):
The deforming network Ma (Deform-Net) takes a uniform 2D lattice as input, and deforms the initial grid points potentially-optimal UV coordinates.
The wrapping network M (Wrap-Net) wraps the eeventially-optimee 2D UV coordinates onto the target 3D geometric surface.
The surface cutting network M, (Cut-Net) finds appropriate cutting seams for transforming the original 3D geometric structure to an open and highly-developable surface manifold.
The unwrapping network Mn. (Unwrap-Net) assumes that its input surface has already been pre-opened, and flattens the 3D points onto the 2D parameter domain.
3.1.1
2D +3-33--00 +2D Cycle Mapping
The upper learning branch begins with pre-defined 2D lattice with N grid points uniformly sampled within |-1, 1] 1 denoted as G Nx2, To perform adaptive UV space deformation, employ offset-based coordinate updating strategy by feeding G into the Deform-Net to produce another set of 2D UV coordinates Q Nx2 which can be formulated as:
M,(G)) (EE( G)) +G,
where x) denotes channel concatenation, & R2 Rd and El (R(h-2) - IR2 are stacked MLPs. Intuitively, the initial 2D grid points first embedded through & into the h-dimensional latent space, concatenated with itself, and then mapped onto the 2D planar domain through t// The learned offsets are point-wisely added to the initial grid points to produce the resulting Q
After that, we perform 2D-to-3D wrapping by feeding the generated Q into the Wrap-Net to produce E Nx3 which is expected to roughly approximate the target 3D geometric structure represented by the input 3D point set P. In the meantime, we use three more output channels to produce normals pa gNx3 The whole network behavior can be described as:
p M, Q <" (IC', Q): Q)),
where E : R2 + h and el (b+2) - 6 are stacked MLPs, and the resulting 6 output channels respectively split into P and Pa.
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In order to construct eycle mapping, we further apply the Cut-Net on the reconstructed P, which is transformed into an open 3D surface manifold Peut E R N as given by:
Paut = (P) ( (P:: PJ) + P. where C. R + R and C. (R+33) - IR33 are stacked MLPs.
(3)
Once we obtain the 3D-to-2D flattening through the Unwrap-Nett which can be formulated as:
3D surface manifold P Pout, it is straightforward to perform
Qayll = (P, (P. rut);
where Çu R + R22 is simply implemented as stacked MIPs, and the resulting Qayele E Nx2 is
expected to be row-wisely equal to Q.
3.1.2 3D +2- +3D Cycle Mapping
The bottom learning branch directly starts by feeding the input 3D point set P into the Cut-Net to produce an open 3D surface manifold Peut. which can be formulated as:
P Pout (P) (P)) +P.
After that, we apply the Unwrap-Net on the generated P Pcut to produce the desired 2D UV coordinates Q, as given by:
Q = MM (M. = (Peut).
(6)
In order to construct cycle mapping, we further apply the Wrap-Net on the generated Q for recovering the target 3D geometric structure, which can be formulated as:
[Paycle; Po Payctel = M. (Q) = <" (Q); Q.),
where P ccycle is expected to be row-wisely equal to P. When the input surface points are accompanied by normals p E IR x3, we also expect that the normal directions of p and the generated Pa Paycle
should be row-wisely consistent.
Remark. Throughout the overall bi-directional cycle mapping framework, only Q is regarded the desired 2D UV coordinates that are row-wisely mapped from the input 3D) surface points P, all the others (e.g.. Q and Qcycle) just serve as intermediate results. Morevoer, is also worth emphasizing that, when the training is finished, any arbitrary cofface-smplee 3D points can be fed into our FAM to obtain point-wise UV coordinates.
Extraction of Cutting Seams. The extraction of points located on the learned cutting seams, which we denote as E- {e; E PP), can be conveniently achieved by comparing the mapping relationships between the input P and the parameterized Q.
Specifically, since P and Q are mapped in a row-wise manner, for each input 3D point p. P and its K-nearest neighbors [p: Kee Jk=1 within P, we can directly know their 2D UV coordinates, denoted
(k) Kent. as qi E Q and {q, /==1- The maximum distance value 77: between qi and its neighboring points can be computed as:
Th max q, (k) K=r).
(8) Then, P; will be determined locate on the cutting seam 77, is larger than a certain threshold Teut- Collecting all such points within P can deduce a set of seam points E, and it is observed that in this way no seam points would be identified if cutting the original 3D surface is actually unnecessary.
3.2 Training Objectives
As an unsupervised learning framework, FAM is trained by minimizing a series of carefully-designed objective functions and constraints, whose formulations and functionalities are introduced below.
Unwrapping Loss. The most fundamental requirement for Q that any two parameterized coor- dinates cannot overlap. Hence, for each 2D UV coordinate qı, we search its K-ncarest-neighher (R) K_. {q: 1km and penalize neighboring points that are too close, given by:
N Kr. max(0, c ||q: q (k) 112))
Lunwrap
(9)
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where the minimum distance value allowed between neighboring points is controlled by threshold
Wrapping Loss. The major supervision of the upper learning branch is that the generated point set P should roughly approximate the original 3D geometric structure represented by P. Since P and P not row-wisely corresponded, we use the commonly-usee Chamfer distance CD(2; *) measure point set similarity, as given by:
L Ewapp CD(P:
(10)
Here, there is no need impose the unwrapping loss over P. since we empirically find that optimizing the CD loss naturally suppresses clustered point distribution. Besides, we do not optimize CD Q: Q). since we observe degraded performances, possibly because in earlier training iterations Q and Q are far from optimal and thus interfere with each other.
