Comparing the Effectiveness of R and S Roughness Parameters on Surfaces Lubricated with Standardized Nominal Particle Size Lubricants

Abstract

The aim of this article is to simulate the distribution of oil or lubricants on the surface by the characterization of the R and S roughness parameters. It is considered that undesirable particles may appear in the oil due to friction, the wear of parts, or dirt in the lubrication circuit. One of the problems that can generate the appearance of particles in the oil is the poor distribution of this along the surface, using standards, standardised particle sizes, and following the standard. This research aims to implement effective algorithms for the calculation of profile and surface roughness parameters following existing standards, as well as the development of a morphological filter that will simulate particles on the surface. The analysis of the surface roughness of the surface in contact with the lubricant can help to decide the maximum particle size that is permissible in the system; therefore, analysing the oil will help to decide when it is necessary to filter or replace that lubricant. This research is carried out form the geometrical point of view of the morphological filter effect. This paper represents a first approach to researching the calculation of all standardized profile and surface roughness parameters. As a result, only a selection of parameters has been chosen from the full range of available ones, in order to improve effectiveness.

1. Introduction

Oils and lubricants serve various essential functions in industry, including reducing friction between moving parts, preventing overheating by acting as heat sinks, and enhancing energy efficiency through friction reduction.

The presence of particles in the oil can lead to increased friction between moving parts and the accumulation of particles in critical areas, which can result in increased wear or even the breakage of parts. It can also lead to an increase in machine temperature due to increased friction and poor heat dissipation, which can result in reduced system performance or increased energy consumption.

For this reason, it is essential to perform an analysis of industrial lubricants to ensure the correct operation of the machines and to extend their useful life. In one of the tests performed in the standards, automatic particle counters (APCs) are used, which are systems that measure the size of the particles and group the quantity of particles in the sample in different size ranges [1]. In this work, those ranges of sizes of particles standardized were used as the starting point for the filter characterization in the evaluation of the parameters.

This paper simulates the issues of oil distribution by applying a morphological filter to a surface, treating it as if it were a particle. Then, the surface is ‘virtually’ measured by the application of a morphological filter parametrized by the particle size. On the filtered ‘surface’, it is evaluated using each of the standardized parameters listed in the following clauses.

Although there are a multitude of standards, in this research, ISO standards have been used, mainly because they are widely accepted, especially in industry, because they provide a standardized and internationally recognized framework. This facilitates comparability of results between different laboratories, equipment, and works. In addition, ISO standards are the result of an international consensus among experts, which ensures that they are practical and effective. Therefore, future works may benefit from this fact, since it will be possible to compare their results with other precedents and, in this way, give more consistency to their own.

This paper does not take into account surface energy, oil temperature, material type, and possible wear on the oil. It is only evaluated from the point of view of the geometrical study of the morphological filter effect.

ISO/TC 213 is the technical committee in charge of creating, updating, and maintaining the ISO GPS standard for the geometrical product specifications and verification [2]. It was created in 1996. It is responsible for the standards ISO 21920, ISO 25178, and ISO 16610 used in this paper.

This research is a continuation of the Master’s Thesis in Metrology, referenced in [3].

Paper [4], by Bartkowiak et al., investigates the correlations between the build-up angle, surface fractal complexity, and wettability on additively manufactured surfaces fabricated with different technologies. This work uses surface data acquired using optical focusing variation, confocal microscopy, confocal fusion, and interferometry technologies. To investigate this correlation, it uses height, hybrid, and spatial parameters. Specifically, the parameters analysed are S_q, S_p, S_z, S_v, S_a, S_dq, S_dr, and S_tr. It does not consider the contacting surfaces and does not evaluate other parameters.

Grzesik’s work in chapters 4 and 20 of [5] have also been consulted on the surface integrity resulting from machining with cutting tools of various materials. An analysis of profile and surface roughness parameters calculated based on ISO 25178-2:2012 [6] and ISO 4287:1999 [7] is performed. Calculations of height, spatial, hybrid, and functional parameters are carried out in the paper. Specifically, the parameters evaluated are S_a, S_q, S_sk, S_ku, S_p, S_v, S_tr, S_al, S_td, S_dq, S_dr, V_vc, V_v, V_mp, and V_mc. Once more, R parameters are not evaluated.

Kim’s work in paper [8] describes the use of a microscope to capture images and evaluate the roughness parameter R_a, comparing the results obtained for shot-blasted and sand-blasted surfaces. In this case, there is no contact, and the S parameters are not measured.

Paper [9], by Lingadurai and Shunmugam, attempts to extend the 2D filter specified in ISO 16610-41:2006 [10] to a three-dimensional surface. Instead of using a spherical shape, as is the case in our study, a toroidal shape has been selected. That paper uses the filter to discriminate between surface waviness and shape defects.

Paper [11], by Lou et al., shows the application process of morphological filters including the scanning process, surface reconstruction, deviation evaluation, filtering for open and closed profiles, contact simulation, the establishment of the uncertainty zone, and the evaluation of the stratified surface. This study is focused on the application of the morphological filter; the roughness parameters are not evaluated as they are in our study.

Paper [12], by Lou et al., shows the application of a morphological filter of a closing disc and a linear segment filter and proposes an algorithm to implement the filter. In our work, we implement the morphological filter, and we evaluate and compare different parameters. Taking into account all the research conducted before, we have first of all selected the size of the morphological filter below the lubricant particle size (100 µm), designed a contacting filter, and evaluated not only the R parameter but also the S parameter. The new application of this paper will be the fictitious change in the surface that may appear when filtering the lubricant and removing bigger lubricant particles. The bigger the size, the smoother the surface will seem to be. That change in “smoothness” is evaluated by the change in some R and S parameters. In our work, we show which R and S parameters will show that change.

