This paper considers the reconstruction of the boundary of a 2D object from one or more digital representations that have been obtained by successive displacements of a digital image sensor. We explain how the reconstruction can be done for known displacements as well as for unknown random displacements that are uniformly distributed. In the latter case we assume that we can solve a certain correspondence problem. The reconstructed boundary can be characterized by a sliding ruler property of which the well-known Lagrange interpolation is a special case. It is also shown that the reconstruction can be exact under certain circumstances. Although we only consider the reconstruction of functions of one variable, the technique can be extended to functions of two or more variables. The technique can also be applied to measurements generated by other digital sensors such as range sensors.