Cycle Consistency Loss. Our bi-directional cycle mapping framework aims promote consistencies within both 3D and 2D domains, which are natural to learn thanks to our symmetric designs of the functionalities of sub-networks. Thus, the cycle consistency loss is formulated as:
laycle P P P | Qcyctell: +
(11)
where CS #; *) computes point- wise cosine similarity between Paycle and ground-trute truth normals P". which can be removed when ground-truth normals are not provided hard to compute.
Mapping Distortion Constraint. We regularize the distortion of the learned neural mapping function by exploiting differential surface properties, which can be conveniently deduced via the automatic differentiation mechanisms in common deep learning programming frameworks.
Throughout the bi-directional cycle mapping framework, 2D UV coordinates Q and Q are respectively mapped to 3D surface points P cyccle and P. For convenience, here we denote the two neural mapping functions as f : R R and g : IR? - R We compute the derivatives of f and g with respect to Q and Q at each 2D UV point (u, v)) to obtain the Jacobian matrices J E R and Jg E R3x2;
J = (fu fu). J, = (gu go)
(12)
where fur for Ju. Iu are dineesiimenii vectors of partial derivatives. Then we further deduce the eigenvalues of J and J Jg. which are denoted as (A}= X3) and (Ag, Ag). Thus, we can promote conformal (i.e., angle-preserving) parameterizations by constraining the following regularizer:
lcont Σ14 A3112 + Σ1M 2211. qEq
(13)
qEQ
4 Experiments
We collected series of 3D surface models with different types and complexities of geometric and/or topological structures for experimental evaluation. We made qualitative and quantitative comparisons with SLIM [29], a sfate-of-the-ar art and widely-used mesh parameterization algorithm. Additionally, we also compared our FAM with FBCP-PC [4] for unstructured and unoriented (i.e., without normals) 3D point cloud parameterization. Finally, we conducted ablation studies to demonstrate the effectiveness of the proposed bi-directional cycle mapping framework and the necessity of Cut-Net, together with different aspects verifications for comprehensively understanding the characteristics of our approach. Due to page limitations, detailed technical implementations and additional experimental results are presented in our Appendix
Comparison with SLIM [29]. Considering that SLIM is only able to be applied on disk-topologies or surfaces with open boundaries, we collected 8 such testing models for parameterization, using its officially-relessee code. For the training of our FAM, we uniformly sample 10, 000 points from the original mesh vertices at each optimization iteration. During testing, the whole vertex set is fed into our FAM to produce per-vertex UV coordinates. Then, we can perform checker-mai texture mappings for qualitative evaluation. For quantitative evaluation, we computed conformality metrics (the lower, the better) by measuring the average of the absolute angle differences between each corresponding angle of the 3D triangles and the 2D parameterized triangles. As shown Figure and Table 1, our FAM outperforms SLIM both qualitatively and quantitatively on all testing models.
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humansfax
hamorchand
hamat-head
Figure 3: Comparison of UV unwrapping and texture mapping results different (a) open surface models produced by (b) our FAM and (c) SLIM, where the 2D UV coordinates are color-coded by ground-truth point-wise normals to facilitate visualization.
Table Quantitative comparisons of our FAM and SLIM in terms of parameterization conformality.
Model human-face human head car-shell spiral human-hand shirt three-men camel-head
SLIM
0.635
0.254
0,411
0.114
0.609
0.443
0.645
0.349
FAM
0.074
0.094
0.037
0.087
0.145
0.166
0.162
0.088
Table 2: Quantitative conformality metrics of our parameterization results.
Model
brain
cow
fandisk
human-body
lion
nail
nefertiti
Conf. Metric
0.263
0.210
0.105
0.198
0.192
0.162
0.117
Model
bucket
dragon
mobius-strip
rocker-arm
torus- three-holes
fertility
Conf. Metric
0.179
0.311
0.062
0.143
0.126
0.142
0.166
Table 3: Conformality metrics of our FAM and FBCP-PC for point cloud parameterization.
Model
cloth-pts (#Pts=7K)
julius-pts (#Pts=11K) spiral-pts (#Pts=28K)
FBCP-PC
0.021
0.019
0.023
FAM
0.037
0.058
0.117
Parameterization with Different Geometric and Topological Complexities. We conducted more comprehensive experimental evaluations, as displayed in Figure and Table 2, showing the universal- ity and robustness of our approach.
Comparison with FBCP-PC [4]. Since FAM operates on discrete surface-sampled points without any dependence on connectivity information, applying our approach to parameterize unstructured and unoriented 3D point clouds is straightforward and basically seamless. The only modification to remove the supervision of point-wise normals, i.e., the last term CS(P", P P in Eqn. (11). As compared in Figure and Table 3, our performances are not better than FBCP-PC. However, we must point out that FBCP-PC requires manually specifying indices of boundary points (arranged in order) additional inputs. Hence, the comparisons are actually quite unfair to us.
Ablation Studies. We evaluated the necessity of our two-branch joint learning architecture. First, removed the upper 2D-3D branch to show the resulting Q. Second, we removed the bottom 3D-2D-+3D branch to show UV coordinates Q by performing ncarest-neighbor matching between P and P. As illustrated in Figure 6, removing any of the two branches will cause different degrees of performance degradation. Furthermore, we verified the necessity of Cut-Net in the whole learning pipeline. As shown in Figure. 7, removing Cut-Net does not lead to obvious performance degradation for models that are simpler to open (e.g., human-fuce and mobius-srip), yet for the other complex models the removal of Cut-Net causes highly-distortes surface flattening. Without learning offsets, the inherent smoothness of neural works can impede "tearing" the originally-continoorr areas on the 3D surface, thus hindering the creation of cutting seams.

