Is the real problem in resolving correspondence using current stereo algorithms the lack of the "right" matching criterion? In studying the related task of reconstructing three-dimensional space curves from their projections in multiple views, we suggest that the problem is more basic: matching and reconstruction are coupled, and so reconstruction algorithms should exploit this rather than assuming that matching can be successfully performed before reconstruction. To realize this coupling, a generative model of curves is introduced which has two key components: (i) a prior distribution of general space curves and (ii) an image formation model which describes how 3D curves are projected onto the image plane. A novel aspect of the image formation model is that it uses an exact description of the gradient field of a piecewise constant image. Based on this forward model, a fully automatic algorithm for solving the inverse problem is developed for an arbitrary number of views. The resulting algorithm is robust to partial occlusion, deficiencies in image curve extraction and it does not rely on photometric information. The relative motion of the cameras is assumed to be given. Several experiments are carried out on various realistic scenarios. In particular, we focus on scenes where traditional correlation-based methods would fail.