The paper introduces the Coherent Point Drift (CPD) algorithm, a probabilistic method for both rigid and non-rigid point set registration. The goal is to align two sets of points by estimating the transformation that maps one set to the other, while handling issues such as noise, outliers, and missing points. The CPD algorithm treats the alignment problem as a probability density estimation, fitting Gaussian Mixture Model (GMM) centroids to the data points by maximizing the likelihood. The key innovation is forcing the GMM centroids to move coherently as a group to preserve the topological structure of the point sets. For rigid transformations, the algorithm imposes coherence constraints by re-parametrizing GMM centroid locations with rigid parameters and deriving a closed-form solution for the EM algorithm. For non-rigid transformations, coherence is enforced through regularization of the displacement field using variational calculus. The paper also presents a fast implementation of the CPD algorithm, reducing computational complexity to linear. Experimental results demonstrate that the CPD algorithm outperforms state-of-the-art methods in various challenging scenarios, including noisy and non-overlapping point sets.The paper introduces the Coherent Point Drift (CPD) algorithm, a probabilistic method for both rigid and non-rigid point set registration. The goal is to align two sets of points by estimating the transformation that maps one set to the other, while handling issues such as noise, outliers, and missing points. The CPD algorithm treats the alignment problem as a probability density estimation, fitting Gaussian Mixture Model (GMM) centroids to the data points by maximizing the likelihood. The key innovation is forcing the GMM centroids to move coherently as a group to preserve the topological structure of the point sets. For rigid transformations, the algorithm imposes coherence constraints by re-parametrizing GMM centroid locations with rigid parameters and deriving a closed-form solution for the EM algorithm. For non-rigid transformations, coherence is enforced through regularization of the displacement field using variational calculus. The paper also presents a fast implementation of the CPD algorithm, reducing computational complexity to linear. Experimental results demonstrate that the CPD algorithm outperforms state-of-the-art methods in various challenging scenarios, including noisy and non-overlapping point sets.