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Braiin
bucket
dragon
COU
fandiak
human body
rocker-amm
torus-
lion
three-hole
nei
nefertit
fertility
Figure 4: Display of surface parameterization results produced by our FAM. (a) input 3D models; (b) learned UV coordinates; (c) texture mappings; (d) learned cutting seams.
only points (without normals) Inputs
additional inputss required: orlented boundory Indices
dc
[b]
(d)
clothpps
Jullus-pts
spirat-pts
cloth-pts
jullus-pts
spirol-pts
Figure 5: Point cloud parameterization achieved by our FAM (left) and FBCP-PC (right).
In addition, we conducted quantitative evaluations self- siff-intersection. Given a triangular mesh, we check if any pair of triangles overlaps in the UV space, and then measure the proportion of overlapped pairs to the total amount of triangle pairs. As reported in Table 4, self-intersection are inevitable both in FAM and SLIM, but the proportions of seff-intersected triangles hard to apply post-processing refinement to slightly adjust the distribution UV coordinates to further relieve or even eliminate self-intersection issues. For point cloud parameterization, we evaluated our robustness to noises in Figure 8, where we can observe that FAM shows stable performances to noisy input conditions (with 1%, 2%, and 4% Gaussian noises for point position perturbation).
very small. In practice, it
Finally, we performed stress tests on a Hilbert-space-flle filli-iiiiinn----eee filling-shaped cylinder model in Figure 9a. It is observed that FAM obtains the basically optimal solution. Still, shown in Figure 9b, 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 generally reasonable and texture mapping looks relatively regular from outside, is hard to deal with the complicated interior structures.

