# Surface Approximation by Means of Gaussian Process Latent Variable Models and Line Element Geometry

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## Abstract

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## 1. Introduction

- We expand existing methods based on kinematic surfaces and line element geometry by introducing GPLVM to describe surfaces in a non-linear way.
- We apply our method to surface approximation.
- We test our method to perform unsupervised surface segmentation.
- We demonstrate our method to perform surface denoising.

## 2. Materials and Methods

#### 2.1. Line Element Geometry

#### 2.2. Kinematic Surfaces

**Theorem**

**1.**

- $\gamma =0$:
- -
- $\mathbf{c}=0,\overline{\mathbf{c}}=0$: $M\left(t\right)$ is the identical motion (no motion at all).
- -
- $\mathbf{c}=0,\overline{\mathbf{c}}\ne 0$: $M\left(t\right)$ is a translation along $\overline{\mathbf{c}}$ and $\Phi $ is a cylinder (not necessarily of revolution).
- -
- $\mathbf{c}\ne 0,\mathbf{c}\xb7\overline{\mathbf{c}}=0$: $M\left(t\right)$ is a rotation about an axis parallel to $\mathbf{c}$ and $\Phi $ is a surface of revolution.
- -
- $\mathbf{c}\ne 0,\mathbf{c}\xb7\overline{\mathbf{c}}\ne 0$: $M\left(t\right)$ is a helical motion about an axis parallel to $\mathbf{c}$ and $\Phi $ is a helical surface.

- $\gamma \ne 0$:
- -
- $\mathbf{c}\ne 0$: $M\left(t\right)$ is a spiral motion and $\Phi $ is a spiral surface.
- -
- $\mathbf{c}=0$: $M\left(t\right)$ is a central similarity, and $\Phi $ is a conical surface (not necessarily of revolution).

#### 2.3. Approximating the Complex

- $k=4$: Only a plane is invariant to four independent uniform motions.
- $k=3$: A sphere is invariant to three independent uniform motions (all rotations).
- $k=2$: The surface is either a cylinder of revolution, a cone of revolution or a spiral cylinder.
- $k=1$: The surface is either a cylinder without revolution (pure translation), a cone without revolution (central similarity), a surface of revolution, a helical surface or a spiral surface.

#### 2.4. Gaussian Processes

#### 2.5. Gaussian Process Latent Variable Models

#### 2.6. Our Approach

## 3. Results

#### 3.1. Surface Approximation

#### 3.2. Surface Segmentation

#### 3.3. Surface Denoising

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**ARD contributions for the dimensions of the latent space for the examples of equiform kinematic surfaces.

**Figure 3.**A 2D representation of the points of the kinematic surfaces in their latent space. The amount of black in the background indicates the posterior uncertainty of the BGPLVM.

**Figure 6.**Three synthetically generated 3D models by combining primitive surfaces. The BCGPLVM is able to show distinguishable structures for points in latent space.

**Figure 7.**The results for a real-world scanned metal hinge. Again, the BGPLVM is able to separate the points in latent space.

**Figure 8.**A bend torus. Noise is added to the entire surface. Moreover, A hundred vertices were translated. (

**b**) The BGPLVM is able to smooth out the surface. Blue are the noisy points. Red are the denoised points.

**Table 1.**An overview of the surfaces and their corresponding GPLVM and properties. Larger point clouds are subsampled uniformly to a smaller set. The number of points retained for training is given in # Train. Noise is the Gaussian noise variance hyperparameter for the GPLVM model, which is either a fixed value or a value that has to be learned along with the other hyperparameters. The number of inducing points for the Bayesian GPLVM is shown in # Ind. To make sure we do not end up in local minima, we restarted the training of the model a number of times given in # Restarts.

Model | # Vertices | # Train | Noise | # Ind | # Restarts | |
---|---|---|---|---|---|---|

Cylinder of revolution | BGPLVM | 2176 | 2176 | $1\times {10}^{-4}$ | 15 | 10 |

Cone of revolution | BGPLVM | 2176 | 2176 | free | 50 | 10 |

Spiral cylinder | BGPLVM | 2210 | 2210 | free | 25 | 10 |

Cylinder w/o revolution | BGPLVM | 1717 | 1717 | free | 50 | 10 |

Cone w/o revolution | BGPLVM | 2210 | 2210 | free | 50 | 10 |

Surface of revolution | BGPLVM | 2816 | 2500 | free | 50 | 10 |

Helical surface | BGPLVM | 2582 | 2500 | free | 25 | 10 |

Spiral surface | BGPLVM | 1842 | 1842 | free | 50 | 10 |

Torus | BGPLVM | 2048 | 2048 | free | 50 | 10 |

Torus bend | BGPLVM | 2048 | 2048 | free | 25 | 10 |

Pear | BGPLVM | 6356 | 5000 | free | 50 | 5 |

Mixture 1 | BCGPLVM | 10,653 | 1000 | $1\times {10}^{-6}$ | NA | 3 |

Mixture 2 | BCGPLVM | 6555 | 1000 | $1\times {10}^{-6}$ | NA | 3 |

Mixture 3 | BCGPLVM | 15,681 | 1000 | $1\times {10}^{-4}$ | NA | 3 |

Hinge | BGPLVM | 22,282 | 5000 | free | 50 | 3 |

Torus bend noisy | BGPLVM | 8192 | 8192 | free | 50 | 3 |

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**MDPI and ACS Style**

De Boi, I.; Ek, C.H.; Penne, R.
Surface Approximation by Means of Gaussian Process Latent Variable Models and Line Element Geometry. *Mathematics* **2023**, *11*, 380.
https://doi.org/10.3390/math11020380

**AMA Style**

De Boi I, Ek CH, Penne R.
Surface Approximation by Means of Gaussian Process Latent Variable Models and Line Element Geometry. *Mathematics*. 2023; 11(2):380.
https://doi.org/10.3390/math11020380

**Chicago/Turabian Style**

De Boi, Ivan, Carl Henrik Ek, and Rudi Penne.
2023. "Surface Approximation by Means of Gaussian Process Latent Variable Models and Line Element Geometry" *Mathematics* 11, no. 2: 380.
https://doi.org/10.3390/math11020380