### Abstract

We present an efficient and robust algorithm for computing continuous visibility for two- or three-dimensional shapes whose boundaries are NURBS curves or surfaces by lifting the problem into a higher dimensional parameter space. This higher dimensional formulation enables solving for the visible regions over all view directions in the domain simultaneously, therefore providing a reliable and fast computation of the visibility chart, a structure which simultaneously encodes the visible part of the shape's boundary from every view in the domain. In this framework, visible parts of planar curves are computed by solving two polynomial equations in three variables (t and r for curve parameters and θ for a view direction). Since one of the two equations is an inequality constraint, this formulation yields two-manifold surfaces as a zero-set in a 3-D parameter space. Considering a projection of the two-manifolds onto the tθ-plane, a curve's location is invisible if its corresponding parameter belongs to the projected region. The problem of computing hidden curve removal is then reduced to that of computing the projected region of the zero-set in the tθ-domain. We recast the problem of computing boundary curves of the projected regions into that of solving three polynomial constraints in three variables, one of which is an inequality constraint. A topological structure of the visibility chart is analyzed in the same framework, which provides a reliable solution to the hidden curve removal problem. Our approach has also been extended to the surface case where we have two degrees of freedom for a view direction and two for the model parameter. The effectiveness of our approach is demonstrated with several experimental results.

Original language | English |
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Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Pages | 451-464 |

Number of pages | 14 |

Volume | 4077 LNCS |

Publication status | Published - 2006 Oct 9 |

Externally published | Yes |

Event | 4th International Conference on Geometric Modeling and Processing, GMP 2006 - Pittsburgh, PA, United States Duration: 2006 Jul 26 → 2006 Jul 28 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 4077 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 4th International Conference on Geometric Modeling and Processing, GMP 2006 |
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Country | United States |

City | Pittsburgh, PA |

Period | 06/7/26 → 06/7/28 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Biochemistry, Genetics and Molecular Biology(all)
- Theoretical Computer Science

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(Vol. 4077 LNCS, pp. 451-464). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4077 LNCS).

**Simultaneous precise solutions to the visibility problem of sculptured models.** / Seong, Jun Kyung; Elber, Gershon; Cohen, Elaine.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics).*vol. 4077 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 4077 LNCS, pp. 451-464, 4th International Conference on Geometric Modeling and Processing, GMP 2006, Pittsburgh, PA, United States, 06/7/26.

}

TY - GEN

T1 - Simultaneous precise solutions to the visibility problem of sculptured models

AU - Seong, Jun Kyung

AU - Elber, Gershon

AU - Cohen, Elaine

PY - 2006/10/9

Y1 - 2006/10/9

N2 - We present an efficient and robust algorithm for computing continuous visibility for two- or three-dimensional shapes whose boundaries are NURBS curves or surfaces by lifting the problem into a higher dimensional parameter space. This higher dimensional formulation enables solving for the visible regions over all view directions in the domain simultaneously, therefore providing a reliable and fast computation of the visibility chart, a structure which simultaneously encodes the visible part of the shape's boundary from every view in the domain. In this framework, visible parts of planar curves are computed by solving two polynomial equations in three variables (t and r for curve parameters and θ for a view direction). Since one of the two equations is an inequality constraint, this formulation yields two-manifold surfaces as a zero-set in a 3-D parameter space. Considering a projection of the two-manifolds onto the tθ-plane, a curve's location is invisible if its corresponding parameter belongs to the projected region. The problem of computing hidden curve removal is then reduced to that of computing the projected region of the zero-set in the tθ-domain. We recast the problem of computing boundary curves of the projected regions into that of solving three polynomial constraints in three variables, one of which is an inequality constraint. A topological structure of the visibility chart is analyzed in the same framework, which provides a reliable solution to the hidden curve removal problem. Our approach has also been extended to the surface case where we have two degrees of freedom for a view direction and two for the model parameter. The effectiveness of our approach is demonstrated with several experimental results.

AB - We present an efficient and robust algorithm for computing continuous visibility for two- or three-dimensional shapes whose boundaries are NURBS curves or surfaces by lifting the problem into a higher dimensional parameter space. This higher dimensional formulation enables solving for the visible regions over all view directions in the domain simultaneously, therefore providing a reliable and fast computation of the visibility chart, a structure which simultaneously encodes the visible part of the shape's boundary from every view in the domain. In this framework, visible parts of planar curves are computed by solving two polynomial equations in three variables (t and r for curve parameters and θ for a view direction). Since one of the two equations is an inequality constraint, this formulation yields two-manifold surfaces as a zero-set in a 3-D parameter space. Considering a projection of the two-manifolds onto the tθ-plane, a curve's location is invisible if its corresponding parameter belongs to the projected region. The problem of computing hidden curve removal is then reduced to that of computing the projected region of the zero-set in the tθ-domain. We recast the problem of computing boundary curves of the projected regions into that of solving three polynomial constraints in three variables, one of which is an inequality constraint. A topological structure of the visibility chart is analyzed in the same framework, which provides a reliable solution to the hidden curve removal problem. Our approach has also been extended to the surface case where we have two degrees of freedom for a view direction and two for the model parameter. The effectiveness of our approach is demonstrated with several experimental results.

UR - http://www.scopus.com/inward/record.url?scp=33749321584&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33749321584&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:33749321584

SN - 9783540367116

VL - 4077 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 451

EP - 464

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

ER -