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Although convolutional neural networks have achieved remarkable success in analyzing 2D images/videos, it is still non-trivial to apply the well-developed 2D techniques in regular domains to the irregular 3D point cloud data. To bridge this gap, we propose ParaNet, a novel end-to-end deep learning framework, for representing 3D point clouds in a completely regular and nearly lossless manner. To be specific, ParaNet converts an irregular 3D point cloud into a regular 2D color image, named point geometry image (PGI), where each pixel encodes the spatial coordinates of a point. In contrast to conventional regular representation modalities based on multi-view projection and voxelization, the proposed representation is differentiable and reversible. Technically, ParaNet is composed of a surface embedding module, which parameterizes 3D surface points onto a unit square, and a grid resampling module, which resamples the embedded 2D manifold over regular dense grids. Note that ParaNet is unsupervised, i.e., the training simply relies on reference-free geometry constraints. The PGIs can be seamlessly coupled with a task network established upon standard and mature techniques for 2D images/videos to realize a specific task for 3D point clouds. We evaluate ParaNet over shape classification and point cloud upsampling, in which our solutions perform favorably against the existing state-of-the-art methods. We believe such a paradigm will open up many possibilities to advance the progress of deep learning-based point cloud processing and understanding.