Point Cloud GAN

Introduction

A fundamental problem in machine learning is: given a data set, learn a generative model that can efficiently generate arbitrary many new sample points from the domain of the underlying distribution (Bishop, 2006). Deep generative models use deep neural networks as a tool for learning complex data distributions (Kingma and Welling, 2013; Oord et al., 2016; Goodfellow et al., 2014). Especially, Generative Adversarial Networks (GAN; Goodfellow et al. 2014) is drawing attentions because of its success in many applications. Compelling results have been demonstrated on different types of data, including text, images and videos (Lamb et al., 2016; Karras et al., 2017; Vondrick et al., 2016). Their wide range of applicability was also shown in many important problems, including data augmentation (Salimans et al., 2016), image style transformation (Zhu et al., 2017), image captioning (Dai et al., 2017) and art creations (Kang, 2017).

Recently, capturing 3D information is garnering attention. There are many different data types for 3D information, such as CAD, 3D meshes and point clouds. 3D point clouds are getting popular since these store more information than 2D images and sensors capable of collecting point clouds have become more accessible. These include Lidar on self-driving cars, Kinect for Xbox and face identification sensor on phones. Compared to other formats, point clouds can be easily represented as a set of points, which has several advantages, such as permutation invariance. The algorithms which can effectively learn from this type of data is an emerging field (Qi et al., 2017a, b; Zaheer et al., 2017; Kalogerakis et al., 2017; Fan et al., 2017). However, compared to supervised learning, unsupervised generative models for 3D data are still under explored (Achlioptas et al., 2017; Oliva et al., 2018).

Extending existing GAN frameworks to handle 3D point clouds, or more generally set data, is not straightforward. In this paper, we begin by formally defining the problem and discussing the difficulty of the problem (Section 2). Circumventing the challenges, we propose a deep generative adversarial network (PC-GAN) with a hierarchical sampling and inference network for point clouds. The proposed architecture learns a stochastic procedure which can generate new point clouds as well as draw samples from point clouds without explicitly modeling the underlying density function (Section 3). The proposed PC-GAN is a generic algorithm which can incorporate many existing GAN variants. By utilizing the property of point clouds, we further propose a sandwiching objective by considering both upper and lower bounds of Wasserstein distance estimate, which can lead to tighter approximation (Section 4). Evaluation on ModelNet40 shows excellent generalization capability of PC-GAN. We first show we can sample from the learned model to generate new point clouds and the latent representations learned by the inference network provide meaningful interpolations between point clouds. We further show the conditional generation results on unseen classes of objects to demonstrate the superior generalization ability of PC-GAN. Lastly, we also provide several interesting studies, such as classification and point clouds generation from images (Section 6).

Problem Definition and Difficulty

Sample more points for a given set, i.e. $x\sim p(x|X)$.

De-Finetti theorem allows us to express the set probability in a factored format as $p(X)=\int_{\theta}\prod_{i=1}^{n}p(x_{i}|\theta)p(\theta)d\theta$ for some suitably defined $\theta$. In case of point clouds, the latent variable

$\theta$ can be interpreted as an object representation. In this view, the factoring can be understood as follows: Given an object, $\theta$, the points $x_{i}$ in the point cloud can be considered as i.i.d. samples from $p(x|\theta)$, an unknown latent distribution representing object $\theta$. Joint likelihood can be expressed as:

Attempts have been made to characterize (1) with parametric models like Gaussian Mixture Models or parametric hierarchical models (Jian and Vemuri, 2005; Strom et al., 2010; Eckart et al., 2015). However, such approaches have limited success as the point cloud conditional density $p(x|\theta)$ is highly non-linear and complicated (example of point clouds can be seen in Figure 1, 3, 7).

