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This paper provides a comprehensive and quantitative review of spatial transformations models for nonrigid image registration. It explains the theoretical foundation of the models and classifies them according to this basis. This results in two categories, physically based models described by partial differential equations of continuum mechanics (e.g., linear elasticity and fluid flow) and basis function expansions derived from interpolation and approximation theory (e.g., radial basis functions, B-splines and wavelets). Recent work on constraining the transformation so that it preserves the topology or is diffeomorphic is also described. The final section reviews some recent evaluation studies. The paper concludes by explaining under what conditions a particular transformation model is appropriate.