DeepCAD: A Deep Generative Network for Computer-Aided Design Models
DeepCAD: 用于计算机辅助设计模型的深度生成网络

Similar in spirit to the approach in natural language processing [40], we first project every command $C_{i}$ onto a common embedding space. Yet, different from words in natural languages, a CAD command $C_{i}=(t_{i},\bm{p}_{i})$ has two distinct parts: its command type $t_{i}$ and parameters $\bm{p}_{i}$. We therefore formulate a different way of computing the embedding of $C_{i}$: take it as a sum of three embeddings, that is, $\bm{e}(C_{i})=\bm{e}_{i}^{\text{cmd}}+\bm{e}_{i}^{\text{param}}+\bm{e}_{i}^{\text{pos}}\in\mathbb{R}^{d_{\textrm{E}}}$.
与自然语言处理中的方法[40]精神相似，我们首先将每个命令 $C_{i}$ 投影到公共嵌入空间。然而，与自然语言中的单词不同，一个 CAD 命令 $C_{i}=(t_{i},\bm{p}_{i})$ 有两个不同的部分：其命令类型 $t_{i}$ 和参数 $\bm{p}_{i}$ 。因此，我们提出了一种不同的计算 $C_{i}$ 嵌入的方法：将其视为三个嵌入的总和，即 $\bm{e}(C_{i})=\bm{e}_{i}^{\text{cmd}}+\bm{e}_{i}^{\text{param}}+\bm{e}_{i}^{\text{pos}}\in\mathbb{R}^{d_{\textrm{E}}}$ 。

The first embedding $\bm{e}_{i}^{\text{cmd}}$ accounts for the command type $t_{i}$, given by $\bm{e}_{i}^{\text{cmd}}=\mathsf{W}_{\text{cmd}}\bm{\delta}_{i}^{\textrm{c}}$. Here $\mathsf{W}_{\text{cmd}}\in\mathbb{R}^{d_{\textrm{E}}\times 6}$ is a learnable matrix and $\bm{\delta}_{i}^{\textrm{c}}\in\mathbb{R}^{6}$ is a one-hot vector indicating the command type $t_{i}$ among the six command types.
第一个嵌入 $\bm{e}_{i}^{\text{cmd}}$ 考虑了命令类型 $t_{i}$ ，由 $\bm{e}_{i}^{\text{cmd}}=\mathsf{W}_{\text{cmd}}\bm{\delta}_{i}^{\textrm{c}}$ 给出。这里 $\mathsf{W}_{\text{cmd}}\in\mathbb{R}^{d_{\textrm{E}}\times 6}$ 是一个可学习的矩阵，而 $\bm{\delta}_{i}^{\textrm{c}}\in\mathbb{R}^{6}$ 是一个表示在六种命令类型中命令类型 $t_{i}$ 的独热向量。

The second embedding $\bm{e}_{i}^{\text{param}}$ considers the command parameters. As introduced in Sec. 3.1.2, every command has 16 parameters, each of which is quantized into an 8-bit integer. We convert each of these integers into a one-hot vector $\bm{\delta}_{i,j}^{\textrm{p}}$ ($j=1..16$) of dimension $2^{8}+1=257$; the additional dimension is to indicate that the parameter is unused in that command. Stacking all the one-hot vectors into a matrix $\bm{\delta}_{i}^{\textrm{p}}\in\mathbb{R}^{257\times 16}$, we embed each parameter separately using another learnable matrix $\mathsf{W}_{\text{param}}^{b}\in\mathbb{R}^{d_{\textrm{E}}\times 257}$, and then combine the individual embeddings through a linear layer $\mathsf{W}_{\text{param}}^{a}\in\mathbb{R}^{d_{\textrm{E}}\times 16d_{\textrm{E}}}$, namely,
第二个嵌入 $\bm{e}_{i}^{\text{param}}$ 考虑了命令参数。如 Sec. 3.1.2 中所述，每个命令有 16 个参数，每个参数都被量化为 8 位整数。我们将这些整数中的每一个转换为一个维度为 $2^{8}+1=257$ 的独热向量 $\bm{\delta}_{i,j}^{\textrm{p}}$ ( $j=1..16$ )；额外的维度用于指示该参数在该命令中未使用。将所有独热向量堆叠成一个矩阵 $\bm{\delta}_{i}^{\textrm{p}}\in\mathbb{R}^{257\times 16}$ ，我们使用另一个可学习的矩阵 $\mathsf{W}_{\text{param}}^{b}\in\mathbb{R}^{d_{\textrm{E}}\times 257}$ 分别嵌入每个参数，然后通过一个线性层 $\mathsf{W}_{\text{param}}^{a}\in\mathbb{R}^{d_{\textrm{E}}\times 16d_{\textrm{E}}}$ 组合这些独立的嵌入，具体为：

$$\bm{e}_{i}^{\text{param}}=\mathsf{W}_{\text{param}}^{a}\text{flat}(\mathsf{W}_{\text{param}}^{b}\bm{\delta}_{i}^{\textrm{p}}),$$

