### Abstract

The distribution of geometric features is anisotropic by its nature. Intrinsic properties of surfaces such as normal curvatures, for example, varies with direction. In this paper this characteristic of a shape is used to create a new anisotropic geodesic (AG) distance map on parametric surfaces. We first define local distance (LD) from a point as a function of both the surface point and a unit direction in its tangent plane and then define a total distance as an integral of that local distance. The AG distance between points on the surface is then defined as their minimum total distance. The path between the points that attains the minimum is called the anisotropic geodesic path. This differs from the usual geodesic in ways that enable it to better reveal geometric features. Minimizing total distances to attain AG distance is performed by associating the LD function with the tensor speed function that controls wave propagation of the convex Hamilton-Jacobi (H-J) equation solver. We present two different, but related metrics for the local distance function, a curvature tensor and a difference curvature tensor. Each creates a different AG distance. Some properties of both new AG distance maps are presented, including parametrization invariance. We then demonstrate the effectiveness of the proposed geodesic map as a shape discriminator in several applications, including surface segmentation and partial shape matching.

Original language | English |
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Title of host publication | IEEE International Conference on Shape Modeling and Applications 2008, Proceedings, SMI |

Pages | 179-186 |

Number of pages | 8 |

DOIs | |

Publication status | Published - 2008 Sep 9 |

Externally published | Yes |

Event | IEEE International Conference on Shape Modeling and Applications 2008, SMI - Stony Brook, NY, United States Duration: 2008 Jun 4 → 2008 Jun 6 |

### Other

Other | IEEE International Conference on Shape Modeling and Applications 2008, SMI |
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Country | United States |

City | Stony Brook, NY |

Period | 08/6/4 → 08/6/6 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science Applications
- Computational Theory and Mathematics

### Cite this

*IEEE International Conference on Shape Modeling and Applications 2008, Proceedings, SMI*(pp. 179-186). [4547968] https://doi.org/10.1109/SMI.2008.4547968

**Anisotropic geodesic distance computation for parametric surfaces.** / Seong, Jun Kyung; Jeong, Won Ki; Cohen, Elaine.

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

*IEEE International Conference on Shape Modeling and Applications 2008, Proceedings, SMI.*, 4547968, pp. 179-186, IEEE International Conference on Shape Modeling and Applications 2008, SMI, Stony Brook, NY, United States, 08/6/4. https://doi.org/10.1109/SMI.2008.4547968

}

TY - GEN

T1 - Anisotropic geodesic distance computation for parametric surfaces

AU - Seong, Jun Kyung

AU - Jeong, Won Ki

AU - Cohen, Elaine

PY - 2008/9/9

Y1 - 2008/9/9

N2 - The distribution of geometric features is anisotropic by its nature. Intrinsic properties of surfaces such as normal curvatures, for example, varies with direction. In this paper this characteristic of a shape is used to create a new anisotropic geodesic (AG) distance map on parametric surfaces. We first define local distance (LD) from a point as a function of both the surface point and a unit direction in its tangent plane and then define a total distance as an integral of that local distance. The AG distance between points on the surface is then defined as their minimum total distance. The path between the points that attains the minimum is called the anisotropic geodesic path. This differs from the usual geodesic in ways that enable it to better reveal geometric features. Minimizing total distances to attain AG distance is performed by associating the LD function with the tensor speed function that controls wave propagation of the convex Hamilton-Jacobi (H-J) equation solver. We present two different, but related metrics for the local distance function, a curvature tensor and a difference curvature tensor. Each creates a different AG distance. Some properties of both new AG distance maps are presented, including parametrization invariance. We then demonstrate the effectiveness of the proposed geodesic map as a shape discriminator in several applications, including surface segmentation and partial shape matching.

AB - The distribution of geometric features is anisotropic by its nature. Intrinsic properties of surfaces such as normal curvatures, for example, varies with direction. In this paper this characteristic of a shape is used to create a new anisotropic geodesic (AG) distance map on parametric surfaces. We first define local distance (LD) from a point as a function of both the surface point and a unit direction in its tangent plane and then define a total distance as an integral of that local distance. The AG distance between points on the surface is then defined as their minimum total distance. The path between the points that attains the minimum is called the anisotropic geodesic path. This differs from the usual geodesic in ways that enable it to better reveal geometric features. Minimizing total distances to attain AG distance is performed by associating the LD function with the tensor speed function that controls wave propagation of the convex Hamilton-Jacobi (H-J) equation solver. We present two different, but related metrics for the local distance function, a curvature tensor and a difference curvature tensor. Each creates a different AG distance. Some properties of both new AG distance maps are presented, including parametrization invariance. We then demonstrate the effectiveness of the proposed geodesic map as a shape discriminator in several applications, including surface segmentation and partial shape matching.

UR - http://www.scopus.com/inward/record.url?scp=50949117664&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=50949117664&partnerID=8YFLogxK

U2 - 10.1109/SMI.2008.4547968

DO - 10.1109/SMI.2008.4547968

M3 - Conference contribution

SN - 9781424422609

SP - 179

EP - 186

BT - IEEE International Conference on Shape Modeling and Applications 2008, Proceedings, SMI

ER -