Partial integrability in surface reconstruction from a given gradient field

This paper investigates vector space methods of reconstructing a surface from a given needle map in discrete image settings. We first describe the subset of the gradient space corresponding to uniformly integrable surfaces in terms of an orthonormal. set of gradient fields. We then formulate the minimum norm solution to the surface reconstruction problem from a given set of gradients, and show that the solution corresponds to an equivalence class of surfaces. Next, we relax the uniform integrability condition to partial integrability by modifying the feasible subspace. We show that partial integrability enforced as such allows reconstruction of surfaces which are not uniformly integrable.