Consider a simple GAN (Goodfellow et al., 2014) with a DeepSets classifier as the discriminator. In order to generate coherent sets of variable size, we consider a generator $G$ having two noise sources: $u$ and $z_{i}$. To generate a set, $u$ is sampled once and $z_{i}$ is sampled for $i=1,2,...,n$ to produce $n$ points in the generated set. Intuitively, fixing the first noise source $u$ selects a set and ensures the points generated by repeated sampling of $z_{i}$ are coherent and belong to same set. The setup is depicted in Figure 2. In this setup, the GAN minimax problem would be:

Now consider the case, when there exist an ‘oracle’ mapping $T$ which maps each sample point deterministically to the object it originated from, i.e. $\exists T:T(\{x_{i}\})=\theta$. A valid example is when different $\theta$ leads to conditional distribution $p(x|\theta)$ with non-overlapping support. Let $D=D^{\prime}\circ T$ and $G$ ignores $z$ then optimization becomes

Thus, we can achieve the lower bound $-\log(4)$ by only matching the $p(\theta)$ component, while the conditional $p(x|\theta)$ is allowed to remain arbitrary. We note that there still exists good solutions, which can lead to successful training, other than this hand-crafted example. We found empirically that GAN with simple DeepSet-like discriminator most of the times fails to learn to generate point clouds even after converging. However, sometimes it does results in reasonable generations. So simply using DeepSets classifier without any constraints in simple GAN in order to handle sets does not always lead to a valid generative model. We need additional constraints for GANs with simple DeepSet-like discriminator to exclude such bad solutions and lead to a more stable training.

Proposed Method

Although GANs have been extended to learn conditional distributions (Mirza and Osindero, 2014; Isola et al., 2017), they require conditioning variables to be observed, such as the one-hot label or a given image. While in case of point clouds we only have partial knowledge of the conditional, i.e. we only have groupings of point coming from the same object but we have no representation $\theta$ of the conditional or the object other than the points themselves. Naïvely modeling $\theta$ to be a one-hot vector, to indicate which object the points belong to in the training data, cannot generalize to unseen testing data. Instead, we need a richer representation for $\theta$, which is an unobserved random variable. Thus, we need to infer $\theta$ during the training. The proposed algorithm has to concurrently learn the inference network $Q(X)$ which encodes $\theta$ while we learn $p(x|\theta)$.

A major hurdle in taking this path is that $X$ is a set of points, which can vary in size and permutation of elements. Thus, making design of $Q$ complicated as traditional neural network can not handle this and possibly is the reason for absence of such framework in the literature despite being a natural solution for the important problem of generative modeling of point clouds. However, we can overcome this challenge and we propose to construct the inference network by utilizing the recent advance in deep learning for dealing with sets (Qi et al., 2017a; Zaheer et al., 2017). This allows it handle variable number of inputs points in arbitrary order, yet yielding a consistent descriptor $\psi$.

We call the proposed algorithm for point cloud generation as PC-GAN as shown in Figure 3. The conditional distribution matching with a learned inference in PC-GAN can also be interpreted as an encoder-decoder formulation (Kingma and Welling, 2013). The difference between it and the point cloud autoencoder (Achlioptas et al., 2017; Yang et al., 2018) will be discussed in Section 5.

Different Divergences for Matching Point Clouds

To train the generator $G_{x}$ using a GAN-like objective for point clouds, we need a discriminator $f(\cdot)$ which distinguishes between the generated samples and true samples conditioned on $\theta$. Combining with the inference network discussed in Section 3, if we use an IPM-based GAN (Arjovsky et al., 2017; Mroueh and Sercu, 2017; Mroueh et al., 2017), the objective can be written as

where $\Omega_{f}$ is the constraint for different probabilistic distances, such as 1-Lipschitz (Arjovsky et al., 2017), $L^{2}$ ball (Mroueh and Sercu, 2017) or Sobolev ball (Mroueh et al., 2017).