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torus-double
Figure Ablation results produced by (a) our complete FAM framework, (b) FAM without the upper
2D >DD learning branch, (c) FAM without the bottom 3D->2D->> learning branch.
mobina dass
humon -anan-aann
thvee-holes
hone
bucdet
(b]
le]
Figure 7: Ablation study on the necessity of Cut-Net: (a) input meshes; (b) results obtained from our full FAM architecture; (c) results obtained by removing the Cut-Net component.
Testing Models
FAM
SLIM
Open-Surface Models (as in Figure 3)
clean
0.156%
0.133%
Higher-Gest Genus Models (as in Figure 4)
0.204%
N/A
(d)
Table 4: Quantitative self-intersection metrics of Figure 8: Applying FAM to point clouds added our parameterization results.
with different levels of Gaussian noises.
hsf-cylinder
[c]
shapenet-car
(a) Highly challenging stress tests of our FAM.
(b) Failure cases of complicated CAD models. Figure (a) input meshes; (b) UV maps; (c): texture mappings and learned cutting seams.
Conclusion
We proposed FAM, the first neural surface parameterization approach targeted at global free-boundary parameterization. Our approach is universal for dealing with different geometric/topologiaal complex- ities regardless of mesh triangulation quality and even applicable unstructured point clouds. More importantly, our approach automatically learns appropriate cutting seams along the target 3D surface, and adaptively deforms the 2D UV parameter domaín. As an unsupervised learning framework, our FAM shows significant potentials and practical values. The current technical implementations still have several aspects of limitations. Our working mode of per-model overfitting cannot exploit existing UV unwrapping results (despite their limited amounts) as ground- -tunns for training generalizable neural models. Besides, series more advanced properties, such as shape symmetry, cutting seam visibility, seamless parameterization, have not yet been considered.
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A Appendix
A.1
Implementation Details
In our bi-directional cycle mapping framework, all sub-networks are architecturally built upon stacked MLPs without batch normalization. We uniformly configured LeakyReI.U non-linearities with the negative slope of 0.01, except for the output layer. Within the Deform-Net Ed and Ed/ are four-layer MLPs with channels [2, 512, 512, 512, 64] and [66, 512, 512, 512, 2]. Within the Wrap-Net, E and Sw are four-layer MLPs with channels [2, 512, 512, 512, 64] and [66, 512, 512, 512, 6]. Within the Cut-Net, C. and < are three-layer MLPs with channels [3, 512, 512, 64 and [67, 512, 512,3). Within the Unwrap-Nets &u is implemented as three-layer MLPs with channels [3. 512, 512, 2/. In addition network structures, there is also set hyperparameters be configured and tuned. As presented in Eqn. (8) for cutting seam extraction, we chose K = 3. Suppose that, at the current training iteration, the side length of the square bounding box of 2D UV coordinates Q is denoted as (QQ).. Then we set the threshold Teut to be 2% of L Q). Besides, as presented in Eqn. (9) for computing the 0.2 L(Q)/VN and K, = 8. When formulating
unwrapping loss, we choose the threshold the overall training objective, the weights for funwraps Ewraps leycle. and lcont are set as 0.01, 1.0, 0.01, and 0.01. All our experiments are conducted on single NVIDIA GeForce RTX 3090 GPU.
Conf. Metric: 0.089
(a)
given cut
(b)
learned cut
Conf. Metric: 0.117
Figure 10: Experiments the nefertiti testing model whose topology is not homeomorphic disk. (a) The parameterization results of SLIM are obtained by manually-speciffing a high-quality cutting seam, with the conformality metric of 0.089. (b) The parameterization results of our FAM, in which the cutting seam automatically learned, with the conformality metric of 0.117.
Table Optimization time costs (minutes) of our FAM and SLIM.
Model
human-face human-heaad car-shell spiral
human-hand
shirt
three-men
camel-head
SLIM
39 min
38 min
19 min
25 min
28 min
31 min
17 min
40 min
FAM
around 18 min (basically unchanged for different models)
A.2
Discussions about Global and Multi-Chart Surface Parameterization
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 distor- tion 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 worth emphasizing that only obtaining chart-wise 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 known as highly non-trivial problem, which is basically skipped in recent neural learning approaches, such DiftSR and Nuvo. Still, local parameterization does have its suitable application scenarios for processing highly-complicated sur- faces such typical ShapeNet-style CAD meshes with rich interior structures and severe multi-layer
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(o)
Figure 11: Results of Nuvo: (a) input meshes (different colors denote chart assignment); (b) chart- wise UV maps.
Figure 12: Results of OptCuts: (a) input meshes; (b) UV maps.
issues, and per-chart distortion can be reduced since the geometry and topology of cropped surface patches become simpler.
A.3 Comparison with SLIM on Non-Disk-Type Surfaces with Manually- -Seecilt--peeieied Cuts
In the main manuscript, our experimental comparisons with SLIM focus on 3D models with open boundaries and/or disk topologies, on which SLIM be applied. Here, we made additional efforts to conduct comparisons with SLIM on the non-disk-type 3D surface model of nefertiti by manually specifying an optimal cutting seam for SLIM to run. Note that this is quite unfair experimental setting, because finding optimal cuts is known to be critical yet complicated task. As illustrated in Figure 10, our approach still achieves highly competitive performances compared with SLIM even under such an unfair comparison setup.
A.4 Running Efficiency
In addition parameterization quality, we further compared the time costs of our FAM and SLIM for training and optimization. Here, the official code of SLIM runs on the Intel(R) Core(TM) i7-9700 CPU. Table 5 lists the time costs for different testing models. Note that, since our FAM operates on surface-samples sampled points and we uniformly used the same number of input points (i.e., N), the time costs of our FAM basically maintain unchanged for different testing models. It turns out that our approach achieves satisfactory learning efficiency.
A.5 Comparison with Nuvo [36]
We provided typical parameterization results of Nuvo using third-party implementation available at https:/ / /ttg//gttttbbbt wwwccoooiiiiieeeeeeeeoo http//riiiiiii/////e /Nuvo, as presented in Figure 11. We can observe that its chart assignment capability is still not stable. It often occurs that some -iitally-discoomeeted 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 valid packing. real-sense packing should adaptively adjust the positions, poses, and sizes of each UV domain via combinations of translation, rotation, and scaling.
A.6 Comparison with OptCuts [16]
We provided typical parameterization results of OptCuts, as presented in Figure 12. Comparatively, 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 casier for implementing and tuning), and parameterization smoothness (not limited mesh vertices). Although OptCuts is able to jointly obtain reasonable surface cuts and UV unwrapping results, its performances are generally sub-optimal compared with ours.
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21


