# An Approach for the Simulation of Ground and Honed Technical Surfaces for Training Classifiers

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Approach for the Generation of Artificial Ground and Honed Surfaces

**g**is

**G**= F(

**g**), so that:

**g**is convoluted with the square root of the given power spectral density in the frequency domain. To achieve the requested variance ${\sigma}_{tar}^{2}(z)$, the power spectral density has to be normalized. The simulation method used here is similar to the one presented by Wu [26]. Wu uses a complex vector with uniformly distributed phases and constant amplitude instead of the Fourier transform

**G**of a normal distributed matrix

**g**. With such an approach, the given autocorrelation function is exactly mapped into the simulated topographies e.g., it does not contain any stochastic variations as in the approach chosen for this paper (cf. Figure 2b,d)). The difference in the generated artificial surfaces is subtle (cf. Figure 2a,c).

_{1}= 100 µm, l

_{2}= 5 µm, a = 2, and ${\sigma}^{2}$ = 1 µm. Δx and Δy are chosen as 0.5 µm, the size of the simulated surface is 1000 × 1000 points.

## 3. Simulation of Directional Structures (Grinding Surfaces)

_{1}and l

_{2}are the correlation lengths in direction $\phi $ and $\phi +90\xb0$. The correlation length describes the distance, where the standardized autocorrelation function falls to h = e

^{−1}. In literature, other definitions for h exist, e.g., h = 0.2 [1]. The continuous autocorrelation function $\tilde{R}\left(x,y\right)$ is discretized and used as a circular autocorrelation function. Depending on the parameter selection, non-differentiable points and discontinuities might occur, and the associated power spectral density may not be real. In the simulation, this leads to deviations from the requested autocorrelation function. Those deviations become smaller, as the simulated topography becomes larger in relation to the chosen autocorrelation length. The parameters for the simulation of a Gaussian distributed surface are the expectation value µ, the variance ${\sigma}^{2}$, the correlation lengths l

_{1}and l

_{2}, the exponent a and the angle $\phi $ (cf. Figure 3).

## 4. Simulation of Honed Structures in Multiple Steps

#### 4.1. One-Step Honing Process

#### 4.2. Multi-Step Honing Processes

## 5. Approach for the Classification of Simulated Ground and Honed Surfaces Using Machine Learning

_{1}and l

_{2}were selected in order to create the process-specific directed structures. The angle, $\phi $, varies freely. For honed structures, the correlation lengths l

_{1}and l

_{2}and variances ${\sigma}_{1}^{2}$ and ${\sigma}_{2}^{2}$ were chosen in such a way that they correspond to a coarse and a fine process step, similar to plateau honing. The honing angle $\phi $ was varied in a wide range. The simulated data was converted into grayscale images and compressed to 50 × 50 pixels to reduce the computational effort. Examples of the simulated ground and honed surfaces used for training can be found in Figure 5.

- A two-layer CNN with 8 filters of the size 5 × 5 in the first layer and 4 filters of the size 10 × 10.
- A two-layer CNN with 6 filters of the size 5 × 5 in the first layer and 6 filters of the size 5 × 5.
- A three-layer CNN with 8 filters of the size 5 × 5 in the first layer and 4 filters of the size 5 × 5 in the 2nd and third layer.
- A three-layer CNN with 6 filters of size 5 × 5 in the first, second and third layer.

^{+}being the number of correct classified data and N = 2000 the overall data set size.

## 6. Verification of the Classifier Using Real Ground and Honed Surfaces

## 7. Summary and Outlook

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**(

**a**,

**b**): ground surface and its corresponding autocorrelation function simulated with the method of Wu (

**c**,

**d**): ground surface and corresponding autocorrelation function simulated with the proposed method.

**Figure 3.**Simulation of a grinding surface. Parameters: Δx = Δy = 0.5 µm; l

_{1}= 500 µm; l

_{2}= 5 µm; ϕ = 25°; σ = 5.68 µm; µ = 3.2 µm; Measurement field: 4000 × 4000 points; Field of view: 1000 × 1000 points.

**Figure 4.**(

**a**) Simulation of a one-step honed surface as the minimum of two simulated grinding surfaces. Parameters: As in Figure 3; (

**b**) Simulation of a plateau honed surface (Δx = Δy = 0.5 µm). Parameters step 1: l

_{1}= 500 µm; l

_{2}= 5 µm; ϕ = 25°; σ = 5.68 µm; µ = 3.2 µm. Parameters step 2: l

_{1}= 20 µm; l

_{2}= 2 µm; ϕ = 25°; σ = 1.8 µm; µ = 2.01 µm.

**Figure 5.**Examples of simulated ground (

**top**) and honed (

**bottom**) surfaces as compressed grayscale images.

**Figure 7.**Visualization of CNN weights for CNN architecture 1 (

**left**) and 3 (

**right**). Eventually untrained weights are marked in red, trained or interesting weights are marked green.

**Table 1.**Selected parameters for generating the training and test data. For parameters with a range, a uniform distribution over the range is assumed.

Parameter | Value |
---|---|

General parameters | |

Δx | 0.5 µm |

Δy | 0.5 µm |

nx | 500 |

ny | 500 |

a | 2 |

Parameters for ground surfaces simulation | |

l_{1} | 400 ± 100 µm |

l_{2} | 5 ± 0.8 µm |

${\sigma}^{2}$ | 22 ± 3 |

$\phi $ | 90 ± 90° (integer steps) |

Parameters for honed surfaces simulation | |

l_{1} (first processing step) | 500 ± 200 µm |

l_{2} (first processing step) | 30 ± 10 µm |

l_{1} (2nd processing step) | 5 ± 1 µm |

l_{2} (2nd processing step) | 3 ± 1 µm |

µ_{1} | 0 µm |

µ_{2} | 0 µm |

${\sigma}_{1}^{2}$ | 45 ± 5 |

${\sigma}_{2}^{2}$ | 0.2 ± 0.1 |

${\phi}_{1}$, ${\phi}_{2}$ | 45 ± 25° (integer steps) |

CNN Number | Detection Rate in % | |
---|---|---|

Simulated Test Data | Real Test Data | |

1. | 96.32 | 64.88 |

2. | 92.55 | 84.50 |

3. | 99.15 | 86.88 |

4. | 99.05 | 86.88 |

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**MDPI and ACS Style**

Rief, S.; Ströer, F.; Kieß, S.; Eifler, M.; Seewig, J.
An Approach for the Simulation of Ground and Honed Technical Surfaces for Training Classifiers. *Technologies* **2017**, *5*, 66.
https://doi.org/10.3390/technologies5040066

**AMA Style**

Rief S, Ströer F, Kieß S, Eifler M, Seewig J.
An Approach for the Simulation of Ground and Honed Technical Surfaces for Training Classifiers. *Technologies*. 2017; 5(4):66.
https://doi.org/10.3390/technologies5040066

**Chicago/Turabian Style**

Rief, Sebastian, Felix Ströer, Simon Kieß, Matthias Eifler, and Jörg Seewig.
2017. "An Approach for the Simulation of Ground and Honed Technical Surfaces for Training Classifiers" *Technologies* 5, no. 4: 66.
https://doi.org/10.3390/technologies5040066