  Most of today's state-of-the-art methods for perspective shape from shading
are modelled in terms of partial differential equations (PDEs) of
Hamilton-Jacobi type. To improve the robustness of such methods w.r.t. noise
and missing data, first approaches have recently been proposed that seek to
embed the underlying PDE into a variational framework with data and smoothness
term. So far, however, such methods either make use of a radial depth
parametrisation that makes the regularisation hard to interpret from a
geometrical viewpoint or they consider indirect smoothness terms that require
additional consistency constraints to provide valid solutions. Moreover the
minimisation of such frameworks is an intricate task, since the underlying
energy is typically non-convex. In our paper we address all three of the
aforementioned issues. First, we propose a novel variational model that
operates directly on the Cartesian depth. In this context, we also point out a
common mistake in the derivation of the surface normal. Moreover, we employ a
direct second-order regulariser with edge-preservation property. This direct
regulariser yields by construction valid solutions without requiring additional
consistency constraints. Finally, we also propose a novel coarse-to-fine
minimisation framework based on an alternating explicit scheme. This framework
allows us to avoid local minima during the minimisation and thus to improve the
accuracy of the reconstruction. Experiments show the good quality of our model
as well as the usefulness of the proposed numerical scheme.