Reviews:
[{"review_id": "gD7SbdynJ1", "reviewer": "Reviewer_eo5W", "summary": "This paper proposes a novel neural-network based optimization framework for obtaining surface parameterization of arbitrary 3D meshes.\nComparing to traditional methods like SLIM, this work can work for possibly low quality meshes of arbitrary topology.\n\nThe core of the proposed pipeline are 4 simple MLPs: \nGiven a point cloud possibly sampled from a 3D mesh, (1) the Deform-Net first deforms a square 2D uv grid to an irregular 2D shape with free boundaries to facilitate lowering the parameterization distortion. Then (2) the Wrap-Net lifts the deformed 2D grid to 3D to match the shape of the input point cloud. (3) The Cut-Net then tries to deform the lifted points to cut the shape to a disk topology. After this (4) the Unwrap-Net further flattens the points in the 2D uv plane.\n\nTo better train the model, the paper also proposes an interesting cycle-consistency loss that aligns the 2D->3D->2D process and 3D->2D->3D process.", "questions": "What is the actual advantage of this work over multi-chart based approach like Nuvo?\n\nWhat would be the case that the proposed method fails but the traditional methods like SLIM work?\nFor example, the Fig. 10 of SLIM shows an example of Tutte’s embedding as a stress test. It works very well. Does the proposed method also work for this case? If not, why?\n\nWhy Cut-net is a neccessary component?", "limitations": "The authors discussed the limitation in tthe conclusion session.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- **Simple method to solve a challenging problem**.\nTraditional algorithms usually involve designing sophisticated energy functions where this method is straight forward and effective. \nIt seems the method is also easiler to implement using python deep learning libraries than the traditional methods that typically requires C++ programming.\n\n\n- **Good results and support arbitrary topology**. The proposed method can be applied to arbitrary shapes and achieve similar or better results than traditional methods like SLIM. It also works for shapes with very high genus (i.e. the 'complex topology' example in Fig. 1).", "weaknesses": "- **Advantages over multi-chart methods are unclear**. \nAlthough this work targets at a global parameterization, its advantage over multi-chart based local parameterization is not thoroughly discussed. This is especially important to convince people that this work is indeed more useful than works liek NUVO in some scenarios.\nTo me, the NUVO method already does a pretty good job at unsupervised learning for surface parameterization using neural networks. Although there is only a third-party implementation available (https://github.com/ruiqixu37/Nuvo), I believe some qualitative analysis is needed to highlight the pros/cons of the two different paradigms.\n\n- **Missing discussion on the very related work of  'OptCuts', which is also a baseline of Nuvo.**\n\n- **It's unclear how the Cut-net is effective.**\nThe purpose of cut-net is to get the seam, but the network design is just a simple MLP to deform the lifted 3D points. Also, there are no losses functions specifically designed for the cut-net. It is unclear how the network will do a reasonable cut to the surface. So I doubt that it is actually a neccessary component. Maybe the Unwrap-net along is good enough.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "### **[Rebuttal to Reviewer eo5W]**\n\n\n### **[W1\\&Q1]** *Discussing the advantages of global parameterization over multi-chart parameterization (e.g., Nuvo).*\n\n**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).\n\nFor 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.\n\nOriginally, 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.\n\n### **[W2]** *Discussing the highly-related work of OptCuts ([Li et al., TOG 2018]).*\n\n**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).\n\nIn 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).\n\n### **[W3\\&Q3]** *Clarifying the working mechanism and necessity of Cut-Net.*\n\n**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.\n\nActually, 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.\n\nWe 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.\n\n### **[Q2]** *Stress test and potential failure cases.*\n\n**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.\n\nAs 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.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "YtOilxI4nZ", "reviewer": "Reviewer_4cq8", "summary": "This paper introduces the Flatten Anything Model (FAM), an unsupervised neural architecture designed to achieve global free-boundary surface parameterization by learning point-wise mappings between 3D points on the target geometric surface and adaptively-deformed UV coordinates within the 2D parameter domain. The proposed FAM incorporates specific functionalities such as surface cutting, UV deforming, unwrapping, and wrapping, which are integrated into a bi-directional cycle mapping framework.\n\nThe FAM offers the following features:\n1. It directly operates on discrete surface points and jointly learns surface cutting.\n2. It exploits the inherent smoothness property of neural networks to learn arbitrary point-to-point mappings.