This paper introduces a procedure for the calculation
of the vertex positions in Marching-Cubes-like surface
reconstruction methods, when the surface to reconstruct is
characterised by a discrete indicator function. Linear or higher
order methods for the vertex interpolation problem require a
smooth input function. Therefore, the interpolation methodology
to convert a discontinuous indicator function into a triangulated
surface is non-trivial. Analytical formulations for this specific
vertex interpolation problem have been derived for the 2D case by
Manson et al. [Eurographics (2011) 30, 2] and the straightforward
application of their method to a 3D case gives satisfactory
visual results. A rigorous extension to 3D, however, requires
a least-squares problem to be solved for the discrete values of
a symmetric neighbourhood. It thus relies on an extra layer
of information, and comes at a significantly higher cost. This
paper proposes a novel vertex interpolation method which yields
second-order-accurate reconstructed surfaces in the general 3D
case, without altering the locality of the method. The associated
errors are analysed and comparisons are made with linear vertex
interpolation and the analytical formulations of Manson et al.
[Eurographics (2011) 30, 2].