Point Cloud Sampling and Recovery (PCSR) is critical for massive real-time point cloud collection and processing since raw data usually requires large storage and computation. In this paper, we address a fundamental problem in PCSR: How to downsample the dense point cloud with arbitrary scales while preserving the local topology of discarding points in a case-agnostic manner (i.e. without additional storage for point relationship)? We propose a novel Locally Invertible Embedding for point cloud adaptive sampling and recovery (PointLIE). Instead of learning to predict the underlying geometry details in a seemingly plausible manner, PointLIE unifies point cloud sampling and upsampling to one single framework through bi-directional learning. Specifically, PointLIE recursively samples and adjusts neighboring points on each scale. Then it encodes the neighboring offsets of sampled points to a latent space and thus decouples the sampled points and the corresponding local geometric relationship. Once the latent space is determined and that the deep model is optimized, the recovery process could be conducted by passing the recover-pleasing sampled points and a randomly-drawn embedding to the same network through an invertible operation. Such a scheme could guarantee the fidelity of dense point recovery from sampled points. Extensive experiments demonstrate that the proposed PointLIE outperforms state-of-the-arts both quantitatively and qualitatively. Our code is released through https://github.com/zwb0/PointLIE.
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