1 Introduction
Significant progress has been made on learning good representations for images, allowing impressive applications in image generation [17, 35]. However, learning a representation for generating highresolution 3D shapes remains an open challenge. Representing a shape as a volumetric function [7, 13, 31] only provides voxelscale sampling of the underlying smooth and continuous surface. In contrast, a point cloud [25, 26] provides a representation for generating onsurface details [9], efficiently leveraging sparsity of the data. However, points do not directly represent neighborhood information, making it difficult to approximate the smooth lowdimensional manifold structure with high fidelity.
To remedy shortcomings of these representations, surfaces are a popular choice in geometric modeling. A surface is commonly modeled by a polygonal mesh: a set of vertices, and a list of triangular or quad primitives composed of these vertices, providing piecewise planar approximation to the smooth manifold. Each mesh vertex contains a 3D (XYZ) coordinate, and, frequently, a 2D (UV) embedding to a plane. The UV parameterization of the surface provides an effective way to store and sample functions on surfaces, such as normals, additional geometric details, textures, and other reflective properties such as BRDF and ambient occlusion. One can imagine converting point clouds or volumetric functions produced with existing learned generative models as a simple postprocess. However, this requires solving two fundamental, difficult, and longstanding challenges in geometry processing: global surface parameterization and meshing.
In this paper we explore learning the surface representation directly. Inspired by the formal definition of a surface as a topological space that locally resembles the Euclidean plane, we seek to approximate the target surface locally by mapping a set of squares to the surface of the 3D shape. The use of multiple such squares allows us to model complex surfaces with nondisk topology. Our representation of a shape is thus extremely similar to an atlas, as we will discuss in Section 3. The key strength of our method is that it jointly learns a parameterization and an embedding of a shape. This helps in two directions. First, by ensuring that our 3D points come from 2D squares we favor learning a continuous and smooth 2manifold structure. Second, by generating a UV parameterization for each 3D point, we generate a global surface parameterization, which is key to many applications such as texture mapping and surface meshing. Indeed, to generate the mesh, we simply transfer a regular mesh from our 2D squares to the 3D surface, and to generate a regular texture atlas, we simply optimize the metric of the square to become asisometricaspossible to the corresponding 3D shape (Fig. LABEL:fig:teaser_fig).
Since our work deforms primitive surface elements into a 3D shape, it can be seen as bridging the gap between the recent works that learn to represent 3D shapes as a set of simple primitives, with a fixed, low number of parameters [32] and those that represent 3D shapes as an unstructured set of points [9]
. It can also be interpreted as learning a factored representation of a surface, where a point on the shape is represented jointly by a vector encoding the shape structure and a vector encoding its position. Finally, it can be seen as an attempt to bring to 3D the power of convolutional approaches for generating 2D images
[17, 35] by sharing the network parameters for parts of the surface.Our contributions. In this paper:

We propose a novel approach to 3D surface generation, dubbed AtlasNet
, which is composed of a union of learnable parametrizations. These learnable parametrizations transform a set of 2D squares to the surface, covering it in a way similar to placing strips of paper on a shape to form a papiermâché. The parameters of the transformations come both from the learned weights of a neural network and a learned representation of the shape.

We show that the learned parametric transformation maps locally everywhere to a surface, naturally adapts to its underlying complexity, can be sampled at any desired resolution, and allows for the transfer of a tessellation or texture map to the generated surface.

We demonstrate the advantages of our approach both qualitatively and quantitatively on high resolution surface generation from (potentially low resolution) point clouds and 2D images

