Min-Max Gradient Surface Reconstruction Via Absolute Minimization

Abstract

We consider a surface reconstruction problem with incomplete data. This problem arises on many applications in topography, bathymetry and 3D objects visualization. We adopt a reconstruction criterium that choose as solution the Lipschitz continuous function that extends the data with minimum Lipschitz constant. This yields a Lipschitzian extension problem (LEP), for which the classical minimization approach has some drawbacks as the lack of canonical solutions. By using the stronger concept of absolute minimizer, it is possible to derive an Euler-Lagrange equation, called Aronsson equation, that in the case of LEP’s turns to be the infinity Laplacian partial differential equation (PDE). In particular, the absolute minimizers coincide with the viscosity solutions of the infinity Laplacian and there exists a convergent finite differences scheme to approximate them. We apply this approach on the resolution of practical problems in bathymetry and video images reconstruction. We discuss the implementation and show our numerical results.