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Deformable models, which include deformable contours (the popular snakes) and deformable surfaces, are a powerful model-based medical image analysis technique. The authors develop a new class of deformable models by formulating deformable surfaces in terms of an affine cell image decomposition (ACID). The authors' approach significantly extends standard deformable surfaces, while retaining their interactivity and other desirable properties. In particular, the ACID induces an efficient reparameterization mechanism that enables parametric deformable surfaces to evolve into complex geometries, even modifying their topology as necessary. The authors demonstrate that their new ACID-based deformable surfaces, dubbed T-surfaces, can effectively segment complex anatomic structures from medical volume images.