Efficient Computation of the Inverse Gradient on Irregular Domains

The inverse gradient problem, finding a scalar field f with a gradient near a given vector field
g
on some bounded and connected domain Ω ϵ R
^𝓃
, can be solved by means of a Poisson equation with inhomogeneous Neumann boundary conditions. We present an elementary derivation of this partial differential equation and an efficient multigrid-based method to numerically compute the inverse gradient on non-rectangular domains. The utility of the method is demonstrated by a range of important medical applications such as phase unwrapping, pressure computation, inverse deformation fields, and fiber bundle tracking.