  Many surface reconstruction methods incorporate normal integration, which is
a process to obtain a depth map from surface gradients. In this process, the
input may represent a surface with discontinuities, e.g., due to
self-occlusion. To reconstruct an accurate depth map from the input normal map,
hidden surface gradients occurring from the jumps must be handled. To model
these jumps correctly, we design a novel discretization scheme for the domain
of normal integration. Our key idea is to introduce auxiliary edges, which
bridge between piecewise-smooth patches in the domain so that the magnitude of
hidden jumps can be explicitly expressed. Using the auxiliary edges, we design
a novel algorithm to optimize the discontinuity and the depth map from the
input normal map. Our method optimizes discontinuities by using a combination
of iterative re-weighted least squares and iterative filtering of the jump
magnitudes on auxiliary edges to provide strong sparsity regularization.
Compared to previous discontinuity-preserving normal integration methods, which
model the magnitudes of jumps only implicitly, our method reconstructs subtle
discontinuities accurately thanks to our explicit representation of jumps
allowing for strong sparsity regularization.
