Characterization of aperiodic surfaces with mesh-based parameters

For technical surfaces, it is important to know their functional purpose and to characterize them accordingly. Therefore, ISO 21920–2 in 2D and ISO 25178–2 in 3D offer parameters that can assess surface functional properties. The topographic portions of a surface, for example hills and dales, can be classified as features and evaluated using feature parameters. However, no parameter exists to describe the spatial distribution of features with regard to the degree of homogeneity for aperiodic surfaces. Here we show the application of the Delaunay triangulation to quantify the spatial distribution respectively the geometric relationship of features. Therefore, the feature points are determined by watershed analysis and the resulting point cloud is meshed in 2D. Based on that mean and standard deviation of the triangle side lengths and the area disorder (AD) are calculated as new parameters. The method is demonstrated for sandblasted and chrome-plated specimens. In addition simulation is used to generate more data for analysis. With the proposed approach the distinction and extent of uniform, homogeneous or inhomogeneous spatial distributions of features with parameter AD can be determined.


Introduction
The surface and its texture have a direct impact on functions like optical appearance or tribological behaviour. Therefore it is important to describe the surface properties according to the function.
In this paper two different examples of manufacturing processes, that are applied to ensure special functions of the surface, are analyzed. The first one are chrome-plated surfaces with spherical particles that are used for transportation rolls e.g. in carbon processing, printing or for injection moulds. Here the application are rope pulleys (figure 1), where friction between the rope and the groove is needed to ensure power transmission. Simultaneously the abrasive wear of the rope has to be minimized to increase fatigue life. In regard to the function the contact area formed by the peaks is important and the distribution of those has to be described.
The second example are surfaces treated with abrasive blasting. One application of blasting is to fulfill optical requirements. Different parts of assemblies like watchcase and wristband of watches or frame and temple of glasses have to look equal to satisfy customer requirements. Another application are functional surfaces for surgical instruments. The requirements are not reflective, smooth to be sterilizable and good feel to prevent slipping. For implants the surface has to be rough to enhance the ingrowth and also sterilizable to prevent nucleation. Another application is to generate a homogenized surface for following post-treatment steps. For all applications it is important that the resulting surface shows a homogeneous pattern that has to be evaluated and described.
Therefore the subject of characterization is the spatial distribution of features. Complete homogeneity without variation equals the distribution in form of a uniform lattice structure see figure 2(a). If the points diverge from this ideal position the disorder increases. Depending on the dispersion of the points in different regions of the underlaying space the distribution can be described as homogeneous (figure 2(b)) or inhomogeneous (figure 2(c)). Where an inhomogenous point process tends to have more or less points on average in certain regions.
ISO 21920-2 [1] offers various parameters for the analysis of 2D profiles. ISO 25178-2 [2] expands the possible parameters for the analysis of 3D surfaces. Both standards do not offer parameters for the description of the distribution of features. For 3D the feature parameter Spd exists which is the number of peaks in relation to the area. But there is still no knowledge if these peaks are evenly distributed or concentrated in a specific region. The homogeneity of the features cannot be described using standard parameters. We present a new analysis method for the homogeneity of aperiodic surfaces based on the segmentation of features and Delaunay triangulation.
In general the homogeneity of a point cloud is mathematically described [3]. However, there is no accepted metric for the level of inhomogeneity of a given pattern [3]. Therefore no generalization can be made and results are only valid for the particular case [3].
There is research in the field of distribution of Nanoparticles in composites. Zhang et al present an overview of methods to characterize the homogeneity of inclusions in composites [4]. These are based either on inclusion features, on inclusion position or based on both, such as the Delaunay triangulation. Various articles describe the application of Delaunay triangulation for the dispersion of particles in composites [5][6][7][8].
Other examples of the application of the proposed method are the use for the classification of polyps [9], the dispersion of nanosilica in cement [10], the characterization of nanopores [11] or recycled thermoplastic composites [12]. It is proven, that Delaunay triangulation and the calculated parameters can detect clustering [4]. It is possible to distinguish between poor and good dispersion and to describe homogeneity. However it has to be kept in mind, that it is insufficent for characterizing small variations of inhomogeneity [4].
The scope of the paper is to demonstrate the new method for specimens manufactured by two different processes, chrome-plating and abrasive blasting. In addition the method is validated with simulated point processes. The first section describes the procedure in general. Afterwards the details and results for the different examples and the simulation are presented. The discussion summarizes all findings and the topics for further investigation.

