The three-dimensional dense point cloud model of cultural relics obtained by laser scanners can easily lead to excessive consumption of resources in data storage, remote transmission, and processing. To solve this problem, the paper proposes a fast compression and recovery framework based on greedy algorithm. First, the point cloud model is regarded as a three-dimensional discrete geometric signal, and the octree method based on Hash function is used to construct the neighborhood constraint relationship for the dense point cloud. Then, the point cloud adjacency matrix is calculated and a discrete Laplacian is constructed to sparse the original signal, and the original signal is randomly sampled through a random Gaussian matrix to complete signal compression. Finally, the L₀ regularization operator and four classical greedy algorithms are introduced to solve the problem quickly. The simulation test is carried out with the point cloud model of the terracotta warriors head and the three-dimensional cultural relic point cloud model of the Tang Sancai Huren figurines. The results show that this framework can complete the effective compression of the dense point cloud model and the rapid reconstruction of the model.