We demonstrate the potential of our method for several applications, including shape interpolation, parameterization, and shape collections alignment.
All the code is available at the project webpage^{1}^{1}1https://github.com/ThibaultGROUEIX/AtlasNet..
2 Related work
3D shape analysis and generation has a long history in computer vision. In this section, we only discuss the most directly related works for representation learning for 2manifolds and 3D shape generation using deep networks.
Learning representations for 2manifolds. A polygon mesh is a widelyused representation for the 2manifold surface of 3D shapes. Establishing a connection between the surface of the 3D shape and a 2D domain, or surface parameterization, is a longstanding problem in geometry processing, with applications in texture mapping, remeshing, and shape correspondence [15]. Various related representations have been used for applying neural networks on surfaces. The geometry image representation [11, 28] views 3D shapes as functions (e.g., vertex positions) embedded in a 2D domain, providing a natural input for 2D neural networks [29]. Various other parameterization techniques, such as local polar coordinates [23, 5] and global seamless maps [22]have been used for deep learning on 2manifolds. Unlike these methods, we do not need our input data to be parameterized. Instead, we learn the parameterization directly from point clouds. Moreover, these methods assume that the training and testing data are 2manifold meshes, and thus cannot easily be used for surface reconstructions from point clouds or images.
Deep 3D shape generation. Nonparametric approaches retrieve shapes from a large corpus [2, 21, 24], but require having an exact instance in the corpus. One of the most popular shape representation for generation is the voxel representation. Methods for generating a voxel grid have been demonstrated with various inputs, namely one or several images [7, 10], full 3D objects in the form of voxel grids [10, 34], and 3D objects with missing shape parts [34, 12]. Such direct volumetric representation is costly in term of memory and is typically limited to coarser resolutions. To overcome this, recent work has looked at a voxel representation of the surface of a shape via octtrees [13, 27, 31]. Recently, Li et al. also attempted to address this issue via learning to reason over hierarchical procedural shape structures and only generating voxel representations at the part level [20]. As an alternative to volumetric representations, another line of work has learned to encode [25, 26] and decode [9] a 3D point representation of the surface of a shape. A limitation of the learned 3D point representation is there is no surface connectivity (e.g., triangular surface tessellation) embedded into the representation.
Recently, Sinha et al. [30]
proposed to use a spherical parameterization of a single deformable mesh (if available) or of a few base shapes (composed with authalic projection of a sphere to a plane) to represent training shapes as parameterized meshes. They map vertex coordinates to the resulting UV space and use 2D neural networks for surface generation. This approach relies on consistent mapping to the UV space, and thus requires automatically estimating correspondences from training shapes to the base meshes (which gets increasingly hard for heterogeneous datasets). Surfaces generated with this method are also limited to the topology and tessellation of the base mesh. Overall, learning to generate surfaces of arbitrary topology from unstructured and heterogeneous input still poses a challenge.
3 Locally parameterized surface generation
In this section, we detail the theoretical motivation for our approach and present some theoretical guarantees.
We seek to learn to generate a surface of a 3D shape. A subset of is a 2manifold if, for every point , there is an open set in and an open set in containing such that is homeomorphic to . The set homeomorphism from to is called a chart, and its inverse a parameterization. A set of charts such that their images cover the 2manifold is called an atlas of the 2manifold. The ability to learn an atlas for a 2manifold would allow a number of applications, such as transfer of a tessellation to the 2manifold for meshing and texture mapping (via texture atlases). In this paper, we use the word surface in a slightly more generic sense than 2manifold, allowing for selfintersections and disjoint sets.
We consider a local parameterization of a 2manifold and explain how we learn to approximate it. More precisely, let us consider a 2manifold , a point and a parameterization of in a local neighborhood of . We can assume that is defined on the open unit square by first restricting to an open neighborhood of with disk topology where it is defined (which is possible because is continuous) and then mapping this neighborhood to the unit square.
We pose the problem of learning to generate the local 2manifold previously defined as one of finding a parameterizations with parameters which map the open unit 2D square to a good approximation of the desired 2manifold . Specifically, calling , we seek to find parameters minimizing the following objective function,
(1) 
where is a loss over 2manifolds, is a regularization function over parameters , and is a scalar weight. In practice, instead of optimizing a loss over 2manifolds , we optimize a loss over point sets sampled from these 2manifolds such as Chamfer and EarthMover distance.
One question is, how do we represent the functions ? A good family of functions should (i) generate 2manifolds and (ii) be able to produce a good approximation of the desired 2manifolds
. We show that multilayer perceptrons (MLPs) with rectified linear unit (ReLU) nonlinearities almost verify these properties, and thus are an adequate family of functions. Since it is difficult to design a family of functions that always generate a 2manifold, we relax this constraint and consider functions that locally generate a 2manifold.
Proposition 1.
Let be a multilayer perceptron with ReLU nonlinearities. There exists a finite set of polygons , such that on each is an affine function: , where are matrices. If for all , , then for any point in the interior of one of the there exists a neighborhood of such that is a 2manifold.
Proof.
The fact that is locally affine is a direct consequence of the fact that we use ReLU nonlinearities. If the inverse of is well defined on the surface and continuous, thus the image of the interior of each is a 2manifold. ∎
To draw analogy to texture atlases in computer graphics, we call the local functions we learn to approximate a 2manifold learnable parameterizations and the set of these functions a learnable atlas. Note that in general, an MLP locally defines a rank 2 affine transformation and thus locally generates a 2manifold, but may not globally as it may intersect or overlap with itself. The second reason to choose MLPs as a family is that they can allow us to approximate any continuous surface.
Proposition 2.
Let be a 2manifold that can be parameterized on the unit square. For any there exists an integer such that a multilayer perceptron with ReLU non linearities and hidden units can approximate with a precision .
Proof.
This is a consequence of the universal representation theorem [16] ∎
In the next section, we show how to train such MLPs to align with a desired surface.
4 AtlasNet
In this section we introduce our model, AtlasNet, which decodes a 3D surface given an encoding of a 3D shape. This encoding can come from many different representations such as a point cloud or an image (see Figure LABEL:fig:teaser_fig for examples).
