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Recent advances in single-view reconstruction (SVR) have been in modelling power (curved 2.5D surfaces) and automation (automatic photo pop-up). We extend SVR along both of these directions. We increase modelling power in several ways: (i) We represent general 3D surfaces, rather than 2.5D Monge patches; (ii) We describe a closed-form method to reconstruct a smooth surface from its image apparent contour, including multilocal singularities ("kidney-bean" self-occlusions); (iii) We show how to incorporate user-specified data such as surface normals, interpolation and approximation constraints; (iv) We show how this algorithm can be adapted to deal with surfaces of arbitrary genus. We also show how the modelling process can be automated for simple object shapes and views, using a-priori object class information. We demonstrate these advances on natural images drawn from a number of object classes.