(1)

where $\text{flat}(\cdot)$ flattens the input matrix to a vector.
其中 $\text{flat}(\cdot)$ 将输入矩阵展平为向量。

Lastly, similar to [40], the positional embedding $\bm{e}_{i}^{\text{pos}}$ is to indicate the index of the command $C_{i}$ in the whole command sequence, defined as $\bm{e}_{i}^{\text{pos}}=\mathsf{W}_{\text{pos}}\bm{\delta}_{i}$, where $\mathsf{W}_{\text{pos}}\in\mathbb{R}^{d_{\textrm{E}}\times N_{c}}$ is a learnable matrix and $\bm{\delta}_{i}\in\mathbb{R}^{N_{c}}$ is the one-hot vector filled with $1$ at index $i$ and $0$ otherwise.
最后，类似于 [ 40] ，位置嵌入 $\bm{e}_{i}^{\text{pos}}$ 用于指示命令 $C_{i}$ 在整个命令序列中的索引，定义为 $\bm{e}_{i}^{\text{pos}}=\mathsf{W}_{\text{pos}}\bm{\delta}_{i}$ ，其中 $\mathsf{W}_{\text{pos}}\in\mathbb{R}^{d_{\textrm{E}}\times N_{c}}$ 是一个可学习的矩阵，而 $\bm{\delta}_{i}\in\mathbb{R}^{N_{c}}$ 是一个独热向量，在索引 $i$ 处填充 $1$ ，其余位置为 $0$ 。

Our encoder $E$ is composed of four layers of Transformer blocks, each with eight attention heads and feed-forward dimension of $512$. The encoder takes the embedding sequence $[\bm{e}_{1},..,\bm{e}_{N_{c}}]$ as input, and outputs vectors $[\bm{e}^{\prime}_{1},..,\bm{e}^{\prime}_{N_{c}}]$; each has the same dimension $d_{\textrm{E}}=256$. The output vectors are finally averaged to produce a single $d_{\textrm{E}}$-dimensional latent vector $\bm{z}$.
我们的编码器 $E$ 由四个 Transformer 模块层组成，每个模块层有八个注意力头，前馈维度为 $512$ 。编码器以嵌入序列 $[\bm{e}_{1},..,\bm{e}_{N_{c}}]$ 为输入，输出向量 $[\bm{e}^{\prime}_{1},..,\bm{e}^{\prime}_{N_{c}}]$ ；每个向量具有相同的维度 $d_{\textrm{E}}=256$ 。输出向量最终被平均以产生一个 $d_{\textrm{E}}$ 维度的潜在向量 $\bm{z}$ 。

Also built on Transformer blocks, our decoder $D$ has the same hyper-parameter settings as the encoder. It takes as input learned constant embeddings while also attending to the latent vector $\bm{z}$—similar input structure has been used in [9, 10]. Output from the last Transformer block is fed into a linear layer to predict a CAD command sequence $\hat{M}=[\hat{C}_{1},..,\hat{C}_{N_{c}}]$, including both the command type $\hat{t}_{i}$ and parameters $\hat{\bm{p}}_{i}$ for each command. As opposed to the autoregressive strategy commonly used in natural language processing [40], we adopt the feed-forward strategy [9, 10], and the prediction of our model can be factorized as
我们的解码器 $D$ 也是基于 Transformer 模块构建的，其超参数设置与编码器相同。它以学习到的常量嵌入作为输入，同时关注潜在向量 $\bm{z}$ —这种类似的输入结构在 [ 9, 10] 中已被使用。最后一个 Transformer 模块的输出被输入到一个线性层，以预测 CAD 命令序列 $\hat{M}=[\hat{C}_{1},..,\hat{C}_{N_{c}}]$ ，包括每个命令的类型 $\hat{t}_{i}$ 和参数 $\hat{\bm{p}}_{i}$ 。与自然语言处理中常用的自回归策略 [ 40] 不同，我们采用了前馈策略 [ 9, 10]，我们模型的预测可以被分解为

$$p(\hat{M}|z,\Theta)=\overset{N_{c}}{\underset{i=1}{\prod}}p(\hat{t}_{i},\hat{\bm{p}}_{i}|z,\Theta),$$