4.1 Learning to decode a surface
Our goal is, given a feature representation for a 3D shape, to generate the surface of the shape. As shown in Section 3, an MLP with ReLUs with parameters can locally generate a surface by learning to map points in to surface points in . To generate a given surface, we need several of these learnable charts to represent a surface. In practice, we consider learnable parameterizations for . To train the MLP parameters , we need to address two questions: (i) how to define the distance between the generated and target surface, and (ii) how to account for the shape feature in the MLP? To represent the target surface, we use the fact that, independent of the representation that is available to us, we can sample points on it. Let be a set of points sampled in the unit square and a set of points sampled on the target surface. Next, we incorporate the shape feature by simply concatenating them with the sampled point coordinates before passing them as input to the MLPs. Our model is illustrated in Figure 0(b). Notice that the MLPs are not explicitly prevented from encoding the same area of space, but their union should cover the full shape. Our MLPs do depend on the random initialization, but similar to convolutional filter weights the network learns to specialize to different regions in the output without explicit biases. We then minimize the Chamfer loss between the set of generated 3D points and ,
(2) 
4.2 Implementation details
We consider two tasks: (i) to autoencode a 3D shape given an input 3D point cloud, and (ii) to reconstruct a 3D shape given an input RGB image. For the autoencoder, we used an encoder based on PointNet [25], which has proven to be state of the art on point cloud analysis on ShapeNet and ModelNet40 benchmarks. This encoder transforms an input point cloud into a latent vector of dimension . We experimented with input point clouds of 250 to 2500 points. For images, we used ResNet18 [14]
as our encoder. The architecture of our decoder is 4 fullyconnected layers of size 1024, 512, 256, 128 with ReLU nonlinearities on the first three layers and tanh on the final output layer. We always train with output point clouds of size 2500 evenly sampled across all of the learned parameterizations – scaling above this size is timeconsuming because our implementation of Chamfer loss has a compute cost that is quadratic in the number of input points. We experimented with different basic weight regularization options but did not notice any generalization improvement. Sampling of the learned parameterizations as well as the ground truth pointclouds is repeated at each training step to avoid overfitting. To train for singleview reconstruction, we obtained the best results by training the encoder and using the decoder from the point cloud autoencoder with fixed parameters. Finally, we noticed that sampling points regularly on a grid on the learned parameterization yields better performance than sampling points randomly. All results used this regular sampling.
4.3 Mesh generation
The main advantage of our approach is that during inference, we can easily generate a mesh of the shape.
Propagate the patchgrid edges to the 3D points. The simplest way to generate a mesh of the surface is to transfer a regular mesh on the unit square to 3D, connecting in 3D the images of the points that are connected in 2D. Note that our method allows us to generate such meshes at very high resolution, without facing memory issues, since the points can be processed in batches. We typically use 22500 points. As shown in the results section, such meshes are satisfying, but they can have several drawbacks: they will not be closed, may have small holes between the images of different learned parameterizations, and different patches may overlap.
Generate a highly dense point cloud and use Poisson surface reconstruction (PSR) [18]. To avoid the previously mentioned drawbacks, we can additionally densely sample the surface and use a mesh reconstruction algorithm. We start by generating a surface at a high resolution, as explained above. We then shoot rays at the model from infinity and obtain approximately 100000 points, together with their oriented normals, and then can use a standard oriented cloud reconstruction algorithm such as PSR to produce a triangle mesh. We found that high quality normals as well as high density point clouds are critical to the success of PSR, which are naturally obtained using this method.
Sample points on a closed surface rather than patches. To obtain a closed mesh directly from our method, without requiring the PSR step described above, we can sample the input points from the surface of a 3D sphere instead of a 2D square. The quality of this method depends on how well the underlying surface can be represented by a sphere, which we will explore in Section 5.1.
5 Results
In this section we show qualitative and quantitative results on the tasks of autoencoding 3D shapes and singleview reconstruction and compare against several baselines. In addition to these tasks, we also demonstrate several additional applications of our approach. More results are available in the supplementary material [1].
Data. We evaluated our approach on the standard ShapeNet Core dataset (v2) [6]. The dataset consists of 3D models covering 13 object categories with 1K10K shapes per category. We used the training and validation split provided by [7] for our experiments to be comparable with previous approaches. We used the rendered views provided by [7] and sampled 3D points on the shapes using [33].
Evaluation criteria. We evaluated our generated shape outputs by comparing to ground truth shapes using two criteria. First, we compared point sets for the output and groundtruth shapes using Chamfer distance (“CD”). While this criteria compares two point sets, it does not take into account the surface/mesh connectivity. To account for mesh connectivity, we compared the output and groundtruth meshes using the “Metro” criteria using the publicly available METRO software [8], which is the average Euclidean distance between the two meshes.
Points baseline. In addition to existing baselines, we compare our approach to the multilayer perceptron “Points baseline” network shown in Figure 0(a)
. The Points baseline network consists of four fully connected layers with output dimensions of size 1024, 512, 256, 7500 with ReLU nonlinearities, batch normalization on the first three layers, and a hyperbolictangent nonlinearity after the final fully connected layer. The network outputs 2500 3D points and has comparable number of parameters to our method with 25 learned parameterizations. The baseline architecture was designed to be as close as possible to the MLP used in AtlasNet. As the network outputs points and not a mesh, we also trained a second network that outputs 3D points and normals, which are then passed as inputs to Poisson surface reconstruction (PSR)
[18] to generate a mesh (“Points baseline + normals”). The network generates outputs in representing both the 3D spatial position and normal. We optimized Chamfer loss in this sixdimensional space and normalized the normals to 0.1 length as we found this tradeoff between the spatial coordinates and normals in the loss worked best. As density is crucial to PSR quality, we augmented the number of points by sampling 20 points in a small radius in the tangent plane around each point [18]. We noticed significant qualitative and quantitative improvements and the results shown in this paper use this augmentation scheme.








