# Sharp Feature Detection as a Useful Tool in Smart Manufacturing

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## Abstract

**:**

## 1. Introduction

#### Prior Work

## 2. Methods and Materials

#### 2.1. Edge Points Detection

#### 2.1.1. PCA—Normal Estimation

- $({\lambda}_{0}\le {\lambda}_{1}\le {\lambda}_{2})\wedge ({\lambda}_{1}\approx {\lambda}_{2})$
- $({\lambda}_{0}\le {\lambda}_{1}\le {\lambda}_{2})\wedge ({\lambda}_{0}\approx {\lambda}_{1})$
- $({\lambda}_{0}\le {\lambda}_{1}\le {\lambda}_{2})\wedge ({\lambda}_{0}\approx {\lambda}_{1}\approx {\lambda}_{2})$.

#### 2.1.2. Region Growing—Initial Normal Threshold Value $th{r}_{{\theta}_{N}}$

#### 2.2. B-Spline Edge Representation

## 3. Results and Discussion

#### 3.1. Estimation of Normal Threshold $th{r}_{{\theta}_{N}}$

#### 3.2. B-Spline Curve Fitting

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

RP | Rapid Prototyping |

RE | Reverse Engineering |

CAD | Computer Aided Design |

PCA | Principal Component Analysis |

STL | Standard Tessellation Language |

NURBS | Non-Uniform Rational B-spline |

## Appendix A

${\mathit{thr}}_{{\mathit{\theta}}_{\mathit{N}}}$ | 0.5 | 2 | 4 | 6 | 14 | |
---|---|---|---|---|---|---|

${M}_{1}$ | ||||||

$per{c}_{N}$ | 4.71 | 1.57 | 0.47 | 0.05 | 0.02 | |

${M}_{2}$ | ||||||

$per{c}_{N}$ | 4.34 | 3.36 | 1.71 | 0.85 | 0.85 | |

${M}_{3}$ | ||||||

$per{c}_{N}$ | 26.54 | 16.38 | 3.24 | 0.56 | 0.01 | |

${M}_{4}$ | ||||||

$per{c}_{N}$ | 29.27 | 18.65 | 2.87 | 0.38 | ≈ 0 | |

${M}_{5}$ | ||||||

$per{c}_{N}$ | 10.41 | 6.56 | 4.04 | 2.03 | 0.06 | |

${M}_{6}$ | ||||||

$per{c}_{N}$ | 67.14 | 37.25 | 14.10 | 5.09 | 0.09 |

Model | Chart | Regression |
---|---|---|

${M}_{1}$ | $-2.06\xb7\mathrm{ln}\left(x\right)+4.10$ | |

${M}_{2}$ | $-4.87\xb7\mathrm{ln}\left(x\right)+8.39$ | |

${M}_{3}$ | $-1.50\xb7\mathrm{ln}\left(x\right)+5.57$ | |

${M}_{4}$ | $-1.28\xb7\mathrm{ln}\left(x\right)+5.21$ | |

${M}_{5}$ | $-2.43\xb7\mathrm{ln}\left(x\right)+7.01$ | |

${M}_{6}$ | $-1.92\xb7\mathrm{ln}\left(x\right)+9.07$ |

**Table A3.**Comparison of B-spline curve approximation of testing clouds, CP is number of control points, ${w}_{\phi}$ is average weighted error.

${C}_{1}$ | |||||

CP | 4 | 10 | 20 | ||

${w}_{\phi}$[mm] | 2.35 | 1.15 | 0.83 | ||

${C}_{2}$ | |||||

CP | 4 | 10 | 20 | 40 | 80 |

${w}_{\phi}$[mm] | 14.01 | 5.54 | 2.02 | 1.25 | 0.99 |

${C}_{3}$ | |||||

CP | 20 | 40 | 80 | ||

${w}_{\phi}$[mm] | 3.96 | 0.96 | 0.85 | ||

${C}_{4}$ | |||||

CP | 80 | 100 | 160 | ||

${w}_{\phi}$[mm] | 0.41 | 0.37 | 0.27 | ||

${C}_{5}$ | |||||

CP | 10 | 40 | 80 | ||

${w}_{\phi}$[mm] | 2.19 | 0.67 | 0.33 |

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**Figure 5.**Testing point clouds: (

**a**) ${M}_{\mathrm{ffl}}$, (

**b**) ${M}_{2}$, (

**c**) ${M}_{3}$, (

**d**) ${M}_{4}$, (

**e**) ${M}_{5}$, (

**f**) ${M}_{6}$.

**Figure 7.**(

**a**) Normal vector is on the opposite side of the point ${Q}_{k}$, distance ${d}_{k}$ is negative, (

**b**) the graphs of the functions ${w}_{\phi}$ (red line) and ${f}_{\phi}$ (blue line).

**Figure 9.**Set of testing point clouds: (

**a**) ${C}_{1}$, (

**b**) ${C}_{2}$, (

**c**) ${C}_{3}$, (

**d**) ${C}_{4}$, (

**e**) ${C}_{5}$.

**Figure 10.**(

**a**) The initial B-spline curve of the cloud points (black crosses) and new control points (red crosses), (

**b**) the distance of the point cloud points and the computed curve line.

Model | ${\mathit{M}}_{1}$ | ${\mathit{M}}_{2}$ | ${\mathit{M}}_{3}$ | ${\mathit{M}}_{4}$ | ${\mathit{M}}_{5}$ | ${\mathit{M}}_{6}$ |
---|---|---|---|---|---|---|

${s}_{in}$ | 87,848 | 67,394 | 195,815 | 123,112 | 103,534 | 90,209 |

Input | Output | Input | Output | ||
---|---|---|---|---|---|

${M}_{1}$ | ${M}_{2}$ | ||||

${M}_{3}$ | ${M}_{4}$ | ||||

${M}_{5}$ | ${M}_{6}$ |

(a) | (b) | (c) | (d) |
---|---|---|---|

$th{r}_{{\theta}_{N}}=4$ | $th{r}_{{\theta}_{N}}=1.2$ | $th{r}_{{\theta}_{N}}=1$ | $th{r}_{{\theta}_{N}}=1.5$ |

Model Curve | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ | ${\mathit{C}}_{5}$ |
---|---|---|---|---|---|

${s}_{in}$ | 1632 | 1307 | 2560 | 2477 | 3130 |

**Table 5.**Computation of approximation error ${E}_{m}$ and ${E}_{gold}$ using different values of $\sigma $.

$\mathit{\sigma}$ | 0.02 | 0.09 | 0.1 |
---|---|---|---|

${E}_{m}$ | 0.9387 | 0.9661 | 0.9702 |

${E}_{gold}$ | 0.9382 | 0.9560 | 0.9604 |

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## Share and Cite

**MDPI and ACS Style**

Prochazkova, J.; Procházka, D.; Landa, J. Sharp Feature Detection as a Useful Tool in Smart Manufacturing. *ISPRS Int. J. Geo-Inf.* **2020**, *9*, 422.
https://doi.org/10.3390/ijgi9070422

**AMA Style**

Prochazkova J, Procházka D, Landa J. Sharp Feature Detection as a Useful Tool in Smart Manufacturing. *ISPRS International Journal of Geo-Information*. 2020; 9(7):422.
https://doi.org/10.3390/ijgi9070422

**Chicago/Turabian Style**

Prochazkova, Jana, David Procházka, and Jaromír Landa. 2020. "Sharp Feature Detection as a Useful Tool in Smart Manufacturing" *ISPRS International Journal of Geo-Information* 9, no. 7: 422.
https://doi.org/10.3390/ijgi9070422