We propose to combine, in general, a lower bound and upper bound estimate by sandwiching the solution between the two, i.e. we solve the following minimization problem:

The problem can be simplified and solved using method of Lagrange multipliers as follows:

By solving the new sandwiched problem (6), we show that under certain conditions we obtain a better estimate of Wasserstein distance in the following lemma:

The primal form of Wasserstein distance is defined as

Finding a modified primal form with low sample complexity, especially for high dimensional data, is still an open research problem (Cuturi, 2013; Genevay et al., 2018). Combining those into the proposed sandwiching objective for high dimensional data is left for future works.

2 Lower Bound Implementation

The dual form of Wasserstein distance is defined as

where $\mathcal{L}_{k}$ is the set of $k$-Lipschitz functions whose Lipschitz constant is no larger than $k$. In practice, deep neural networks parameterized by $\phi$ with constraints $f_{\phi}\in\Omega_{\phi}$ (Arjovsky et al., 2017), result in a distance approximation

In practice, choosing clipping range $c$ is non-trivial. Small ranges limit the capacity of networks, while large ranges result in numerical issues during the training. On the other hand, in addition to weight clipping, several constraints (regularization) have bee proposed with better empirical performance, such as gradient penalty (Gulrajani et al., 2017) and $L^{2}$ ball (Mroueh and Sercu, 2017). However, there is no guarantee the resulted functions are still Lipschitz or the resulted distances are lower bounds of Wasserstein distance. To take the advantage of those regularization with the Lipschitz guarantee, we propose a simple variation by combining weight clipping, which always ensures Lipschitz functions.

Note that, if $c\rightarrow\infty$, then $\Omega_{c}\cap\Omega_{\phi}=\Omega_{\phi}$. Therefore, from Proposition 2, for any regularization of discriminator (Gulrajani et al., 2017; Mroueh and Sercu, 2017; Mroueh et al., 2017), we can always combine it with a weight clipping constraint $\Omega_{c}$ to ensure a valid lower bound estimate of Wasserstein distance and enjoy the advantage that it is numerically stable when we use large $c$ compared with original weight-clipping WGAN (Arjovsky et al., 2017).

Related Works

Generative Adversarial Network (Goodfellow et al., 2014) aims to learn a generator that can sample data followed by the data distribution. Compelling results on learning complex data distributions with GAN have been shown on images (Karras et al., 2017), speech (Lamb et al., 2016), text (Yu et al., 2016; Hjelm et al., 2017), vedio (Vondrick et al., 2016) and 3D voxels (Wu et al., 2016). However, the GAN algorithm on 3D point cloud is still under explored (Achlioptas et al., 2017). Many alternative objectives for training GANs have been studied. Most of them are the dual form of $f$-divergence (Goodfellow et al., 2014; Mao et al., 2017; Nowozin et al., 2016), integral probability metrics (IPMs) (Zhao et al., 2016; Li et al., 2017a; Arjovsky et al., 2017; Gulrajani et al., 2017) or IPM extensions (Mroueh and Sercu, 2017; Mroueh et al., 2017). Genevay et al. (2018) learn the generative model by the approximated primal form of Wasserstein distance (Cuturi, 2013).

Instead of training a generative model on the data space directly, one popular approach is combining with autoencoder (AE), which is called adversarial autoencoder (AAE) (Makhzani et al., 2015). AAE constrain the encoded data to follow normal distribution via GAN loss, which is similar to VAE (Kingma and Welling, 2013) by replacing the KL-divergence on latent space via any GAN loss. Tolstikhin et al. (2017) provide a theoretical explanation for AAE by connecting it with the primal form of Wasserstein distance. The other variant of AAE is training the other generative model to learn the distribution of the encoded data instead of enforcing it to be similar to a known distribution (Engel et al., 2017; Kim et al., 2017). Achlioptas et al. (2017) explore a AAE variant for point cloud. They use a specially-designed encoder network (Qi et al., 2017a) for learning a compressed representation for point clouds before training GAN on the latent space. However, their decoder is restricted to be a MLP which generates $m$ fixed number of points, where $m$ has to be pre-defined. That is, the output of their decoder is fixed to be $3m$ for 3D point clouds, while the output of the proposed $G_{x}$ is only 3 dimensional and $G_{x}$ can generate arbitrarily many points by sampling different random noise $z$ as input. Yang et al. (2018); Groueix et al. (2018b) propose similar decoders to $G_{x}$ with fixed grids to break the limitation of Achlioptas et al. (2017) aforementioned, but they use heuristic Chamfer distance without any theoretical guarantee and do not exploit generative models for point clouds. The proposed PC-GAN can also be interpreted as an encoder-decoder formulation. However, the underlying interpretation is different. We start from De-Finetti theorem to learn both $p(X|\theta)$ and $p(\theta)$ with inference network interpretation of $Q$, while Achlioptas et al. (2017) focus on learning $p(\theta)$ without modeling $p(X|\theta)$.