\n3. It learns the global parameterization.\n4. It learns the free-boundary surface parameterization.", "questions": "1. Do self-intersections exist in the results produced by FAM when the topology connectivity is recovered?\n2. In real-world data, point clouds are often noisy. Can FAM handle this type of point cloud data effectively?\n3. Global parameterization with a regular boundary can be applied in image or surface registration. However, if the boundary is free, it appears to reduce the potential application scenarios. Additionally, global parameterization may introduce more distortion than multi-charts parameterization. What are the advantages of global parameterization with a free boundary in FAM?", "limitations": "Yes, in the conclusion section, the authors discuss the limitations of the proposed model.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The quality and clarity is good enough, it is easy to understand the motivation, outline and the proposed method.\n\nThe proposed FAM operates directly on the point cloud instead of a mesh, thereby significantly reducing the stringent requirements for mesh quality. Additionally, the FAM can automatically compute cutting seams for meshes with complex topologies, eliminating the need for pre-cutting, which can be challenging to compute.", "weaknesses": "The work does not evaluate self-intersection, which is a concern for users in downstream geometry processing applications.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "### **[Rebuttal to Reviewer 4cq8]**\n\n### **[W1\\&Q1]** *Evaluation of self-intersection.*\n\n**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.\n\n### **[Q2]** *Handling noisy point clouds.*\n\n**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.\n\n### **[Q3]** *1) Advantages of free boundary over fixed regular boundary; 2) Advantages of global parameterization over multi-chart parameterization.*\n\n**Response:** We are sorry for not adequately explaining these issues in our paper. Here we will present more detailed discussions.\n\n1) 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.\n\n2) 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.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "5kuzVgS95x", "reviewer": "Reviewer_Hnxu", "summary": "This paper proposes an unsupervised neural surface parameterization method, named FAM, which maps 3D surface points to adaptively deformed UV coordinates in the 2D parameter domain. Inspired by the actual physical procedures, the neural architecture includes several sub-networks for surface cutting, UV deforming, unwrapping, and wrapping, respectively. These sub-networks are assembled into a bi-directional cycle mapping framework, with carefully designed loss functions as constraints. The proposed FAM is the first method to achieve global free-boundary parameterization.", "questions": "Please refer to “Weaknesses” part for suggestions.", "limitations": "The authors have clearly demonstrated the limitations of this work.", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. The proposed FAM is the first method to achieve global free-boundary parameterization. Compared to existing methods, FAM can 1) directly deal with global parameterization, without needing manual effort for surface cutting; 2) map 3D points to an adaptively deformed 2D domain with free boundary, thus reducing distortions.\n\n2. The design of several sub-networks and objective functions make sense to me.\n\n3. The extracted cutting seams of the surface look reasonable.", "weaknesses": "1. Missing analysis about the Cut-Net. From my point of view, Cut-Net is a very important module as it enables the joint learning of surface cutting without any supervision or manual effort. However, this module is not analyzed in ablation studies. Specifically, I am curious about whether this Cut-Net can be removed by directly mapping 3D surface points to 2D UV coordinates with the Unwrap-Net. If not, why can cutting seams be learned without any constraints on the intermediate P_cut?\n\n2. In experiments on surfaces with disk-topologies or open boundaries, only one baseline SLIM is evaluated. I wonder why the authors do not evaluate the more recent DiffSR and Nuvo, since these local parameterization methods should also be applicable to such surfaces.\n\n3. In experiments on surfaces with more complex topologies, only one baseline FBCP-PC is evaluated. The authors should also try to include other recent methods, even though they need manual surface cutting. Despite the unfair comparison, we can still see the gap and potential of the proposed automatic surface cutting. Besides, the authors can brute-forcely apply local parameterization methods without providing manual surface cutting if possible, therefore further demonstrate the advantages of the proposed automatic surface cutting.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "### **[Rebuttal to Reviewer Hnxu]**\n\n\n### **[W1]** *Clarifying the working mechanism and necessity of Cut-Net.*\n\n**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.\n\nActually, 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.\n\nWe 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.