Axis-aligned height-field block decomposition of 3D shapes

We propose a novel algorithm for decomposing general three-dimensional geometries into a small set of overlap-free height-field blocks, volumes enclosed by a flat base and a height-field surface defined with respect to this base. This decomposition is useful for fabrication methodologies such as 3-axis CNC milling, where a single milling pass can only carve a single height-field surface defined with respect to the machine tray but can also benefit other fabrication settings. Computing our desired decomposition requires solving a highly constrained discrete optimization problem, variants of which are known to be NP-hard. We effectively compute a high-quality decomposition by using a two-step process that leverages the unique characteristics of our setup. Specifically, we notice that if the height-field directions are constrained to the major axes, then we can always produce a valid decomposition starting from a suitable surface segmentation. Our method first produces a compact set of large, possibly overlapping, height-field blocks that jointly cover the model surface by recasting this discrete constrained optimization problem as an unconstrained optimization of a continuous function, which allows for an efficient solution. We then cast the computation of an overlap-free, final decomposition as an ordering problem on a graph and solve it via a combination of cycle elimination and topological sorting. The combined algorithm produces a compact set of height-field blocks that jointly describe the input model within a user given tolerance. We demonstrate our method on a range of inputs and showcase a number of real life models manufactured using our technique.