5.1 Autoencoding 3D shapes
In this section we evaluate our approach to generate a shape given an input 3D point cloud and compare against the Points baseline. We evaluate how well our approach can generate the shape, how it can generalize to object categories not seen during training, and its sensitivity to the number of patches.
Method  CD  Metro 

Oracle 2500 pts  0.85  1.56 
Oracle 125K pts    1.26 
Points baseline  1.91   
Points baseline + normals  2.15  1.82 (PSR) 
Ours  1 patch  1.84  1.53 
Ours  1 sphere  1.72  1.52 
Ours  5 patches  1.57  1.48 
Ours  25 patches  1.56  1.47 
Ours  125 patches  1.51  1.41 
Evaluation on surface generation. We report quantitative results for shape generation from point clouds in Table 1, where each approach is trained over all ShapeNet categories and results are averaged over all categories. Notice that our approach outperforms the Points baseline on both the Chamfer distance and Metro criteria, even when using a single learned parameterization (patch). Also, the Points baseline + normals has worse Chamfer distance than the Points baseline without normals indicating that predicting the normals decreases the quality of the point cloud generation.
We also report performance for two “oracle” outputs indicating upper bounds in Table 1. The first oracle (“Oracle 2500 pts”) randomly samples 2500 points+normals from the ground truth shape and applies PSR. The Chamfer distance between the random point set and the ground truth gives an upper bound on performance for pointcloud generation. Notice that our method outperforms the surface generated from the oracle points. The second oracle (“Oracle 125K pts”) applies PSR on all 125K points+normals from the groundtruth shape. It is interesting to note that the Metro distance from this result to the ground truth is not far from the one obtained with our method.
We show qualitative comparisons in Figure 2. Notice that the PSR from the baseline point clouds (Figure 1(b)) look noisy and lower quality than the meshes produced directly by our method and PSR performed on points generated from our method as described in Section 4.3 (Figure 1(c)).
Sensitivity to number of patches. We show in Table 1 our approach with varying number of learnable parameterizations (patches) in the atlas. Notice how our approach improves as we increase the number of patches. Moreover, we also compare with the approach described in Section 4.3 which samples points on the 3D unit sphere instead of 2D patches to obtain a closed mesh. Notice that sampling from a sphere quantitatively outperforms a single patch, but multiple patches perform better.
We show qualitative results for varying number of learnable parameterizations in Figure 2. As suggested by the quantitative results, the visual quality improves with the number of parameterizations. However, more artifacts appear with more parameterizations, such as closebutdisconnected patches (e.g., sail of the sailboat) . We thus used 25 patches for the singleview reconstruction experiments (Section 5.2)
Category  Points  Ours  Ours  