Chrome-plated probes
The special Topocrom ® coating system generates a texture with spherical particles and results in roundridged profiles. Depending on the process parameters and duration the shape and number of those particles varies. The growth of the particles is dependent on the chemical composition of the electrolyte, the distance and the design of anode and cathode and the resultant current density. Probe 2 is produced with electrolyte temperature 37°and an anode distance of 100 mm. The result is an open structure where the chrome balls have no contact to each other. The parameter for probe 6 are an electrolyte temperature of 39°and anode distance of 110 mm. In contrast to probe 2 the  chrome balls have contact to each other and the number of peaks is higher. The surface of the specimens is shown in figure 3.

Blasted probes
The blasted surface is created by the impact of the blast media and consists of overlapping dents. The surfaces are generated with different blast media. S is a spherical media named Chronital and the other one is Grittal GM with an angular broken shape (figure 4). The number describes the size range which is 0,7-1,25 mm for size 60 and 0,05-0,2 mm for size 10. One surface is produced for each shape and size of blast media.
The company BMF GmbH employs a machine called Twister to carry out the manufacturing process, which involves accelerating an abrasive material using a central spin wheel that rotates at approximately 9000 rpm. The probes are made of chrome-nickel steel and are placed on a special workpiece carrier that rotates around itself and the spin wheel (figure 5). The duration of the blasting process is 10 min. The resulting surfaces are shown in figure 6.

Measurement and data processing
The optical measurement system Confovis Duo Vario proceeds the areal measurement. The specimens are all measured using structured illumination, which is the confocal modus of the instrument. The chromeplated probes are measured with a 20x NA0.6 (measurement field 562 × 562 μm) and at the same position with 50× NA0.95 objective (measurement field of 225 × 225 μm). The blasted surfaces are measured with the 20× objective and the covered surface area is 1,1 × 1,1 mm.
The processing of the data is done with MountainsMap ® . First, the topography layer is extracted from the multi-channel data that consists of topography and intensity. Then the surface is aligned by LS-plane leveling to remove the tilt of the probes. The next step is applying a Gaussian-Filter (S-Filter) to remove the micro roughness by cutting the short wavelengths. The cut-off wavelength depends on the manufacturing process of the specimen (chrome-plated 2,5 μm and blasted 8 μm). The resulting S-surface is used for detecting the features. The last step of preparation is filling the non-measured points by calculation of a smoothed form out of the neighbouring points. Figure 7 illustrates the processing of the surface data.
Before the triangulation takes place the points of interest have to be identified. Jiang et al [13] and Leach et al [14] give an overview of the feature based characterisation. To determine the features segmentation, that is the mathematical and automated process of partitioning the surface, is applied [13,15]. In this  paper segmentation by watershed analysis together with Wolf pruning is used according to ISO 16610-85 [16]. The process is robust and deterministic [17,18]. Watershed analysis partitions the topography into hills and dales by simulating the filling of the surface with water. First the water fills the insignificant dale. On the saddle point the water flows into the adjacent dale. If that one is significant the two dales are combined. Otherwise the filling is continued until the water flows in a significant dale. A change tree represents the results (figure 8). In a next step Wolf pruning combines small motifs that are not significant to avoid over-segmentation. Wolf pruning of heights means that for each peak or pit the height difference to the saddle point is calculated. For heights smaller than a specific threshold the peak or pit is combined with the adjacent saddle point beginning from the smalles (figure 8). It is also possible to use pruning of the areas. The size of the threshold depends on the surface and is chosen by subjective analysis of the generated motifs.
For the chrome-plated surfaces hills are detected and the centers of the spheres are exported as point cloud. Both smoothing (mean filter with a 3×3 kernel) and Wolf pruning (5% of Sz) are applied. For the measurements with 20 times objective additional pruning of areas is applied with 0,1% (probe 2) and 0,05% (probe 6) of the surface area. The reason are false measurement points that appear around steep structures. These do not belong to a peak, but due to the height difference are detected as one. This effect results from the different NA of the objective. The 20 times objective with NA of 0.6 has a smaller acceptance angle of around 37°in comparison to the 50 times one with NA of 0.95 which results in an angle of 71°. By cutting out motifs with small areas these false peak points are not further considered when meshing the points.
In contrast to the chrome-plated surfaces the features on the blasted surface have a different geometry. According to the manufacturing process here the valleys, which are the dents from the blast media, are the point of interest. So these are identified and the extreme points of the motifs are exported. In each case the x and y values create the extracted point cloud which is the input for the next step to calculate the mesh.