(2)

where $\Theta$ denotes network parameters of the decoder.
其中 $\Theta$ 表示解码器的网络参数。

Leveraging the dataset, we train our autoencoder network using the standard Cross-Entropy loss. Formally, we define the loss between the predicted CAD model $\hat{M}$ and the ground truth model $M$ as
利用数据集，我们使用标准的交叉熵损失训练我们的自动编码器网络。形式上，我们将预测的 CAD 模型 $\hat{M}$ 与真实模型 $M$ 之间的损失定义为

$$\mathcal{L}=\overset{N_{c}}{\underset{i=1}{\sum}}\ell(\hat{t}_{i},t_{i})+\beta\overset{N_{c}}{\underset{i=1}{\sum}}\overset{N_{P}}{\underset{j=1}{\sum}}\ell(\hat{\bm{p}}_{i,j},\bm{p}_{i,j}),$$

(3)

where $\ell(\cdot,\cdot)$ denotes the standard Cross-Entropy, $N_{p}$ is the number of parameters ($N_{p}=16$ in our examples), and $\beta$ is a weight to balance both terms ($\beta=2$ in our examples). Note that in the ground-truth command sequence, some commands are empty (i.e., the padding command $\langle\mathtt{EOS}\rangle$) and some command parameters are unused (i.e., labeled as $-1$). In those cases, their corresponding contributions to the summation terms in (3) are simply ignored.
其中 $\ell(\cdot,\cdot)$ 表示标准的交叉熵， $N_{p}$ 是参数数量（在我们的示例中为 $N_{p}=16$ ）， $\beta$ 是平衡两个项的权重（在我们的示例中为 $\beta=2$ ）。请注意，在真实命令序列中，有些命令为空（即填充命令 $\langle\mathtt{EOS}\rangle$ ），有些命令参数未使用（即标记为 $-1$ ）。在这些情况下，它们对（3）中求和项的贡献简单地被忽略。

The training process uses the Adam optimizer [22] with a learning rate $0.001$ and a linear warm-up period of $2000$ initial steps. We set a dropout rate of $0.1$ for all Transformer blocks and apply gradient clipping of $1.0$ in back-propagation. We train the network for $1000$ epochs with a batch size of $512$.
训练过程使用 Adam 优化器[22]，学习率为 $0.001$ ，并有一个线性预热期为 $2000$ 初始步。我们为所有 Transformer 块设置 dropout 率为 $0.1$ ，并在反向传播中应用梯度裁剪 $1.0$ 。我们使用批量大小 $512$ 训练网络 $1000$ 个 epoch。

Once the autoencoder is well trained, we can represent a CAD model using a $256$-dimensional latent vector $\bm{z}$. For automatic generation of CAD models, we employ the latent-GAN technique [6, 12, 50] on our learned latent space. The generator and discriminator are both as simple as a multilayer perceptron (MLP) network with four hidden layers, and they are trained using Wasserstein-GAN training strategy with gradient penalty [7, 18]. In the end, to generate a CAD model, we sample a random vector from a multivariate Gaussian distribution and feeding it into the GAN’s generator. The output of the GAN is a latent vector $\bm{z}$ input to our Transformer-based decoder.
一旦自动编码器训练良好，我们可以使用一个 $256$ 维潜在向量 $\bm{z}$ 来表示一个 CAD 模型。为了自动生成 CAD 模型，我们在我们的学习潜在空间中采用了潜在 GAN 技术[6, 12, 50]。生成器和判别器都像是一个具有四个隐藏层的多层感知器（MLP）网络，并且它们使用 Wasserstein-GAN 训练策略和梯度惩罚进行训练[7, 18]。最后，为了生成一个 CAD 模型，我们从多元高斯分布中采样一个随机向量并将其输入到 GAN 的生成器中。GAN 的输出是一个潜在向量 $\bm{z}$ ，该向量输入到我们的基于 Transformer 的解码器中。