baseline  1 patch  125 patches  
chair  LOO  3.66  3.43  2.69 
All  1.88  1.97  1.55  
car  LOO  3.38  2.96  2.49 
All  1.59  2.28  1.56  
watercraft  LOO  2.90  2.61  1.81 
All  1.69  1.69  1.23  
plane  LOO  6.47  6.15  3.58 
All  1.11  1.04  0.86 


Generalization across object categories. An important desired property of a shape autoencoder is that it generalizes well to categories it has not been trained on. To evaluate this, we trained our method on all categories but one target category (“LOO”) for chair, car, watercraft, and plane categories, and evaluated on the heldout category. The corresponding results are reported in Table 2 and Figure 3. We also include performance when the methods are trained on all of the categories including the target category (“All”) for comparison. Notice that we again outperform the pointgenerating baseline on this leaveoneout experiment and that performance improves with more patches. The car category is especially interesting since when trained on all categories the baseline has better results than our method with 1 patch and similar to our method with 125 patches. If not trained on cars, both our approaches clearly outperform the baseline, showing that at least in this case, our approach generalizes better than the baseline. The visual comparison shown Figure 3 gives an intuitive understanding of the consequences of not training for a specific category. When not trained on chairs, our method seems to struggle to define clear thin structures, like legs or armrests, especially when they are associated to a change in the topological genus of the surface. This is expected as these types of structures are not often present in the categories the network was trained on.
5.2 Singleview reconstruction
pla.  ben.  cab.  car  cha.  mon.  lam.  spe.  fir.  cou.  tab.  cel.  wat.  mean  
Ba CD  2.91  4.39  6.01  4.45  7.24  5.95  7.42  10.4  1.83  6.65  4.83  4.66  4.65  5.50 
PSG CD  3.36  4.31  8.51  8.63  6.35  6.47  7.66  15.9  1.58  6.92  3.93  3.76  5.94  6.41 
Ours CD  2.54  3.91  5.39  4.18  6.77  6.71  7.24  8.18  1.63  6.76  4.35  3.91  4.91  5.11 
Ours Metro  1.31  1.89  1.80  2.04  2.11  1.68  2.81  2.39  1.57  1.78  2.28  1.03  1.84  1.89 





