## 2 Method
### MU-GAN Architecture  
The hole area can be regarded as a region with sparse distribution of points. The hole-filling task can be accomplished by increasing the density of the point cloud in the hole area, which can also be regarded as upsampling. In this section, an MU-GAN is designed based on the PU-Net [38]. As depicted in Figure 3, the architecture contains a generator and a discriminator. The purpose of the generator is to generate a point cloud that is denser than the input point cloud, and the purpose of the discriminator is to determine whether the input point cloud is real data or generated by the generator. Figure 3: Overview of MU-GAN. (1) We first obtain k (k = 3) point cloud data (\(P_{1}\), \(P_{2}\), \(P_{3}\) ) of different points of an artifact model \(P\). A generator is used to generate up-sampled 2\(\times\) clean, dense complete point cloud data for each noisy, sparse point cloud data. The generated multi-scale point cloud data are used as input to train the discriminator. (2) To generate uniform patches, the multi-scale feature-fused  
Figure 2: Hole detection results; the hole is marked by the red line. point cloud data are generated through the stages of feature extraction and point cloud reconstruction, and the missing regions are extracted as filler patches. (3) The input data to be repaired and the filler patches are merged to obtain the final point cloud repair results. The generator is shown in the upper part of Figure 3. It has three components: the feature extraction module, the feature expansion module, and the point set reconstruction module. For the feature extraction module, the intra-level dense connections method in [39] is applied to extract the features from the input point cloud with data size \((N\times d)\), where \(d=3\). The method defines the local neighborhood in feature space, groups the features via KNN based on the feature similarity, and finally refines the features of each group through multi-layer perceptrons (MLPs). After the feature extraction, the size of the input \(P\), the original complete point cloud, is converted to \((N\times C)\), where \(C\) denotes the number of extraction features. The feature expansion module serves the purpose to increase the number of points (from \(N\) to \(rN\)). It should be noted that two key steps in the feature expansion module are the up and down modules, which can enable the generator to produce more diverse point distributions and are depicted as \"up\" and \"down\" operators in Figure 3. The detailed structure of these two modules is illustrated in Figure 4. More details about these modules can be referenced in [40]. In the point cloud reconstruction module, the 3D coordinate reconstruction of per-point features (generated from the feature expansion module) is carried out by a series of fully connected layers, and the dimension of the features is changed from \(rN\times C\) to \(rN\times 3\). Finally, the dense point cloud with the size of \(rN\times 3\) is generated. In the training stage, the original complete point cloud is \(P\) which has 2048 points. The input data consist of three parts, which are \(P_{1}\) (1024 points), \(P_{2}\) (512 points), and \(P_{3}\) (256 points) are the downsampled point cloud from \(P\), \(P_{1}\), and \(P_{2}\), respectively. In other words, the inputs are different scales of the original point cloud. After using the three-point cloud as input of the generator, three dense generated point clouds can be obtained. The discriminator is shown at the bottom of Figure 3. The input is the generated dense point cloud. Additionally, the target point cloud with \(rN\) points is downsampled at different scales, and then think of \(P\), \(P_{1}\) and \(P_{2}\) as ground truth \(P_{gt}\). The output of the discriminator is a value between 0 and 1. If the confidence value is closer to 1, it indicates that the input is more realistic with the target distribution. Based on the network proposed in [35], the global features are obtained through a set of fully connected and max-pooling layers.