To thoroughly understand our autoencoder’s performance, we measure the difference between $M$ and $\hat{M}$ in terms of both the CAD commands and the resulting 3D geometry. We propose to evaluate command accuracy using two metrics, namely Command Accuracy ($\text{ACC}_{\text{cmd}}$) and Parameter Accuracy ($\text{ACC}_{\text{param}}$). The former measures the correctness of the predicted CAD command type, defined as
为了全面了解我们的自编码器的性能，我们分别从 CAD 指令和最终 3D 几何形状两个方面测量 $M$ 和 $\hat{M}$ 之间的差异。我们提出使用两个指标来评估指令精度，即指令精度（ $\text{ACC}_{\text{cmd}}$ ）和参数精度（ $\text{ACC}_{\text{param}}$ ）。前者测量预测 CAD 指令类型的正确性，定义为

$$\text{ACC}_{\text{cmd}}=\frac{1}{N_{c}}\sum_{i=1}^{N_{c}}\mathbb{I}[t_{i}=\hat{t}_{i}].$$

(4)

Here the notation follows those in Sec. 3. $N_{c}$ denote the total number of CAD commands, and $t_{i}$ and $\hat{t}_{i}$ are the ground-truth and recovered command types, respectively. $\mathbb{I}[\cdot]$ is the indicator function (0 or 1).
此处符号遵循第 3 节中的符号。 $N_{c}$ 表示 CAD 命令的总数， $t_{i}$ 和 $\hat{t}_{i}$ 分别是真实命令类型和恢复的命令类型。 $\mathbb{I}[\cdot]$ 是指示函数（0 或 1）。

Once the command type is correctly recovered, we also evaluate the correctness of the command parameters. This is what Parameter Accuracy ($\text{ACC}_{\text{param}}$) is meant to measure:
一旦命令类型被正确恢复，我们还评估命令参数的正确性。这就是参数准确率（ $\text{ACC}_{\text{param}}$ ）所要衡量的内容：

$$\text{ACC}_{\text{param}}=\frac{1}{K}\sum_{i=1}^{N_{c}}\sum_{j=1}^{|\hat{\bm{p}}_{i}|}\mathbb{I}[|\bm{p}_{i,j}-\hat{\bm{p}}_{i,j}|<\eta]\mathbb{I}[t_{i}=\hat{t}_{i}],$$

(5)

where $K=\sum_{i=1}^{N_{c}}\mathbb{I}[t_{i}=\hat{t}_{i}]|\bm{p}_{i}|$ is the total number of parameters in all correctly recovered commands. Note that $\bm{p}_{i,j}$ and $\hat{\bm{p}}_{i,j}$ are both quantized into 8-bit integers. $\eta$ is chosen as a tolerance threshold accounting for the parameter quantization. In practice, we use $\eta=3$ (out of 256 levels).
其中 $K=\sum_{i=1}^{N_{c}}\mathbb{I}[t_{i}=\hat{t}_{i}]|\bm{p}_{i}|$ 是所有正确恢复的命令中的参数总数。请注意， $\bm{p}_{i,j}$ 和 $\hat{\bm{p}}_{i,j}$ 都量化为 8 位整数。 $\eta$ 被选为考虑参数量化的容差阈值。在实践中，我们使用 $\eta=3$ （256 个级别中的某个级别）。

To measure the quality of recovered 3D geometry, we use Chamfer Distance (CD), the metric used in many previous generative models of discretized shapes (such as point clouds) [6, 17, 12]. Here, we evaluate CD by uniformly sampling $2000$ points on the surfaces of reference shape and recovered shape, respectively; and measure CD between the two sets of points. Moreover, it is not guaranteed that the output CAD command sequence always produces a valid 3D shape. In rare cases, the output commands may lead to an invalid topology, and thus no point cloud can be extracted from that CAD model. We therefore also report the Invalid Ratio, the percentage of the output CAD models that fail to be converted to point clouds.
为了衡量恢复的 3D 几何质量，我们使用 Chamfer Distance（CD），这是许多先前用于离散形状（如点云）的生成模型中使用的度量标准 [ 6, 17, 12]。在这里，我们通过在参考形状和恢复形状的表面上均匀采样 $2000$ 个点来评估 CD；并测量这两组点之间的 CD。此外，输出 CAD 命令序列并不总是能生成有效的 3D 形状。在罕见的情况下，输出命令可能导致无效的拓扑结构，因此无法从该 CAD 模型中提取点云。因此，我们还报告无效比例，即无法转换为点云的输出 CAD 模型的比例。

Due to the lack of existing CAD generative models, we compare our model with several variants in order to justify our data representation and training strategy. In particular, we consider the following variants.
由于缺乏现有的 CAD 生成模型，我们将我们的模型与几种变体进行比较，以证明我们的数据表示和训练策略。特别是，我们考虑以下几种变体。