We evaluate the potential of our method for singleview reconstruction. We compare qualitatively our results with three stateoftheart methods, PointSetGen [9], 3DR2N2 [7] and HSP [13] in Figure 4. To perform the comparison for PointSetGen [9] and 3DR2N2 [7], we used the trained models made available online by the authors. For HSP [13], we asked the authors to run their method on the images in Fig. 4. Note that since their model was trained on images generated with a different renderer, this comparison is not absolutely fair. To remove the bias we also compared our results with HSP on real images for which none of the methods was trained (Fig. 5) which also demonstrates the ability of our network to generalize to real images.
Figure 4 emphasizes the importance of the type of output (voxels for 3DN2D2 and HSP, point cloud for PointSetGen, mesh for us) for the visual appearance of the results. Notice the small details visible on our meshes that may be hard to see on the unstructured point cloud or volumetric representation. Also, it is interesting to see that PointSetGen tends to generate points inside the volume of the 3D shape while our result, by construction, generates points on a surface.
To perform a quantitative comparison against PointSetGen [9], we evaluated the Chamfer distance between generated points and points from the original mesh for both PointSetGen and our method with 25 learned parameterizations. However, the PointSetGen network was trained with a translated, rotated, and scaled version of ShapeNet with parameters we did not have access to. We thus first had to align the point clouds resulting from PointSetGen to the ShapeNet models used by our algorithm. We randomly selected 260 shapes, 20 from each category, and ran the iterative closest point (ICP) algorithm [3] to optimize a similarity transform between PointSetGen and the target point cloud. Note that this optimization improves the Chamfer distance between the resulting point clouds, but is not globally convergent. We checked visually that the point clouds from PointSetGen were correctly aligned, and display all alignments on the project webpage^{2}^{2}2http://imagine.enpc.fr/~groueixt/atlasnet/PSG.html.. To have a fair comparison we ran the same ICP alignment on our results. In Table 3 we compared the resulting Chamfer distance. Our method provides the best results on 6 categories whereas PointSetGen and the baseline are best on 4 and 3 categories, respectively. Our method is better on average and generates point clouds of a quality similar to the state of the art. We also report the Metro distance to the original shape, which is the most meaningful measure for our method.
To quantitatively compare against HSP [13], we retrained our method on their publicly available data since train/test splits are different from 3DR2N2 [7] and they made their own renderings of ShapeNet data. Results are in Table 4. More details are in the supplementary [1].
Chamfer  Metro  