GAN for learning conditional distribution (conditional GAN) has been studied in images with single conditioning (Mirza and Osindero, 2014; Pathak et al., 2016; Isola et al., 2017; Chang et al., 2017) or multiple conditioning (Wang and Gupta, 2016). The case on point cloud is still under explored. Also, most of the works assume the conditioning is given (e.g. labels and base images) without learning the inference during the training. Training GAN with inference is studied by Dumoulin et al. (2016); Li et al. (2017b); however, their goal is to infer the random noise $z$ of generators instead of semantic latent variable of the data. Li et al. (2018) is a parallel work aiming to learn GAN and unseen latent variable simultaneously, but they only study image and video datasets.

Lastly, we briefly review some recent development of deep learning on point clouds. Instead of transforming into 3D voxels and projecting objects into different views to use convolution (Su et al., 2015; Maturana and Scherer, 2015; Wu et al., 2015; Qi et al., 2016; Tatarchenko et al., 2017), which have the concern of memory usage, one direction is designing permutation invaraint operation for dealing with set data directly Qi et al. (2017a); Zaheer et al. (2017); Qi et al. (2017b). Wang et al. (2018) use graph convolution (Bronstein et al., 2017) to utilize local neighborhood information. Most of the application studied by those works focus on classification and segmentation tasks, but they can be used to implement the inference network $Q$ of PC-GAN.

Experiments

In this section we demonstrate the point cloud generation capabilities of PC-GAN. As discussed in Section 5, we refer Achlioptas et al. (2017) as AAE as it could be treated as an AAE extension to point clouds and we use the implementation provided by the authors for experiments. The sandwitching objective $W_{s}$ for PC-GAN combines $W_{L}$ and $W_{U}$ with the mixture 1:20 without tunning for all experiment. $W_{L}$ is a GAN loss by combining Arjovsky et al. (2017) and Mroueh and Sercu (2017) and we adopt (Bertsekas, 1985) for $W_{U}$ as discussed in Section 4.2 and Section 4.1.1. We parametrize $Q$ in PC-GAN by DeepSets (Zaheer et al., 2017). The review of DeepSets is in Appendix B. Other detailed configurations of each experiment can be found in Appendix C. Next, we study both synthetic 2D point cloud and ModelNet40 benchmark datasets.

We created a simple 2D synthetic point cloud datasets from parametric distributions on which we can carry out thorough evaluations of the prposed PC-GAN and draw comparisons with AAE Achlioptas et al. (2017). We generate 2D point clouds for circles, where the center of circles is followed a mixture of four Gaussians with means equal to $\{\pm 16\}\times\{\pm 16\}$. The covariance matrices were set to be $16I$ and we used equal mixture weights. The radius of the circles was drawn from a uniform distribution $\mbox{\it Unif}(1.6,6.4)$. One sampled circile is shown in Figure 5(a). We sampled $10,000$ circles for the training and testing data, respectively.

We evaluated the conditional distributions on the $10,000$ testing circles. For the proposed PC-GAN, we pass the same points into the inference network $Q$, then sample $500$ points with the conditional generator $G_{x}$ to match the output number of AAE. We measured the empirical distributions of the centers and the radius of the generated circles conditioning on the testing data for PC-GAN. Similarly, we measured the reconstructed circles of the testing data for AAE. The results are shown in Figure 5.