$\mathtt{Alt}\text{-}\mathtt{Rel}$ represents curve positions relative to the position of its predecessor curve in the loop. It contrasts to our model, which uses absolute positions in curve specification.
$\mathtt{Alt}\text{-}\mathtt{Rel}$ 表示曲线相对于其在环中前一个曲线的位置。这与我们的模型形成对比，我们的模型在曲线规范中使用绝对位置。

$\mathtt{Alt}\text{-}\mathtt{Trans}$ includes in the extrusion command the starting point position of the loop (in addition to the origin of the sketch plane). Here the starting point position and the plane’s origin are in the world frame of reference of the CAD model. In contrast, our proposed method includes only the sketch plane’s origin, and the origin is translated to the loop’s starting position—it is therefore more compact.
$\mathtt{Alt}\text{-}\mathtt{Trans}$ 在拉伸命令中包含环路的起始点位置（除了草图平面的原点）。在这里，起始点位置和平面的原点位于 CAD 模型的世界参考系中。相比之下，我们提出的方法仅包含草图平面的原点，并将原点平移到环路的起始位置——因此更加紧凑。

$\mathtt{Alt}\text{-}\mathtt{ArcMid}$ specifies an arc using its ending and middle point positions, but not the sweeping angle and the counter-clockwise flag used in Table 1.
$\mathtt{Alt}\text{-}\mathtt{ArcMid}$ 使用弧线的终点和中点位置来指定弧线，但未指定表 1 中使用的扫掠角度和逆时针标志。

$\mathtt{Alt}\text{-}\mathtt{Regr}$ regresses all parameters of the CAD commands using the standard mean-squared error in the loss function. Unlike the model we propose, there is no need to quantize continuous parameters in this approach.
$\mathtt{Alt}\text{-}\mathtt{Regr}$ 使用损失函数中的标准均方误差回归所有 CAD 命令参数。与我们所提出的模型不同，这种方法无需量化连续参数。

$\mathtt{Ours}\text{+}\mathtt{Aug}$ uses the same data representation and training objective as our proposed solution, but it augment the training dataset by including randomly composed CAD command sequences (although the augmentation may be an invalid CAD sequence in few cases).
$\mathtt{Ours}\text{+}\mathtt{Aug}$ 使用与我们所提出的解决方案相同的数据表示和训练目标，但通过包含随机组合的 CAD 命令序列来增强训练数据集（尽管在少数情况下，这种增强可能构成无效的 CAD 序列）。

More details about these variants are described in Sec. D of the supplementary document.
这些变体的更多细节在补充文档的 D 节中有描述。

The quantitative results are report in Table 2, and more detailed CD scores are given in Table 4 of the supplementary document. In general, $\mathtt{Ours}\text{+}\mathtt{Aug}$ (i.e., training with synthetic data augmentation) achieves the best performance, suggesting that randomly composed data can improve the network’s generalization ability. The performance of $\mathtt{Alt}\text{-}\mathtt{ArcMid}$ is similar to $\mathtt{Ours}$. This means that middle-point representation is a viable alternative to represent arcs. Moreover, $\mathtt{Alt}\text{-}\mathtt{Trans}$ performs slightly worse in terms of CD than $\mathtt{Ours}$ (e.g., see the green model in Fig. 4).
定量结果报告在表 2 中，补充文档的表 4 给出了更详细的 CD 分数。总的来说， $\mathtt{Ours}\text{+}\mathtt{Aug}$ （即使用合成数据增强进行训练）取得了最佳性能，这表明随机组合的数据可以提高网络的泛化能力。 $\mathtt{Alt}\text{-}\mathtt{ArcMid}$ 的性能与 $\mathtt{Ours}$ 相似。这意味着中间点表示是表示弧线的可行替代方案。此外， $\mathtt{Alt}\text{-}\mathtt{Trans}$ 在 CD 方面比 $\mathtt{Ours}$ 稍差（例如，参见图 4 中的绿色模型）。

