In this paper, we propose a new surface registration approach using a generic deformation model, which is efficient to compute and flexible to represent arbitrary local shape deformations. From Riemannian geometry, local deformation at each point of a surface can be characterized by the eigenvalues of a special transformation matrix between two canonically parameterized domains. This local transformation specifies all the deformations (i.e., diffeomorphisms) between surfaces while being independent of both intrinsic (parametrization) and extrinsic (embedding) representations. In particular, we show that existing deformation representations (e.g., isometry or conformality) can be viewed as special cases of the proposed local deformation model. Furthermore, a computationally efficient, closed-form solution is derived in the discrete setting via finite element discretization. Based on the proposed deformation model, the shape registration problem is formulated as a high-order Markov Random Field (MRF) defined on the simplicial complex (e.g., planar or tetrahedral mesh). An efficient high-order MRF optimization algorithm is designed in the paper for such a special structured MRF-MAP problem, which can be implemented in a distributed fashion and requires minimal memory. Finally, we demonstrate the speed and accuracy performance of the proposed approach in the applications of shape registration and tracking.