The point clouds obtained directly from three-dimensional scanning devices are often sparse and noisy. Therefore, point cloud upsampling plays an increasingly crucial role in fields such as point cloud reconstruction and rendering. However, point cloud upsampling methods are primarily supervised and fixed-rate, which restricts their applicability in various scenarios. In this paper, we propose a novel point cloud upsampling method, named RE-PU, which is based on the point cloud reconstruction and achieves self-supervised upsampling at arbitrary rates. The proposed method consists of two main stages: the first stage is to train a network to reconstruct the original point cloud from a prior distribution, and the second stage is to upsample the point cloud data by increasing the number of sampled points on the prior distribution with the trained model. The experimental results demonstrate that the proposed method can achieve comparable outcomes to supervised methods in terms of both visual quality and quantitative metrics.