Meshing with delaunay
The theory of Delaunay triangulation is described in various papers [5,19,20]. The triangulation creates a mesh, which is based on the perimeter condition. Therefore, no point is inside the circumcircle of any triangle and all minimum angles of the triangles are maximized. The triangles reflect the relationship between the features since their edges represent the spacing between nearest neighbours. The Delaunay triangulation with a random set of points is unique and therefore the method is well suited for the analysis [5]. Here the triangulation is applied in 2D.
The point cloud of the extracted features is imported in MatLab where the Delaunauy triangulation is processed. An example of the point cloud and the generated mesh is shown in figure 7.

Parameter
The standard roughness parameters, Sa, Sq and S10z from ISO 25178-2, are quantified from the S-surface [2]. For the chrome-plated surface Spd is calculated additionally [2].
New parameters are developed based on the triangles of the Delaunay triangulation. Different geometric parameters can be calculated for each triangle to describe the distribution of the segmented features. Subsequently statistical analysis for those parameters is performed.
The literature review reveals a wide use of side lengths especially the mean and standard deviation [6,10,11,21,22]. Histograms are also used to give an overview of the distribution of side lengths [9,21,23]. Some papers also calculate the areas and their statistics [23,24]. According to Bakshi et al [6] a higher mean and smaller standard deviation of the side lengths indicate better and more uniform distributions. In contrast Sargam et al [10] describe that a smaller mean and smaller standard deviation show better dispersion. Since there is a contradiction in the literature these two parameters are considered here. In regard to the areas the calculation of area disorder AD (formula 1) which was first described by Marcelpoil et al [25] is more common than the statistical analysis of areas [4,7,8,22,24,[26][27][28].
AD is a unitless quantity that characterizes the regularity of positions, where s A is the standard deviation andĀ the mean of Delaunay triangle areas [27]. It equals zero when the standard deviation of the triangle areas is zero, which resembles polygons that are equal in shape and size and therefore describes a uniform system. The function AD has an upper limit of one for a high standard deviation to mean ratio of the triangle areas. Therefore it is expected to get a larger disorder and irregularity with rising AD values. Because of these characteristics it is well suited to become a measure for the spatial distribution of features.

Simulated point clouds
To enhance the limited dataset of measured surfaces, poisson point processes are used to generate two dimensional point sets with different spatial distributions. A simulation is employed to vary the degree of disorder for the various point processes. Subsequently the findings of the measured surfaces are compared using these and methods to distinguish spatial distributions are validated.
To evaluate the complete range of AD from zero to close to one, four different point processes are constructed, namely the uniform, homogen, inhomogen and clustered point process [25]. To be comparable, the point processes share the same base arguments. These are the area of the underlaying space of the point set with a length L x and L y and a poisson distributed random variable, which defines the number of the simulated points. With a area disorder of zero a perfectly ordered lattice of points serves as the definition of the uniform point process. A poisson like noise factor is calculated independently for each dimension for each point on that grid. The maximum value for that factor is ±L x and ±L y , which equals 100% noise. Therefore any point at 100% noise has at least the ability to reach the border of the underlaying space. In progress of the simulation the noise factor is increased from 0% to 100% to evaluate rising area disorder values ( figure 9).
For the case of complete spatial randomness a homogeneous poisson point process is created [3,29]. During the simulation the poisson distributed random variable for the number of points is seeded again in every run with the same intensity λ (mean density of the poisson point process). The positions of the points vary independently but every region in the area L x * L y contains the same number of points on average, thus homogeneous. For the opposite case, the inhomogeneous poisson point process, the dispersion of the points vary on average at certain regions of the simulated area. Therefore the constant intensity λ is converted to a nonconstant function that changes its value for every processed point ( figure 10). An extrem example of inhomogeneous distributions, the cluster point process, is simulated to better distinguish between homogeneous and inhomogeneous pattern ( figure 11). Similar to the characteristics of the inhomogeneous poisson point process, it is anticipated to produce greater AD values. During the simulation the number of clusters is randomized independently from the number of points. For all point processes the mean density λ and the area is the same, therefore any process starts nearly with the same number of points. While increasing the noise factor on uniform point processes the probability to lower the number of points rise.