HSP [13]  11.6  1.49 
Ours (25 patches)  9.52  1.09 
5.3 Additional applications
Shape interpolation. Figure 5(a) shows shape interpolation. Each row shows interpolated shapes generated by our AtlasNet, starting from the shape in the first column to the shape in the last. Each intermediate shape is generated using a weighted sum of the latent representations of the two extreme shaped. Notice how the interpolated shapes gradually add armrests in the first row, and chair legs in the last.
Finding shape correspondences. Figure 5(b) shows shape correspondences. We colored the surface of reference chair (left) according to its 3D position. We transfer the surface colors from the reference shape to the inferred atlas (middle). Finally, we transfer the atlas colors to other shapes (right) such that points with the same color are parametrized by the same point in the atlas. Notice that we get semantically meaningful correspondences, such as the chair back, seat, and legs without any supervision from the dataset on semantic information.
Mesh parameterization Most existing rendering pipelines require an atlas for texturing a shape (Figure 5(c)). A good parameterization should minimize amount of area distortion () and stretch () of a UV map. We computed average pertriangle distortions for 20 random shapes from each category and found that our inferred atlas usually has relatively high texture distortion (, where undistorted map has ). Our result, however, is wellsuited for distortion minimization because all meshes have disklike topology and inferred map is bijective, making it easy to further minimize distortion with offtheshelf geometric optimization [19], yielding small distortion (, see bottom row for example).
Limitations and future work are detailed in the supplementary materials [1].
6 Conclusion
We have introduced an approach to generate parametric surface elements for 3D shapes. We have shown its benefits for 3D shape and singleview reconstruction, outperforming existing baselines. In addition, we have shown its promises for shape interpolation, finding shape correspondences, and mesh parameterization. Our approach opens up applications in generation and synthesis of meshes for 3D shapes, similar to still image generation [17, 35].
Acknowledgments. This work was partly supported by ANR project EnHerit ANR17CE230008, Labex Bézout, and gifts from Adobe to École des Ponts. We thank Aaron Herzmann for fruitful discussions, Christian Häne for his help in comparing to [13] and Kevin Wampler for helping with geometric optimization for surface parameterization.
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7 Supplementary
7.1 Overview
This document provides more detailed quantitative and qualitative results highlighting the strengths and limitations of AtlasNet.
Detailed results, per category, for the autoencoder
These tables report the metro reconstruction error and the chamfer distance error. It surprisingly shows that our method with 25 learned parameterizations outperforms our method with 125 learned parameterizations in 7 categories out of 13 for the metro distance, but is significantly worse on the cellphone category, resulting in the 125 learned parameterizations approach being better on average. This is not mirrored in the Chamfer distance.
Regularisation
In the autoencoder experiment, we tried using weight decay with different weight. The best results were obtained without any regularization.
Limitations
We describe two limitations with our approach. First, when a small number of learned parameterizations are used, the network has to distort them too much to recreate the object. This leads, when we try to recreate a mesh, to small triangles in the learned parameterization space being distorted and become large triangles in 3D covering undesired regions. On the other hand, as the number of learned parameterization increases, errors in the topology of the reconstructed mesh can be sometimes observed. In practice, it means that the reconstructed patches overlap, or are not stiched together.
Additional Single View Reconstruction qualitative results
In this figure, we show one example of singleview reconstruction per category and compare with the state of the art, PointSetGen and 3DR2N2. We consistently show that our method produces a better reconstruction.
Additional Autoencoder qualitative results
In this figure, we show one example per category of autoencoder reconstruction for the baseline and our various approaches to reconstruct meshes, detailed in the main paper. We show how we are able to recreate fine surfaces.
Additional Shape Correspondences qualitative results
We color each vertex of the reference object by its distance to the gravity center of the object, and transfer these colors to the inferred atlas. We then propagate them to other objects of the same category, showing semantically meaningful correspondences between them. Results for the plane and watercraft categories are shown and generalize to all categories.
Deformable shapes.
We ran an experiment on human shape to show that our method is also suitable for reconstructing deformable shapes. The FAUST dataset [4] is a collection of meshes representing several humans in different poses. We used 250 shapes for training, and 50 for validation (without using the ground truth correspondences in any way). In table 5, we report the reconstruction error in term of Chamfer distance and Metro distance for our method with 25 squarred parameterizations, our methods with a sphere parametrization, and for the baseline. We found results to be consistent with the analysis on ShapeNet. Qualitative results are shown in figure 13, revealing that our method leads to qualitatively good reconstructions.
Chamfer  Metro  

25 patches  15.47  11.62 
1 Sphere  15.78  15.22 
1 Ref. Human  16.39  13.46 
Point cloud superresolution
AtlasNet can generate pointclouds or meshes of arbitrary resolution simply by sampling more points. Figure 7 shows qualitative results of our approach with 25 patches generating high resolution meshes with 122500 points. Moreover, PointNet is able to take an arbitrary number of points as input and encodes a minimal shape based on a subset of the input points. This is a doubleedged sword : while it allows the autoencoder to work with varying number of input points, it also prevent it from reconstructing very fine details, as they are not used by PointNet and thus not present in the latent code. We show good results using only 250 input points, despite the fact that we train using 2500 input points which shows the capacity of our decoder to interpolate a surface from a small number of input points, and the flexibility of our pipeline.


Details on the comparison against HSP [13]
We perform a quantitative comparison against an octreebased state of the art method. AtlasNet is trained with 25 learned parameterizations on the same data as their publicly available trained model^{3}^{3}3https://github.com/chaene/hsp.. 100 random samples are drawn from each category from the test split. We evaluated the the quality of the reconstruction using the Chamfer distance on the unnormalized meshes, and the metro distance. Voxelised versions of meshes often appear inflated. This bias can appear for HSP, where we observed that the generated meshes were slightly larger than the original meshes. We ran an ICP alignment procedure on the generated meshes for both methods to remove this bias. In table 8, we report per category results. As AtlasNet was specifically trained to optimise the chamfer distance, we outperform HSP in every category. AtlasNet also outperforms HSP in metro distance in each category for the metro distance, for which none of the two algorithm where trained to optimise. List of sampled used, ans trained model for AtlasNet are available in the github repository.
Limitations and future work
Our results have limitations that lead to many open question and perspective for future work. First, the patches for our generated shapes are not guaranteed to be connected (except if the surface the input points are sampled from is already closed, as in the sphere experiment). An open question is how to effectively stitch the patches together to form a closed shape. Second, we have demonstrated results on synthetic object shapes. Ideally, we would like to extend to entire real scenes. Third, we have optimized the parameterization of the generated meshes posthoc. It would be good to directly learn to generate the surfaces with low distortion parameterizations. Fourth, this work generates surfaces by minimizing an energy computed from point clouds. An open question is how to define a loss on meshes that is easy to optimize? Finally, as the atlases provide promising correspondences across different shapes, an interesting future direction is to leverage them for shape recognition and segmentation.
pla.  ben.  cab.  car  cha.  mon.  lam.  spe.  fir.  cou.  tab.  cel.  wat.  mean  