\n\n### **[W2]** *Evaluating multi-chart local parameterization.*\n\n**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).\n\nFor 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.\n\nStill, 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.\n\n\n### **[W3]** *Experimenting with complex-topology models with manual surface cutting.*\n\n**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. \n\nIn 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. \n\nAs 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.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "oQr3CleN9Y", "reviewer": "Reviewer_FN2a", "summary": "The authors introduce the Flatten Anything Model (FAM), an unsupervised tool for UV mapping 3D geometries. FAM consists of four main components:\n\n1. **Deform-Net**: Adjusts 2D points in an input grid.\n2. **Warp-Net**: Converts these 2D points into 3D space.\n3. **Cut-Net**: Predicts a seam along which the 3D mesh is cut for unwrapping.\n4. **Unwrap-Net**: Projects the cut 3D points back into 2D space.\n\nSince the unsupervised model uses two-cycle consistency losses to ensure the mappings from 2D to 3D and back remain consistent. During inference, the UV mapping process utilizes the Cut-Net and Unwrap-Net modules in sequence. Unlike similar approaches, FAM generates a single, easily usable unwrapped patch and can also work with unstructured point clouds. It outperforms selected baselines on 3D mesh tasks.", "questions": "I reiterate the suggestions mentioned in the weakness for self-consistency:\n> Apart from that observation, it's challenging to identify any major flaws in this work. The resolution and clarity of some figures could be enhanced. For instance, the teaser dedicates excessive space to phrases like \"complex topology\". Instead, enlarging the independent figures would be beneficial. Moreover, Tables 1-3 should clarify whether the metrics are intended to be minimized or maximized. It is important to note that this comment does not affect my overall evaluation.\n\nDiscussing the potential applications of this method, particularly for certain types of objects or its applicability to meshes derived from NeRFs [1] or NeuS [2], which are inherently noisy, would be highly beneficial.\n\n[1] Mildenhall B, Srinivasan PP, Tancik M, Barron JT, Ramamoorthi R, Ng R. Nerf: Representing scenes as neural radiance fields for view synthesis. Communications of the ACM. 2021 Dec 17;65(1):99-106.\n[2] Wang P, Liu L, Liu Y, Theobalt C, Komura T, Wang W. Neus: Learning neural implicit surfaces by volume rendering for multi-view reconstruction. arXiv preprint arXiv:2106.10689. 2021 Jun 20.", "limitations": "The authors have not addressed the limitations of their method. It would be beneficial to include a separate paragraph, possibly in the supplementary materials, detailing the types of objects to which FAM is applicable, including its suitability for simpler CAD objects.", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "- The paper is well-organized and the narrative is clear, with each component having a clear purpose. The authors' efforts to replicate the standard procedures for creating a UV map are commendable.\n- The paper positions itself effectively within the current body of research. Initially, I noticed an absence of comparison with cutting SOTA methods such as Nuvo (or the widely-used xatlas). Nonetheless, the paper justifies this by clearly explaining why a comparison is not feasible.\n- Regarding performance, FAM outshines SLIM and matches FBCP-PC, which is designed specifically for point clouds, unlike FAM.\n- Dividing the pipeline into separate, understandable components simplifies the process of tuning the model and integrating it with different methodologies.", "weaknesses": "- One can argue with the statement \"... global parametrization, a more valuable yet much harder problem setting\". Such a parametrization serves a different purpose than real-world applications.  For instance, implementing this parametrization on a car or any technical CAD object is impractical, whereas an atlas of patches is more applicable.\n- Apart from that observation, it's challenging to identify any major flaws in this work. The resolution and clarity of some figures could be enhanced. For instance, the teaser dedicates excessive space to phrases like \"complex topology\". Instead, enlarging the independent figures would be beneficial. Moreover, Tables 1-3 should clarify whether the metrics are intended to be minimized or maximized. It is important to note that this comment does not affect my overall evaluation.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "### **[Rebuttal to Reviewer FN2a]**\n\n### **[W1]** *Inaccurate claim about the settings of global and local parameterization.*\n\n**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.\n\n### **[W2]** *Enhancing the presentation quality of some figures and table contents.*\n\n**Response:** We are sorry for the figure and table issues. We will carefully fix these problems and enhance the presentation quality.\n\n### **[Q1\\&L1]** *Discussing the applicability of FAM.*\n\n**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.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
### **[Global Response]**