Simulation
For every point process a simulation with 1000 runs is calculated and values are altered according to section 2.5. The statistical results for the simulations are outlined in boxplots. A box shows the quantile range of 0,25 to 0,75 and a blue line represents its median. Green lines mark the quantile range of 0,95 and 0,05. All points outside the whiskers length, which corresponds to approximately ±2.7σ, are expected to be ouliers.
The statistics for mean and standard deviation of side lengths for all Delaunay triangles are summarized in figures 12 and 13. These results demonstrate that it is impossible to differentiate between spatial distributions when considering these two factors indepen-   dently of each other.
The statistical results for parameter AD are summarized in figure 14. With these findings it is easily possible to distinguish between uniform and clustered spatial distributed points. It is not clearly feasible to separate between homogeneous and inhomogeneous poisson distribution. But it is possible to define that any AD value larger than the 95% confidence interval of AD for the homogeneous poisson point process indicates an inhomogeneous spatial distribution. This is true for any AD value greater than 0,497 since the cluster point process is likewise an inhomogeneous spatial distribution.
A completely uniform pattern results in AD equal zero. It is interesting, that there are no results for AD between 0 and 0,25. The uniform point process overlaps with the homogeneous and inhomogeneous poisson point process. In other terms, with increasing noise factor the uniform characteristic transforms to some spatial random pattern. As a result, it is critical to identify limitations that allow a spatial distribution to be defined as uniform. To achieve this, the transition with the lower 95% confidence interval of AD for the homogeneous poisson point process is chosen as the limit ( figure 15). For any AD value below 0,45 the spatial distribution has a uniform characteristic, which reaches its maximum at AD equal zero. Furthermore the transition from uniform to random pattern acts in a nonlinear manner, as shown in the region of interest in figure 15. To describe that behavior and the level of the uniform characteristic with the corresponding AD factor additional research must be conducted.

Chrome-plated surfaces
The chrome-plated surfaces are processed as described in section 2.2. The results of the watershed analysis and the calculated mesh of the surfaces are shown in figure 16.  The standard parameters in table 1 Sa and Sq are comparable across all specimens and measurements. S10z shows an slight increase for the larger measurement area by 20x objective. All these parameters describe the heights of the surface and reveal no information about the spatial distribution. Parameter Spd is supposed to describe the density of peaks, therefore the number of all peaks is divided by area of the underlaying obervation space. Concerning the simulation with different point processes, the value of Spd stays the same, since the number of points and the area do not change. Since Spd does not take any spatial information into account it can not indicate any difference between uniform and random distributed peaks. These findings confirm the need for new parameters that are able to describe the disorder of the spatial distribution.    Table 2 contains the calculated parameters after Delaunay triangulation.
Mean, standard deviation and AD are smaller for Probe 6. Related to the AD values this indicates a stronger uniform spatial distribution compared to probe 2. This is consistent with the visual impression of the triangulated points ( figure 16) and the findings for AD in section 3.1. Figure 17 shows the side length histograms of the two probes. These only provide information about the distances between the points, and thus the density. The peaks are more numerous and closer together in probe 6. Besides that, it is not possible to perceive any difference in their spatial distribution.

Blasted surfaces
The second example are four different blasted surfaces. Results of the segmentation and the meshed point cloud for each sample is shown in figure 18. When the minimum lies on the boundary or near it, it is uncertain if that point is a real minimum since the valley could be lower outside the measured area. So before calculating the mesh from the extracted point cloud the points near the boundary within a distance from 0 to 0,01 mm are deleted. Table 3 summarizes the standard parameters. The roughness parameter Sa, Sq and S10z are higher for larger blast media size. The value of S10z for GM60 is high in comparison to the others. Several outliers can be the reason because only Gauss filtering is applied and no special treatment for outliers. Apart from the size of the blast media, GM results in higher values for all parameters. It is possible to distinguish between the surfaces with the height parameters. However, these values do not correlate with the AD values listed in table 4.    Statistic values for the side lengths and AD increase with larger blast media size. The side length parameters seem to follow the same trend as the AD values, besides that they scale and vary different. The values of AD do not exceed the limit of 0,45 and therefore the spatial distributions of the surface features are uniform to some extent. Both blast medias result in a similar distribution of features.

