# Discrimination of Surface Topographies Created by Two-Stage Process by Means of Multiscale Analysis

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}, when the F-test was applied. This could not be achieved using conventional parameters [7,8].

## 2. Materials and Methods

#### 2.1. Sample Preparations

#### 2.2. Measurements

^{®}software (Digital Surf, Besançon, France).

#### 2.3. Multiscale and Statistical Analysis

_{uc}, followed by filtration with a high-pass filter at the lower wavelength cutoff, λ

_{lc}. The cutoffs refer to the wavelength where the filter has approximately 50% transmission. In this work, we used three different bandwidths, namely 20, 50, and 100 µm, which overlapped each other, which is similar to approach A from [26]. Lower and upper cutoff wavelengths are shown in Table 1. The number of bands depended on their widths and changed from 13 for the narrowest to 5 for the widest. The periodicity of the acquired topographies was evaluated with MountainsMap software to verify if the Gaussian filter could be applied in this study.

_{abs}, Hq

_{abs}, Ka

_{abs}, Kq

_{abs}, κ1a

_{abs}, κ1q

_{abs}, κ2a

_{abs}, κ1q

_{abs}, Ha

_{abs}, Hq

_{abs}, Ka

_{abs}, and Kq

_{abs}. Please note that the term “abs” in the subscript refers to the unsigned curvature. The full list of conventional profile and areal as well as length- and area-scale and curvature parameters used in this study is presented in Table A1 (Appendix A).

## 3. Results and Analysis

#### 3.1. Surface Topographies

#### 3.2. Bandpass Filter

#### 3.3. Length- and Area-Scale Analysis

^{2}for area and <150 μm for length), upper, inner, and outer regions of specimen B took evidently smaller values of the geometric measures when compared to the others. The lower surface of the same ring appeared to be similar to the lower surface of ring A, when considering the same range of scales. At larger scales, the results for both rings could be clearly differentiated. This became even clearer when considering complexity (Figure 9). Surface topographies measured on specimen B exhibited noticeably lower fractal complexity than their counterparts. This might indicate that second-stage processing via mass-finishing plays a dominant role in the formation of distinctive surface topographies. Differences between surfaces expressed through area- and length-scale analysis of the corresponding location on both rings were rather subtle and hard to be visually detected based on the figures.

#### 3.4. Curvature

#### 3.5. Discrimination Analysis

^{2}). The same phenomenon was noted for Lsfc and scales up to 150 μm.

## 4. Discussion

## 5. Conclusions

- All four studied multiscale methods performed generally well in discriminating against each factor and their combinations;
- Bandpass filtration using Sa and Sq exhibited the best performance as p-value < 0.05 for all three bands. Skewness and kurtosis failed to be used as a statistical discriminator against mass finishing for a bandwidth equal to 50 µm. Other height parameters did not generally provide sufficient confidence levels in the studied case. Hybrid and volume group parameters performed well at differentiation between surface topographies considering all factors with the exceptions of Vm and Vmp for the narrowest and shortest bandwidth (60–80 µm) when considering both factors as superpositions. Most feature parameters can be used to discriminate surfaces taking into account only the location on each ring. Spatial parameters were found to perform poorly. Similar conclusions can be drawn for profile characterization counterparts;
- Asfc and Lsfc both exhibited superior performance in discrimination against all factors and their combinations when compared to RelL and RelA. The latter parameters could be used only when finer scales were considered. Discrimination against mass-finishing was confident no matter the parameter derived from those methods;
- Curvature can be used as a discriminant only when considering measures of variability. Unsigned curvature provides significantly confident discrimination for finer scales.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Abbreviation | Full Name |
---|---|

