## Abstract

The framework of boxels is developed to represent 2.5D datasets, such as urban environments. Boxels are axis-aligned non-intersecting boxes which can be used to directly represent objects in the scene or as bounding volumes. Guibas and Yao have shown that axis-aligned disjoint rectangles in the plane can be ordered into four total orders so that any ray meets them in one of the four orders. This is also applicable to boxels, and it is shown that there exist four different partitionings of the boxels into ordered sequences of disjoint sets, called antichains, so that boxels in one antichain can act as occluders of the boxels in subsequent antichains. The expected runtime for the antichain partitioning is O(nlogn), where n is the number of boxels. This partitioning can be used for the efficient implementation of virtual drivethroughs and ray tracing. Boxels can also be easily organized into hierarchies to speed up the rendering. For drivethroughs, the antichains are processed in front-to-back order together with a run-length encoding of the boxel horizon, yielding real-time rendering of scenes with up to 300,000 buildings. For ray tracing, a ray intersects at most one boxel in an antichain, and the time to determine that boxel is O(1) for most "natural" scenes, and at worst, logarithmic in the size of the antichain. Objects which are not axis-aligned can also be handled by a simple modification. Boxel rendering can also be parallelized for multi-core machines.

Original language | English |
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Pages (from-to) | 167-187 |

Number of pages | 21 |

Journal | Computational Geometry: Theory and Applications |

Volume | 41 |

Issue number | 3 |

DOIs | |

State | Published - Nov 2008 |

### Bibliographical note

Funding Information:This research is based on the first author’s MSc thesis in Computer Science, carried out under the second author’s supervision at the University of Haifa. Thanks are due to the Caesarea Rothschild Institute for its generous support. This research was also supported in part by Grant No. 01-01-01509 from the Israel Ministry of Science and Technol- ogy. The authors wish to thank Ilan Newman for helpful discussions concerning the probabilistic model. Thanks are also due to the anonymous referees for their helpful comments.

## Keywords

- 2.5D data
- Antichains
- Axis-aligned
- Boxels
- Drivethrough
- Front-to-back rendering
- Horizon
- Partial order
- Ray-tracing
- Rectangles
- Urban scene
- Virtual reality
- Walkthrough

## ASJC Scopus subject areas

- Computer Science Applications
- Geometry and Topology
- Control and Optimization
- Computational Theory and Mathematics
- Computational Mathematics