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We consider the estimation of diffeomorphic transformations aligning a known shape and its distorted observation. The classical way to solve this registration problem is to find correspondences between the shapes and then compute the transformation parameters from these landmarks. Here we propose a novel framework where the exact transformation is obtained as the solution of a polynomial system of equations. The method has been applied to 2D and 3D medical image registration, industrial inspection, planar homography estimation, etc... and its robustness has also been demonstrated. The advantage of the proposed solution is that it is fast, easy to implement, has linear time complexity, works without established correspondences and provides an exact solution regardless of the magnitude of transformation.