Discussion
The are three main topics covered in the discussion. First the generation of the points and the point cloud itself are discussed. Then the parameters follow. Last the link between the parameters and the function is considered.
Only a small portion of the surface is covered by the measurement and hence the point cloud. To extrapolate from the results of the partial area to the whole surface is only valid if the sample is a representative of the whole. The material has to have the same statistics everywhere. The watershed segmentation is challenging for peaks close to each other, particularly for overlapping features with little height difference. Also feature detection through motif fails sometimes in terms of peak recognition under certain geometric conditions, e.g. minimal height differences between neighboured peaks. Therefore other segmentation methods have to be tested and evaluated in addition to the developed procedure. The selection of filters and settings dependent on the surface assumes detailed knowledge about it. In general all filters have to be applied carefully since they have a big influence on the results. Especially the values for pruning have a direct impact on the number of peaks and so the point cloud. As a result, the robustness of the feature detection process may vary, since only a few surfaces were measured.
In regard to the triangulated point cloud itself the Delaunay algorithm may produce wrong triangulations at the boundary of the mesh. To overcome this problem periodic boundary conditions involving virtual feature points outside the measured area are developed at the moment. This method is also used by Bray et al [7] and Yap and Oh [24] and is described in ISO 16610-85 [16]. Specific details about parameter settings are missing to fine-tune this strategy and produce reliable outcomes. Therefore simulations with realistic synthetic data are planned. The parameter AD is suitable to describe spatial distribution according to the definition in this paper. Especially the simulation with different point processes showed, that uniform, spatial random and clustered patterns are clearly distinguishable. However it still has some potential for improvement. It would be appropriate to differentiate the levels of uniform and clustered distributions more precisely. In addition, AD values for homogeneous and inhomogeneous distribution overlap and might not be reliably detected. Further investigations based on simulated point processes and Delaunay triangulation properties are needed to evolve the AD parameter.
The analysis with the mesh is carried out in two dimensions. By adding height informations to the method, the real distance relationship between features could be calculated. Also enhancing the method with other attributes of a feature, like width, length or diameter, could be crucial for some tasks.
Furthermore there are initial results for the chrome-plated surfaces in relation to the function. The impact of the surface structure is demonstrated in experiments using a textile rope composed of Technora T221 and two distinct grooves. The rope was worn to the point of breaking with constant lifting speed of 25 min −1 in a test bench. Surfaces having a spatial distribution of peaks such as probe 6, took twice as long to break the rope compared to surfaces with a spatial distribution such as in probe 2. This suggests a link between the surface and the function. Figure 19 shows an section of the surface before and Figure 19. Chrome-plated surface before and after the wear investigation.
after the experiments. The isolated peaks are thought to cause the rope to wear out more quickly. The obtained parameters from this study can now be applied to mathematically define the surface and run a test series to thoroughly examine the relationship between the function and the surface.

Conclusion
Our study provides a new method to calculate parameters to describe the spatial distribution of features on aperiodic surfaces. The method has been validated by simulation and applied to six different surfaces, which are manufactured by two different processes and validated by simulations. Features, detected by watershed segmentation and Wolf pruning, are connected with Delaunay triangulation. The normalized and unitless parameter AD is deeply investigated and validated as a parameter to distinguish spatial patterns, for example uniform and inhomogeneous distributions. A range of AD values for the definition of a uniform [0-0,45] and a inhomogeneous [0,497-1] spatial distribution was found and determined by simulation.The study has investigated a small amount of real measured surfaces. Future work will explore more surface data and further development will evolve the Area Disorder parameter by integrating more attributes of a feature and optimizing the boundary conditions of the Delaunay triangulation.

Data availability statement
The data cannot be made publicly available upon publication because they are not available in a format that is sufficiently accessible or reusable by other researchers. The data that support the findings of this study are available upon reasonable request from the authors.

Funding
This Project is supported by the Federal Ministry for Economic Affairs and Climate Action (BMWK) on the basis of a decision by the German Bundestag.