### Abstract

Reconstructing a free-form surface from 3-dimensional (3D) noisy measurements is a central problem in inspection, statistical quality control, and reverse engineering. We present a new method for the statistical reconstruction of a free-form surface patch based on 3D point cloud data. The surface is represented parametrically, with each of the three Cartesian coordinates (x, y, z) a function of surface coordinates (u, v), a model form compatible with computer-aided-design (CAD) models. This model form also avoids having to choose one Euclidean coordinate (say, z) as a "response" function of the other two coordinate "locations" (say, x and y), as commonly used in previous Euclidean kriging models of manufacturing data. The (u, v) surface coordinates are computed using parameterization algorithms from the manifold learning and computer graphics literature. These are then used as locations in a spatial Gaussian process model that considers correlations between two points on the surface a function of their geodesic distance on the surface, rather than a function of their Euclidean distances over the xy plane. We show how the proposed geodesic Gaussian process (GGP) approach better reconstructs the true surface, filtering the measurement noise, than when using a standard Euclidean kriging model of the "heights", that is, z(x, y). The methodology is applied to simulated surface data and to a real dataset obtained with a noncontact laser scanner. Supplementary materials are available online.

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

Pages (from-to) | 87-99 |

Number of pages | 13 |

Journal | Technometrics |

Volume | 57 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2 2015 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Modeling and Simulation
- Applied Mathematics

### Cite this

*Technometrics*,

*57*(1), 87-99. https://doi.org/10.1080/00401706.2013.879075

}

*Technometrics*, vol. 57, no. 1, pp. 87-99. https://doi.org/10.1080/00401706.2013.879075

**Geodesic gaussian processes for the parametric reconstruction of a free-form surface.** / Del Castillo, Enrique; Colosimo, Bianca M.; Tajbakhsh, Sam Davanloo.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Geodesic gaussian processes for the parametric reconstruction of a free-form surface

AU - Del Castillo, Enrique

AU - Colosimo, Bianca M.

AU - Tajbakhsh, Sam Davanloo

PY - 2015/1/2

Y1 - 2015/1/2

N2 - Reconstructing a free-form surface from 3-dimensional (3D) noisy measurements is a central problem in inspection, statistical quality control, and reverse engineering. We present a new method for the statistical reconstruction of a free-form surface patch based on 3D point cloud data. The surface is represented parametrically, with each of the three Cartesian coordinates (x, y, z) a function of surface coordinates (u, v), a model form compatible with computer-aided-design (CAD) models. This model form also avoids having to choose one Euclidean coordinate (say, z) as a "response" function of the other two coordinate "locations" (say, x and y), as commonly used in previous Euclidean kriging models of manufacturing data. The (u, v) surface coordinates are computed using parameterization algorithms from the manifold learning and computer graphics literature. These are then used as locations in a spatial Gaussian process model that considers correlations between two points on the surface a function of their geodesic distance on the surface, rather than a function of their Euclidean distances over the xy plane. We show how the proposed geodesic Gaussian process (GGP) approach better reconstructs the true surface, filtering the measurement noise, than when using a standard Euclidean kriging model of the "heights", that is, z(x, y). The methodology is applied to simulated surface data and to a real dataset obtained with a noncontact laser scanner. Supplementary materials are available online.

AB - Reconstructing a free-form surface from 3-dimensional (3D) noisy measurements is a central problem in inspection, statistical quality control, and reverse engineering. We present a new method for the statistical reconstruction of a free-form surface patch based on 3D point cloud data. The surface is represented parametrically, with each of the three Cartesian coordinates (x, y, z) a function of surface coordinates (u, v), a model form compatible with computer-aided-design (CAD) models. This model form also avoids having to choose one Euclidean coordinate (say, z) as a "response" function of the other two coordinate "locations" (say, x and y), as commonly used in previous Euclidean kriging models of manufacturing data. The (u, v) surface coordinates are computed using parameterization algorithms from the manifold learning and computer graphics literature. These are then used as locations in a spatial Gaussian process model that considers correlations between two points on the surface a function of their geodesic distance on the surface, rather than a function of their Euclidean distances over the xy plane. We show how the proposed geodesic Gaussian process (GGP) approach better reconstructs the true surface, filtering the measurement noise, than when using a standard Euclidean kriging model of the "heights", that is, z(x, y). The methodology is applied to simulated surface data and to a real dataset obtained with a noncontact laser scanner. Supplementary materials are available online.

UR - http://www.scopus.com/inward/record.url?scp=84922972551&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84922972551&partnerID=8YFLogxK

U2 - 10.1080/00401706.2013.879075

DO - 10.1080/00401706.2013.879075

M3 - Article

AN - SCOPUS:84922972551

VL - 57

SP - 87

EP - 99

JO - Technometrics

JF - Technometrics

SN - 0040-1706

IS - 1

ER -