Rp | Maximum peak height of the roughness profile |

Rv | Maximum valley depth of the roughness profile |

Rz | Maximum height of roughness profile |

Rc | Mean height of the roughness profile elements |

Rt | Total height of roughness profile |

Ra | Arithmetic mean deviation of the roughness profile |

Rq | Root-mean-square (RMS) deviation of the roughness profile |

Rsk | Skewness of the roughness profile |

Rku | Kurtosis of the roughness profile |

Sq | Root-mean-square height |

Ssk | Skewness |

Sku | Kurtosis |

Sp | Maximum peak height |

Sv | Maximum pit height |

Sz | Maximum height |

Sa | Arithmetic mean height |

Smr | Areal material ratio |

Smc | Inverse areal material ratio |

Sxp | Extreme peak height |

Sal | Autocorrelation length |

Str | Texture-aspect ratio |

Std | Texture direction |

Sdq | Root-mean-square gradient |

Sdr | Developed interfacial area ratio |

Vm | Material volume |

Vv | Void volume |

Vmp | Peak material volume |

Vmc | Core material volume |

Vvc | Core void volume |

Vvv | Pit void volume |

Spd | Density of peak |

Spc | Arithmetic mean peak curvature |

S10z | Ten point height |

S5p | Five point peak height |

S5v | Five point pit height |

Sda | Mean dale area |

Sha | Mean hill area |

Sdv | Mean dale volume |

Shv | Mean hill volume |

Sku | Core roughness depth |

Spk | Reduced summit height |

Svk | Reduced valley depth |

Smr1 | Upper bearing area |

Smr2 | Lower bearing area |

RelL | Relative length |

RelA | Relative area |

Lsfc | Length-scale fractal complexity |

Asfc | Area-scale fractal complexity |

κ1a | Average maximum curvature |

κ1q | Standard deviation of maximum curvature |

κ2a | Average minimum curvature |

κ2q | Standard deviation of minimum curvature |

Ha | Average mean curvature |

Hq | Standard deviation of mean curvature |

Ka | Average Gaussian curvature |

Kq | Standard deviation of Gaussian curvature |

κ1a_{abs} | Average absolute maximum curvature |

κ1q_{abs} | Standard deviation of absolute maximum curvature |

κ2a_{abs} | Average absolute minimum curvature |

κ2q_{abs} | Standard deviation of absolute minimum curvature |

Ha_{abs} | Average absolute mean curvature |

Hq_{abs} | Standard deviation of absolute mean curvature |

Ka_{abs} | Average absolute Gaussian curvature |

Kq_{abs} | Standard deviation of absolute Gaussian curvature |

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**Figure 2.**Ring during hot-rolling: 1—conical tools shaping upper and lower surfaces, 2—guiding roller, 3—internal roller and 4—external rollers.

**Figure 3.**Ring R1 (

**top row**) and R2 (

**bottom row**); images in right column represent the magnified views of the corresponding rings.

**Figure 4.**Renderings of topographic measurements capturing four different locations: upper, bottom, internal, and external of each of the two rings A and B.

**Figure 5.**Renderings of bandpass filtered topographies at the upper location of Specimen A using three different bandwidths: 20, 50, and 100 µm. Please note that HP and LP stand for cut-off wavelengths of high- and low-pass filters, respectively.

**Figure 6.**Evolution of arithmetic mean height (Sa) with scale for three different bandwidths: (

**a**) 20, (

**b**) 50, and (

**c**) 100 µm.

**Figure 7.**Evolution of texture direction (Std) with scale for three different bandwidths: (

**a**) 20, (

**b**) 50, and (

**c**) 100 µm.

**Figure 9.**Evolution of area-scale (

**a**) and length-scale (

**b**) fractal complexity as a function of scale.

**Figure 11.**Evolution of p-value calculated in the discrimination analysis of two independent formation factors and their product as a function of scale for area- and length-scale parameters: RelA (

**top left**), Asfc (

**top right**), RelL (

**bottom left**), Lsfc (

**bottom right**). Please note that red line indicates p = 0.05 at each graph.

**Figure 12.**Evolution of p-value calculated in the discrimination analysis of two independent formation factors and their product as a function of scale for average absolute maximum curvature (

**left**) and their signed counterpart (

**right**). Please note that the red line indicates p = 0.05 at each graph.

Bandwidth = 20 μm | Bandwidth = 50 μm | Bandwidth = 100 μm | ||||||
---|---|---|---|---|---|---|---|---|

λ_{lc} (μm) | λ_{center} (μm) | λ_{uc} (μm) | λ_{lc} (μm) | λ_{center} (μm) | λ_{uc} (μm) | λ_{lc} (μm) | λ_{center} (μm) | λ_{uc} (μm) |

60 | 70 | 80 | 65 | 90 | 115 | 60 | 110 | 160 |

70 | 80 | 90 | 75 | 100 | 125 | 70 | 120 | 170 |

80 | 90 | 100 | 66 | 110 | 135 | 80 | 130 | 180 |

90 | 100 | 110 | 95 | 120 | 145 | 90 | 140 | 190 |

100 | 110 | 120 | 105 | 130 | 155 | 100 | 150 | 198 |

110 | 120 | 130 | 115 | 140 | 165 | – | – | – |

120 | 130 | 140 | 125 | 150 | 175 | – | – | – |

130 | 140 | 150 | 135 | 160 | 185 | – | – | – |

140 | 150 | 160 | 145 | 170 | 195 | – | – | – |

150 | 160 | 170 | – | – | – | – | – | – |

160 | 170 | 180 | – | – | – | – | – | – |

170 | 180 | 190 | – | – | – | – | – | – |

180 | 190 | 198 | – | – | – | – | – | – |

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**MDPI and ACS Style**

Bartkowiak, T.; Grochalski, K.; Gapiński, B.; Wieczorowski, M. Discrimination of Surface Topographies Created by Two-Stage Process by Means of Multiscale Analysis. *Materials* **2021**, *14*, 7044.
https://doi.org/10.3390/ma14227044

**AMA Style**

Bartkowiak T, Grochalski K, Gapiński B, Wieczorowski M. Discrimination of Surface Topographies Created by Two-Stage Process by Means of Multiscale Analysis. *Materials*. 2021; 14(22):7044.
https://doi.org/10.3390/ma14227044

**Chicago/Turabian Style**

Bartkowiak, Tomasz, Karol Grochalski, Bartosz Gapiński, and Michał Wieczorowski. 2021. "Discrimination of Surface Topographies Created by Two-Stage Process by Means of Multiscale Analysis" *Materials* 14, no. 22: 7044.
https://doi.org/10.3390/ma14227044