Isogeometric finite-elements methods and variational reconstruction tasks in vision — A perfect match

Inverse problems are abundant in vision. A common way to deal with their inherent ill-posedness is reformulating them within the framework of the calculus of variations. This always leads to partial differential equations as conditions of (local) optimality. In this paper, we propose solving such equations numerically by isogeometric analysis, a special kind of finite-elements method. We will expose its main advantages including superior computational performance, a natural ability to facilitate multi-scale reconstruction, and a high degree of compatibility with the spline geometries encountered in modern computer-aided design systems. To animate these fairly general arguments, their impact on the well-known depth-from-gradients problem is discussed, which amounts to solving a Poisson equation on the image plane. Experiments suggest that, by the isogeometry principle, reconstructions of unprecedented quality can be obtained without any prefiltering of the data.