# RE-PU: A Self-Supervised Arbitrary-Scale Point Cloud Upsampling Method Based on Reconstruction
## 1 Introduction  
A point cloud is a collection of points in a three-dimensional coordinate system used to represent the surface of objects or scenes. It serves as the standard output of 3D scanning devices such as laser scanners, structured light cameras, and depth cameras widely applied in fields like autonomous driving, robotics, virtual reality, 3D printing, and computer graphics [1]. However, point clouds obtained directly from 3D devices are often sparse, noisy, and uneven, and they may lose information due to occlusion. These challenges pose difficulties for downstream tasks such as rendering and classification. Therefore, point cloud upsampling has garnered increasing attention in recent years, serving as a crucial step to obtain more detailed and accurate scene representations [2]. Point cloud upsampling aims to increase the density of a point cloud, transforming a sparse and uneven low-quality point cloud into a dense and uniform one while preserving the basic geometric shape represented by the original point cloud [3]. It involves generating new points consistent with the original geometry while avoiding distortions like noise and artifacts. Increasing the point cloud can improve the quality of the point cloud and make it more detailed. This is very important for many applications. For example, in the field of rendering, increasing the point cloud can help the model to render more realistic images. In the field of 3D object detection, increasing the point cloud can help the model to detect smaller objects. Point cloud upsampling is a challenging task due to the unordered, sparse, and irregular nature of point clouds [4]. In the early stages, point cloud upsampling was typically achieved through optimization-based methods using various shape priors to refine the point cloud generation, such as normals and local smoothness [5]. However, the effectiveness of these methods was frequently hindered by the intricate structure of 3D models. With the progression of deep learning, PointNet [6] has emerged as a groundbreaking method that directly employsneural networks for point cloud processing. In recent years, inspired by image super-resolution [7], there has been a growing number of data-driven point cloud upsampling methods proposed, gradually becoming mainstream solutions. Historically, deep learning based point cloud upsampling methods were primarily supervised and fixed-rate [8; 9]. Supervised methods face challenges since they depend on training and testing data with similar distributions. This can be especially challenging in practical applications where achieving identical distributions proves to be difficult. For instance, models trained on synthetic datasets might struggle to perform optimally on real scanned point clouds. On the other hand, fixed-rate upsampling necessitates training separate models for different upsampling rates, leading to considerable inconvenience in practical applications. Additionally, post-processing steps like furthest point sampling were often needed before downstream tasks. Both these characteristics restricted the applicability of conventional methods in various scenarios [10]. In this study, we introduce a novel reconstruction-based point cloud upsampling approach, named RE-PU, which achieves self-supervised upsampling at arbitrary rates. The core idea of this method is to map a prior distribution onto the three-dimensional surface represented by the point cloud. Specifically, we employ an autoencoder or a decoder for self-supervision: the former is for multiple shapes in one model, and the latter is for only one shape. The encoder identifies input point clouds, and the decoder takes the encoder's output along with a prior distribution as input to reconstruct a prior distribution into the input point cloud. Therefore, we can obtain a parametric neural representation of the point cloud surface, which converts a discrete representation into a continuous representation. Subsequently, point cloud upsampling can be achieved by sampling from the prior distribution.