Baseline PSR  2.71  2.12  1.98  2.24  2.68  1.78  2.58  2.29  1.03  1.90  2.66  1.15  2.46  2.12 
Baseline PSR PA  1.38  1.97  1.75  2.04  2.08  1.53  2.51  2.25  1.46  1.57  2.06  1.15  1.80  1.82 
Ours 1 patch  1.11  1.41  1.70  1.93  1.76  1.35  2.01  2.30  1.01  1.46  1.46  0.87  1.46  1.53 
Ours 1 sphere  1.03  1.33  1.64  1.99  1.76  1.30  2.06  2.33  0.93  1.41  1.59  0.79  1.54  1.52 
Ours 5 patch  0.99  1.36  1.65  1.90  1.79  1.28  2.00  2.27  0.92  1.37  1.57  0.76  1.40  1.48 
Ours 25 patch  0.96  1.35  1.63  1.96  1.49  1.22  1.86  2.22  0.93  1.36  1.31  1.41  1.35  1.47 
Ours 125 patch  1.01  1.30  1.58  1.90  1.36  1.29  1.95  2.29  0.85  1.38  1.34  0.76  1.37  1.41 
pla.  ben.  cab.  car  cha.  mon.  lam.  spe.  fir.  cou.  tab.  cel.  wat.  mean  

Baseline  1.11  1.46  1.91  1.59  1.90  2.20  3.59  3.07  0.94  1.83  1.83  1.71  1.69  1.91 
Baseline + normal  1.25  1.73  2.19  1.74  2.19  2.52  3.89  3.51  0.98  2.13  2.17  1.87  1.88  2.15 
Ours 1 patch  1.04  1.43  1.79  2.28  1.97  1.83  3.06  2.95  0.76  1.90  1.95  1.29  1.69  1.84 
Ours 1 sphere  0.98  1.31  2.02  1.75  1.81  1.83  2.59  2.94  0.69  1.73  1.88  1.30  1.51  1.72 
Ours 5 patch  0.96  1.21  1.64  1.76  1.60  1.66  2.51  2.55  0.68  1.64  1.52  1.25  1.46  1.57 
Ours 25 patch  0.87  1.25  1.78  1.58  1.56  1.72  2.30  2.61  0.68  1.83  1.52  1.27  1.33  1.56 
Ours 125 patch  0.86  1.15  1.76  1.56  1.55  1.69  2.26  2.55  0.59  1.69  1.47  1.31  1.23  1.51 
pla.  ben.  cab.  car  cha.  mon.  lam.  spe.  fir.  cou.  tab.  cel.  wat.  mean  

metro  HSP  1.10  1.84  1.28  1.06  1.61  1.66  1.93  1.77  1.05  1.37  1.93  1.39  1.34  1.49 
Ours 25 patch  0.77  1.01  1.04  0.92  1.19  1.22  1.26  1.46  0.95  1.19  1.27  0.83  1.09  1.09  
chamfer  HSP  2.60  17.4  14.3  1.77  10.0  19.4  9.46  21.7  2.34  12.9  20.2  13.2  4.89  11.6 
Ours 25 patch  1.33  14.1  12.5  1.29  7.23  17.5  6.99  17.8  1.69  11.2  17.0  10.6  4.20  9.52 
Weight Decay  Ours : 25 patches 

8.57  
4.84  
3.42  
1.56 










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