### Abstract

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.

Original language | English (US) |
---|---|

Article number | 169 |

Journal | ACM Transactions on Graphics |

Volume | 37 |

Issue number | 5 |

DOIs | |

State | Published - Nov 1 2018 |

### Fingerprint

### Keywords

- Fabrication
- Shape decomposition

### ASJC Scopus subject areas

- Computer Graphics and Computer-Aided Design

### Cite this

*ACM Transactions on Graphics*,

*37*(5), [169]. https://doi.org/10.1145/3204458

**Axis-aligned height-field block decomposition of 3D shapes.** / Muntoni, Alessandro; Livesu, Marco; Scateni, Riccardo; Sheffer, Alla; Panozzo, Daniele.

Research output: Contribution to journal › Article

*ACM Transactions on Graphics*, vol. 37, no. 5, 169. https://doi.org/10.1145/3204458

}

TY - JOUR

T1 - Axis-aligned height-field block decomposition of 3D shapes

AU - Muntoni, Alessandro

AU - Livesu, Marco

AU - Scateni, Riccardo

AU - Sheffer, Alla

AU - Panozzo, Daniele

PY - 2018/11/1

Y1 - 2018/11/1

N2 - 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.

AB - 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.

KW - Fabrication

KW - Shape decomposition

UR - http://www.scopus.com/inward/record.url?scp=85055756106&partnerID=8YFLogxK

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U2 - 10.1145/3204458

DO - 10.1145/3204458

M3 - Article

VL - 37

JO - ACM Transactions on Graphics

JF - ACM Transactions on Graphics

SN - 0730-0301

IS - 5

M1 - 169

ER -