In this paper, we propose and analyze several methods to estimate a rigid transformation from a set of 3D matched points or matched frames, which are important features in geometric algorithms. We also develop tools to predict and verify the accuracy of these estimations. The theoretical contributions are: an intrinsic model of noise for transformations based on composition rather than addition; a unified formalism for the estimation of both the rigid transformation and its covariance matrix for points or frames correspondences, and a statistical validation method to verify the error estimation, which applies even when no "ground truth" is available. We analyze and demonstrate on synthetic data that our scheme is well behaved. The practical contribution of the paper is the validation of our transformation estimation method in the case of 3D medical images, which shows that an accuracy of the registration far below the size of a voxel can be achieved, and in the case of protein substructure matching, where frame features drastically improve both selectivity and complexity.