From Figure 5, both AAE and PC-GAN can successfully recover the center distribution, but AAE does not learn the radius distribution well. Even if we increase number the hidden layer unit of the decoder to be $20$ (AAE-20), which almost doubles the number of parameters, the performance is still not satisfactory. Compared with AAE, the proposed PC-GAN recovers the both center and radius distributions well with less parameters. The gap of memory usage could be larger if we configure AAE to generate more points, while the model size required for PC-GAN is independent of the number of points. The reason is MLP decoder adopted by Achlioptas et al. (2017) wasted parameters for nearby points. A much larger model (more parameters) can potentially boost the performance, yet would be still restricted to generate a fixed number of points for each object as discussed in Section 5.

2 Conditional Generation on ModelNet40

We consider the ModelNet40 (Wu et al., 2015) benchmark, which contains 40 classes of objects. There are $9,843$ training and $2,468$ testing instances. We follow Zaheer et al. (2017) to do pre-processing. For each object, we sampled $10,000$ points from the mesh representation and normalize it to have zero mean (for each axis) and unit (global) variance. During the training, we augment the data by uniformly rotating $0,\pi/8,\dots,7\pi/8$ rad on the $x$-$y$ plane. For PC-GAN, the random noise $z_{2}$ is fixed to be $10$ dimensional for all experiments. For other settings, we follow Achlioptas et al. (2017).

We start from a smaller model which is only trained on single class of objects. For AAE, the latent code size for its encoder is $128$ and the decoder outputs $2,048$ points for each object. The number of parameters for encoder and decoder are $15M$ in total. Similarly, we set the size of PC-GAN latent variable (the output of $Q$) to be $128$ dimensional. The number of parameters for $G_{x}$ and $Q$ is less than $1M$ in total.

We also train the proposed model on all $9,843$ objects in the training set. The size of AAE latent code of is increased to be $256$. The number of parameters of its encoder and decoder is $15.2M$. We set the size of PC-GAN latent variable to be $256$ dimensional as well. The number of parameters for $G_{x}$ and $Q$ are around $3M$ in total.

2.1 Quantitative Comparison

We first evaluate the performance of trained conditional generator $G_{x}$ and the inference network $Q$. We are interested in whether the learned model can model the distribution of the unseen testing data. Therefore, for each testing point cloud, we use $Q$ to infer the latent variable $Q(X)$, then use $G_{x}$ to generate points. We then compare the distribution between the input point cloud and the generated point clouds.

There are many criteria based on finite sample estimation can be used for evaluation, such $f$-divergence and IPM. However, the estimator with finite samples are either biased or with high variance (Peyré et al., 2017; Wang et al., 2009; Póczos et al., 2012; Weed and Bach, 2017). Also, it is impossible to use these estimators with infinitely many samples if they are accessible.

For ModelNet40, the meshes of each object are available. In many statistically guaranteed distance estimates, the adopted statistics are commonly based on distance between nearest neighbors (Wang et al., 2009; Póczos et al., 2012). Therefore, we propose to measure the performance with the following criteria. Given a point cloud $\{x_{i}\}_{i=1}^{n}$ and a mesh, which is a collection of faces $\{F_{j}\}_{j=1}^{m}$, we measure the distance to face (D2F) as

where $\mathcal{D}(x_{i},F_{j})$ is the Euclidean distance from $x_{i}$ to the face $F_{j}$. This distance is similar to Chamfer distance, which is commonly used for measuring images and point clouds (Achlioptas et al., 2017; Fan et al., 2017), with infinitely samples from true distributions (meshes).

