Coherent Point Drift (CPD) is a probabilistic method for both rigid and non-rigid point set registration. The goal is to align two point sets by assigning correspondences and recovering the transformation that maps one set to the other. The method treats the alignment as a probability density estimation problem, fitting Gaussian Mixture Model (GMM) centroids (representing the first point set) to the data (the second point set) by maximizing the likelihood. GMM centroids are forced to move coherently to preserve the topological structure of the point sets. In the rigid case, the coherence constraint is imposed by re-parametrizing GMM centroid locations with rigid parameters, leading to a closed-form solution of the EM algorithm. In the non-rigid case, the constraint is imposed by regularizing the displacement field and using variational calculus to derive the optimal transformation. A fast algorithm reduces the computational complexity to linear. CPD is tested for both rigid and non-rigid transformations in the presence of noise, outliers, and missing points, showing accurate results and outperforming current state-of-the-art methods. The algorithm is robust to degradations such as noise, outliers, and missing points. The method is applicable for large data sets and has a linear computational complexity. The CPD algorithm is related to existing methods like TPS-RPM, but it uses a different regularization term and allows for easier generalization to higher dimensions. The algorithm is implemented efficiently using the fast Gauss transform and low-rank matrix approximation. The CPD algorithm is robust and accurate in various experiments, including registration of point sets with missing parts, corrupted by outliers, and biased to different sides.Coherent Point Drift (CPD) is a probabilistic method for both rigid and non-rigid point set registration. The goal is to align two point sets by assigning correspondences and recovering the transformation that maps one set to the other. The method treats the alignment as a probability density estimation problem, fitting Gaussian Mixture Model (GMM) centroids (representing the first point set) to the data (the second point set) by maximizing the likelihood. GMM centroids are forced to move coherently to preserve the topological structure of the point sets. In the rigid case, the coherence constraint is imposed by re-parametrizing GMM centroid locations with rigid parameters, leading to a closed-form solution of the EM algorithm. In the non-rigid case, the constraint is imposed by regularizing the displacement field and using variational calculus to derive the optimal transformation. A fast algorithm reduces the computational complexity to linear. CPD is tested for both rigid and non-rigid transformations in the presence of noise, outliers, and missing points, showing accurate results and outperforming current state-of-the-art methods. The algorithm is robust to degradations such as noise, outliers, and missing points. The method is applicable for large data sets and has a linear computational complexity. The CPD algorithm is related to existing methods like TPS-RPM, but it uses a different regularization term and allows for easier generalization to higher dimensions. The algorithm is implemented efficiently using the fast Gauss transform and low-rank matrix approximation. The CPD algorithm is robust and accurate in various experiments, including registration of point sets with missing parts, corrupted by outliers, and biased to different sides.