Perhaps more interestingly, while $\mathtt{Alt}\text{-}\mathtt{Rel}$ has high parameter accuracy ($\text{ACC}_{\text{param}}$), even higher than $\mathtt{Ours}$, it has a relatively large CD score and sometimes invalid topology: for example, the yellow model in the second row of Fig. 4 has two triangle loops intersecting with each other, resulting in invalid topology. This is caused by the errors of the predicted curve positions. In $\mathtt{Alt}\text{-}\mathtt{Rel}$ , curve positions are specified with respect to its predecessor curve, and thus the error accumulates along the loop.
或许更有趣的是，虽然 $\mathtt{Alt}\text{-}\mathtt{Rel}$ 具有高参数精度（ $\text{ACC}_{\text{param}}$ ），甚至高于 $\mathtt{Ours}$ ，但它具有相对较大的 CD 分数，并且有时会出现无效拓扑：例如，图 4 第二行的黄色模型有两个三角形环相互交叉，导致无效拓扑。这是由预测曲线位置的误差引起的。在 $\mathtt{Alt}\text{-}\mathtt{Rel}$ 中，曲线位置相对于其前一个曲线指定，因此误差沿着环累积。

Lastly, $\mathtt{Alt}\text{-}\mathtt{Regr}$ , not quantizing continuous parameters, suffers from larger errors that may break curcial geometric relations such as parallel and perpendicular edges (e.g., see the orange model in Fig. 4).
最后， $\mathtt{Alt}\text{-}\mathtt{Regr}$ ，不量化连续参数会导致更大的误差，可能会破坏关键的几何关系，例如平行和垂直的边缘（例如，参见图 4 中的橙色模型）。

We also verify the generalization of our autoencoder: we take our autoencoder trained on our created dataset and evaluate it on the smaller dataset provided in [48]. These datasets are constructed from different sources: ours is based on models from Onshape repository, while theirs is produced from designs in Autodesk Fusion 360. Nonetheless, our network generalizes well on their dataset, achieving comparable quantitative performance (see Sec. E in supplementary document).
我们也验证了我们的自编码器的泛化能力：我们使用在自建数据集上训练的自编码器，并在[48]中提供的较小数据集上评估它。这些数据集来自不同来源：我们的数据集基于 Onshape 仓库中的模型，而他们的数据集则来自 Autodesk Fusion 360 中的设计。尽管如此，我们的网络在他们数据集上表现良好，实现了相当相当的定量性能（见补充文档的 E 节）。

For quantitative comparison with point-cloud generative models, we follow the metrics used in l-GAN [6]. Those metrics measure the discrepancy between two sets of 3D point-cloud shapes, the set $\mathcal{S}$ of ground-truth shapes and the set $\mathcal{G}$ of generated shapes. In particular, Coverage (COV) measures what percentage of shapes in $\mathcal{S}$ can be well approximated by shapes in $\mathcal{G}$. Minimum Matching Distance (MMD) measures the fidelity of $\mathcal{G}$ through the minimum matching distance between two point clouds from $\mathcal{S}$ and $\mathcal{G}$. Jensen-Shannon Divergence (JSD) is the standard statistical distance, measuring the similarity between the point-cloud distributions of $\mathcal{S}$ and $\mathcal{G}$. Details of computing these metrics are present in the supplement (Sec. G).
为了与点云生成模型进行定量比较，我们遵循 l-GAN [6]中使用的指标。这些指标测量两组 3D 点云形状之间的差异，即真实形状的集合 $\mathcal{S}$ 和生成形状的集合 $\mathcal{G}$ 。具体来说，覆盖率（COV）衡量集合 $\mathcal{S}$ 中有多少形状可以被集合 $\mathcal{G}$ 中的形状很好地逼近。最小匹配距离（MMD）通过来自 $\mathcal{S}$ 和 $\mathcal{G}$ 的两个点云之间的最小匹配距离来衡量 $\mathcal{G}$ 的保真度。Jensen-Shannon 散度（JSD）是标准的统计距离，用于测量 $\mathcal{S}$ 和 $\mathcal{G}$ 点云分布之间的相似性。这些指标的计算细节在补充材料（第 G 节）中有所介绍。

Figure 5 illustrates some output examples from our CAD generative model and l-GAN. We then convert ground-truth and generated CAD models into point clouds, and evaluate the metrics. The results are reported in Table 3, indicating that our method has comparable performance as l-GAN in terms of the point-cloud metrics. Nevertheless, CAD models, thanks to their parametric representation, have much smoother surfaces and sharper geometric features than point clouds.
图 5 展示了我们 CAD 生成模型和 l-GAN 的一些输出示例。然后我们将真实 CAD 模型和生成的 CAD 模型转换为点云，并评估指标。结果报告在表 3 中，表明我们的方法在点云指标方面与 l-GAN 具有可比的性能。然而，CAD 模型由于其参数化表示，比点云具有更平滑的表面和更清晰的几何特征。