However, the algorithm can have low or zero D2F by only focusing a small portion of the point clouds (mode collapse). Therefore, we are also interested in whether the generated points recover enough supports of the distribution. We compute the Coverage ratio as follows. For each points, we find the its nearest face, we then treat this face is coveredWe should do thresholding to ignore outlier points. In our experiments, we observe that without excluding outliers does not change conclusion for comparison.. We then compute the ratio of number of faces of a mesh is covered. A sampled mesh is showed in Figure 6, where the details have more faces (non-uniform). Thus, it is difficult to get high coverage for AAE or PC-GAN trained by limited number of sampled points. However, the coverage ratio, on the other hand, serve as an indicator about how much details the model recovers.

The results are reported in Table 1. We compare four different algorithm, AAE and PC-GAN with three objectives, including upper bound $W_{U}$ ( $\epsilon$ approximated Wasserstein distance), lower bound $W_{L}$ (GAN with $L^{2}$ ball constraints and weight clipping), and the sandwiching loss $W_{S}$ as discussed in Section 4.1, The study with $W_{U}$ and $W_{L}$ also serves as the ablation test of the proposed sandwiching loss $W_{s}$.

Since $W_{U}$ directly optimizes distance between training and generated point clouds, $W_{U}$ usually results in smaller D2F than $W_{L}$ in Table 1. One the other hand, although $W_{L}$ only recovers lower bound estimate of Wasserstein distance, its discriminator is known to focus on learning support of the distribution (Bengio, 2018), which results in better coverage (support) than $W_{U}$.

Theoretically, the proposed sandwiching $W_{s}$ results in a tighter Wasserstein distance estimation than $W_{U}$ and $W_{L}$ (Lemma 1). Based on above discussion, it can also be understood as balancing both D2F and coverage by combining both $W_{U}$ and $W_{L}$ to get a desirable middle ground. Empirically, we even observe that $W_{s}$ results in better coverage than $W_{L}$, and competitive D2F with $W_{U}$. The intuitive explanation is that some discriminative tasks are off to $W_{U}$ objective, so the discriminator can focus more on learning distribution supports. We argue that this difference is crucial for capturing the object details. Some reconstructed point clouds of testing data are shown in Figure 7. For aeroplane examples, $W_{U}$ are failed to capture aeroplane tires and $W_{s}$ has better tire than $W_{L}$. For Chair example, $W_{s}$ recovers better legs than $W_{U}$ and better seat cushion than $W_{L}$. Lastly, we highlight $W_{s}$ outperforms others more significantly when training data is larger (ModelNet10 and ModelNet40) in Table 1.

Data PC-GAN $(W_{S})$ AAE PC-GAN $(W_{U})$ PC-GAN $(W_{L})$

In most of cases, PC-GAN with $W_{s}$ has lower D2F in Table 1 with less number of parameters aforementioned. Similar to the argument in Section 6.1, although AAE use larger networks, the decoder wastes parameters for nearby points. AAE only outperforms PC-GAN ($W_{s}$) in Guitar and Sofa in terms of D2F, since the variety of these two classes are low. It is easier for MLP to learn the shared template (basis) of the point clouds. On the other hand, due to the limitation of the fixed number of output points and Chamfer distance objective, AAE has worse coverage than PC-GAN, It can be supported by Figure 7, where AAE is also failed to recover aeroplane tire.

3 Hierarchical Sampling

In Section 3, we propose a hierarchical sampling process for sampling point clouds. In the first hierarchy, the generator $G_{\theta}$ samples a object, while the second generator $G_{x}$ samples points to form the point cloud. The randomly sampled results without given any data as input are shown in Figure 8. The point clouds are all smooth, structured and almost symmetric. It shows PC-GAN captures inherent symmetries and patterns in all the randomly sampled objects, even if overall object is not perfectly formed. This highlights that learning point-wise generation scheme encourages learning basic building blocks of objects.

4 Understand the Learned Manifold

A commonly used method to demonstrate quality of the learned latent space is showing whether the interpolation between two objects on the latent space results in smooth change. We interpolate the inferred representations from two objects by the inference network, and use the generator to sample points. The inter-class result is shown in Figure 9.