We sincerely thank all reviewers for their time and efforts in reviewing our paper, providing constructive comments and valuable suggestions. We are very grateful to reviewers' positive acknowledgment of this work:

-- Reviewer eo5W thinks that our approach is interesting and novel, simple but effective to solve a challenging problem with good results. 

-- Reviewer 4cq8 recognizes the clarity and presentation quality of our manuscript, and thinks that our approach is well-motivated and effective to reduce the stringent requirements for input data quality and to find good surface cuts. 

-- Reviewer Hnxu points out our efforts as the first global free-boundary parameterization method and recognizes our technical soundness and reasonable surface cutting. 

-- Reviewer FN2a thinks that our work is well-organized and well-motivated with clear narrative and good performance, and it is commendable to replicate standard procedures for UV unwrapping.



During the rebuttal period, we made the following efforts to address all the raised concerns for further improving the quality of this work:

-- 1) We clarified the working mechanism of Cut-Net and supplemented ablation studies to verify its necessity and effectiveness.

-- 2) We explained the advantages of global parameterization over multi-chart parameterization and supplemented the results of the latest representative work of Nuvo.

-- 3) We explained the advantages of performing free-boundary parameterization over pursuing fixed regular (i.e., rectangular) boundary.

-- 4) We discussed the highly-related work of OptCuts and supplemented its parameterization results.

-- 5) We conducted stress tests on highly-complicated testing models and analyzed potential failure cases of our approach.

-- 6) We experimented with noisy point clouds to demonstrate the robustness of our approach.

-- 7) We supplemented quantitative evaluations of self-intersection issues.

-- 8) We rectified inaccurate claims made in the manuscript and supplemented more explanations to the applicability of our approach.

For convenience, below we will briefly summarize each raised **Weakness (W)**, **Question (Q)**, and **Limitation (L)**, and provide the corresponding response item by item.

All the newly-added discussions, experiments, and analyses will be supplemented into the final version of our paper.

Thanks again for your time and effort in our submission. We appreciate any further questions and discussions.

Meta-review: The reviewers unanimously agree that the paper's novel neural network-based optimization framework for obtaining surface parameterization of arbitrary 3D meshes is a valuable contribution to 3D computer vision research.

The reviewers posed insightful clarifying questions (e.g., regarding issues with Cut-Net, advantages over multi-chart methods, and evaluations on challenging cases), and the authors' rebuttal provided illuminating and pertinent answers. All reviewers have expressed positive opinions about the submission, with higher scores following the rebuttal.

Given the quality of the work and the authors' satisfactory responses, I recommend accepting this paper as a poster. However, it is crucial that the authors address the concerns raised during the discussion and incorporate the promised revisions in the final version of the paper. This will ensure that the published work reflects the clarifications and improvements discussed during the review process.
Decision: Accept (poster)