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The study of diffeomorphism groups is fundamental to computational anatomy, and in particular to image registration. One of the most developed frameworks employs a Riemannian-geometric approach using right-invariant Sobolev metrics. To date, the computation of the Riemannian log and exponential maps on the diffeomorphism group have been defined implicitly via an infinite-dimensional optimization problem. In this paper we the employ Brenier's (1991) polar factorization to decompose a diffeomorphism h as h(x)=S∘ψ(x), where $\psi=\nabla \rho$ is the gradient of a convex function ρ and S∈SDiff(ℝd) is a volume-preserving diffeomorphism. We show that all such mappings ψ form a submanifold, which we term IDiff(ℝd), generated by irrotational flows from the identity. Using the natural metric, the manifold IDiff(ℝd) is flat. This allows us to calculate the Riemannian log map on this submanifold of diffeomorphisms in closed form, and develop extremely efficient metric-based image registration algorithms. This result has far-reaching implications in terms of the statistical analysis of anatomical variability within the framework of computational anatomy.