It is also popular to show intra-class interpolation. In addition showing simple intra-class interpolations, where the objects are almost aligned, we present an interesting study on interpolations between rotations. During the training, we only rotate data with $8$ possible angles for augmentation, here we show it generalizes to other unseen rotations as shown in Figure 10.

However, if we linearly interpolate the code, the resulted change is scattered and not smooth as shown in Figure 10. Instead of using linear interpolation, We train a 2-layer MLP with limited hidden layer size to be 16, where the input is the angle, output is the corresponding latent representation of rotated object. We then generate the code for rotated planes with this trained MLP. It suggests although the transformation path of rotation on the latent space is not linear, it follows a smooth trajectoryBy the capability of 1-layer MLP.. It may also suggest the geodesic path of the learned manifold may not be nearly linear between rotations. Finding the geodesic path with a principal method Shao et al. (2017) and Understanding the geometry of the manifold for point cloud worth more deeper study as future work.

We evaluate the quality of the representation acquired from the learned inference network $Q$. We train the inference network $Q$ and the generator $G_{x}$ on the training split of ModelNet40 with data augmentation as mentioned above for learning generative models without label information. We then extract the latent representation $Q(X)$ for each point clouds and train linear SVM on the that with its label. We apply the same setting to a linear classifier on the latent code of Achlioptas et al. (2017).

We only sample $1000$ as input for our inference network $Q$. As the Deep Sets architecture for the inference network is invariant to number of points, we can sample different number of points as input to the trained inference network for evaluation. Because of the randomness of sampling points for extracting latent representation, we repeat the experiments $20$ times and report the average accuracy and standard deviation on the testing split in Table 2. By using $1000$ points, we are already better than unsupervised algorithms, Achlioptas et al. (2017) with $2048$ points and 3D Voxel GAN (Wu et al., 2016), and competitive with the supervised learning algorithm Deep Sets.

In above, we studied the reconstruction of unseen testing objects, while PC-GAN still saw the point clouds from the same class during training. Here we study a more challenging task. We train PC-GAN on first 30 (Alphabetic order) class, and test on the other fully unseen 10 classes. Some reconstructed (conditionally generated) point clouds are shown in Figure 11. More (larger) results can be found in Appendix LABEL:sec:larger. For the object from the unseen classes, the conditionally generated point clouds still recovers main shape and reasonable geometry structure, which confirms the advantage of the proposed PC-GAN: by enforcing the point-wise transformation, the model is forced to learn the underlying geometry structure and the shared building blocks, instead of naively copying the input from the conditioning. The resulted D2F and coverage are $57.4$ and $0.36$, which are only slightly worse than $48.4$ and $0.38$ by training on whole 40 classes in Table 1 (ModelNet40), which also supports the claims of the good generalization ability of PC-GAN.

5 Images to Point Cloud

Here we demonstrate a potential extension of the proposed PC-GAN for images to point cloud applications. After training $Q$ as described in 6.3, instead of learning $G_{\theta}$ for hierarchical sampling, we train a regressor $R$, where the input is the different views of the point cloud $X$, and the output is $Q(X)$. In this proof of concept experiment, we use the $12$ view data and the Res18 architecture in Su et al. (2015), while we change the output size to be $256$. Some example results on reconstructing testing data is shown in Figure 12. A straightforward extension is using end-to-end training instead of two-staged approached adopted here. Also, after aligning objects and take representative view along with traditional ICP techniques, we can also do single view to point cloud transformation as Choy et al. (2016); Fan et al. (2017); Häne et al. (2017); Groueix et al. (2018a), which is not the main focus of this paper and we leave it for future work.

Conclusion

In this paper, we first showed a straightforward extension of existing GAN algorithm is not applicable to point clouds. We then proposed a GAN modification (PC-GAN) that is capable of learning to generate point clouds by using ideas both from hierarchical Bayesian modeling and implicit generative models. We further propose a sandwiching objective which results in a tighter Wasserstein distance estimate theoretically and better performance empirically.

In contrast to some existing methods (Achlioptas et al., 2017), PC-GAN can generate arbitrary as many i.i.d. points as we need to form a point clouds without pre-specification. Quantitatively, PC-GAN achieves competitive or better results using smaller network than existing methods. We also demonstrated that PC-GAN can capture delicate details of point clouds and generalize well even on unseen data. Our method learns “point-wise” transformations which encourage the model to learn the building components of the objects, instead of just naively copying the whole object. We also demonstrate other interesting results, including point cloud interpolation and image to point clouds.

Although we only focused on 3D applications in this paper, our framework can be naturally generalized to higher dimensions. In the future we would like to explore higher dimensional applications, where each 3D point can have other attributes, such as RGB colors and 3D velocity vectors.

References

Appendix A Technical Proof

We prove the claim by show that LHS is at most $\epsilon_{1}$, which is the lower bound for RHS.

Without loss of generality we can assume $\lambda<0.5$, which brings us to

Appendix B Permutation Equivariance Layers

We briefly review the notion of Permutation Equivariance Layers proposed by Zaheer et al. (2017) as a background required for this paper. For more details, please refer to Zaheer et al. (2017).

Zaheer et al. (2017) propose a generic framework of deep learning for set data. The building block which can be stacked to be deep neural networks is called Permutation Equivariance Layer. One Permutation Equivariance Layer example is defined as

where $\sigma$ can be any functions (e.g. parametrized by neural networks) and $X={x_{1},\dots,x_{n}}$ is an input set. Also, the mox pooling operation can be replaced with mean pooling. We note that PointNetQi et al. (2017a) is a special case of using Permutation Equivariance Layer by properly defining $\sigma(\cdot)$. In our experiments, we follow Zaheer et al. (2017) to set $\sigma$ to be a linear layer with output size $h$ followed by any nonlinear activation function.

Appendix C Experiment Settings

The batch size is fixed to be $64$. We sampled 10,000 samples for training and testing.

For the inference network, we stack $3$ mean Permutation Equivariance Layer (Zaheer et al., 2017), where the hidden layer size (the output of the first two layers ) is $30$ and the final output size is $15$. The activation function are used SoftPlus. For the generater is a $5$ layer MLP, where the hidden layer size is set to be $30$. The discirminator is $4$ layer MLP with hidden layer size to be $30$. For Achlioptas et al. (2017), we change their implementation by replcing the number of filters for encoder to be $$, while the hidden layer width for decoder is$10$or$20$ except for the output layer. The decoder is increased from 3 to 4 layers to have more capacity.

C.2 ModelNet40

We follow Zaheer et al. (2017) to do pre-processing. For each object, we sampled $10,000$ points from the mesh representation and normalize it to have zero mean (for each axis) and unit (global) variance. During the training, we augment the data by uniformly rotating $0,\pi/8,\dots,7\pi/8$ rad on the $x$-$y$ plane. The random noise $z_{2}$ of PC-GAN is fixed to be $10$ dimensional for all experiments.

For $Q$ of single class model, we stack $3$ max Permutation Equivariance Layer with output size to be $128$ for every layer. On the top of the satck, we have a $2$ layer MLP with the same width and the output . The generator $G_{x}$ is a $4$ layer MLP where the hidden layer size is $128$ and output size is $3$. The discirminator is $4$ layer MLP with hidden layer size to be $128$.

For training whole ModelNet40 training set, we increae the width to be $256$. The generator $G_{x}$ is a $5$ layer MLP where the hidden layer size is $256$ and output size is $3$. The discirminator is $5$ layer MLP with hidden layer size to be $256$. For hirarchical sampling, the top generator $G_{\theta}$ and discriminator are all $5$-layer MLP with hidden layer size to be $256$.

For AAE, we follow every setting used in Achlioptas et al. (2017), where the latent code size is $128$ and $256$ for single class model and whole ModelNet40 models.