Paper [13], by Tóth et al., highlights the important role of anti-wear additives in protecting contact surfaces even with a thin film of lubricant. This study is carried out with particles between 1 nm and 100 nm. Tribological measurements reveal that the optimum particle concentration of 0.5 wt.% reduces the wear scarring produced by 45% and 90% of the pure wear volume compared to pure oil. This study complements the results of our study obtained from the comparison of the areal and profile roughness parameters, as it can be deduced from them that the smallest particles studied do not present major changes in the results of the calculated parameters.

The primary objective of this paper is to determine the critical impact of lubricant particle size behaviour and performance on the contacting surface by evaluating its effect on areal and roughness parameters.

2. Materials and Methods

2.1. Preliminary Research of the State-of-the-Art on Particle Size in Oils

In the last years, many industrial parts have been measured with roughness instruments fulfilling ISO requirements. Therefore, it is a fact that for the final user, it will be easier and competitive to improve their parts if they can take decisions from ISO parameters. It is out of the scope of this paper to define new parameters.

Thorough research has been performed into the current standards and regulations regarding the presence of particles in oils and lubricants. These are as follows:

• ISO 4406:2021 Hydraulic fluid power—Fluids—Method for coding the level of contamination by solid particles [14];
• NAS 1638:1964 (National Aerospace Standard)—International particle count standards [15,16,17];
• SAE AS4059 Rev. E. (2020) SAE Aerospace Standard [16,17,18];
• GOST 17216-2001 Gosudarstvenny Standard [16,17,19];
• NAV AIR 10-1A17 (1989) Navy Standard for Hydraulic Fluids used for aircraft hydraulic systems [16,17,20].

Table 1. Table of particle sizes for each of the standards.

Standards with Particle Sizes	Standards with Calibration Procedures	Size [µm]
ISO 4406:2021 [14]	ISO 4407:2002 [21]	4, 6, 14
	ISO 11171:2022 [22]	5, 15
NAS 1638:1964 [15,16,17];	ISO 4402:1977 [23]	[5–15], [15–25], [25–50], [50–100]
SAE AS4059 Rev. E [16,17,18]	ISO 4402:1991 [24,25]	4, 6, 14, 21, 38, 70
	ISO 11171:2022 [22]	2, 5, 15, 25, 50, 100
GOST 17216:2001 [16,17,19];	ISO 4402:1991 [24,25]	[5–10], [10–25], [25–50], [50–100], [100–200]
NAV AIR 10-1A17 (1989) [16,17,20]	-	[5–10], [10–25], [25–50], [50–100], >100

shows the particle size defined by each standard and the calibration standard related to the first. In all of them, particle sizes for lubricants are somehow specified, and are therefore used as reference for this work.

It is noted that the particle size limit used in these standards is based on the standard used for the calibration of the particle measurement system. There are two types for use: ISO 4407:2002 [21] (MTD) and ISO 11171:2022 [22] (ACFTD). The first uses the diameter equivalent to the projected area of particles and the second uses the maximum length of the particles. This divergence approach needs to have very close size values, but not identical. Some standards refer to ISO 4402:1991 [24] instead of ISO 11171:1999 [26], as ISO 11171:1999 [26] replaced ISO 4402:1991 [24] in 1999.

All the values referenced Table 1 were used for this paper. In cases where the standards gave a range instead of a specific value, the limit values of the ranges were selected, as will be shown.

Therefore, the following particle sizes were selected: 2 µm, 4 µm, 5 µm, 6 µm, 14 µm, 15 µm, 21 µm, 25 µm, 38 µm, 50 µm, 70 µm, and 100 µm.

The 200 µm size has not been taken into account due to the high processing time necessary to apply the filtering. Because the filter size is too large for the chosen surface area, representing as much as 20% of the final surface, this was considered excessive. Moreover, normally, lubricated surfaces do not work under those extreme conditions.

2.2. Programmed Algorithms

The algorithms were developed using Matlab 2024B [27] programming and another numerical calculation platform. Matlab 2024B was selected because of its high flexibility and easy programming. Due to its computational power and high-level programming, it eases code debugging and high-quality graphical representation.

The general flow diagram of the programming it shown in Figure 1.

The description of each box in the diagram above and the specific functions programmed are explained below.

• Variables of parametrization programming: The parameters of the point cloud and the standard are set.
• Surface generation: MySimLab is used to generate the point cloud and developed functions are used to generate the standards and combine point clouds. These were the functions:

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  Gen_triangular_standard.m

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  Gen_surface.m

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  Gen_total_surface.m

• 2D morphological interaction: The surface is measured as a profile, which is filtered with a profile morphological closure disk filter, simulating the path of a disk over the profile. It was programmed using the scripts FPMCD.m and app_filter_profile.m.
• 3D morphological interaction: The surface is filtered with an areal morphological closure ball filter, simulating the path of a particle or stylus through the surface. It uses the scripts FAMCB.m and app_filter_surface.m.
• R parameter calculation: Profile roughness parameters were evaluated on the profile selected on the surface with the following scripts programmed as defined in the standards referenced at the end of the papers. These were the programmed functions:

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  Calc_Rq.m

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  Calc_Rsk.m

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  Calc_Rku.m

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  Calc_Rp.m

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  Calc_Rv.m

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  Calc_Rzt.m

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  Calc_Ra.m

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  Calc_Rmc.m

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  Calc_Rcm.m

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  Calc_Rdc.m

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  Calc_Rdq.m

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  Calc_Rdr.m

  ○

  Calc_Rk.m

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  Calc_Rvkx.m

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  Calc_Rvk.m

  ○

  Calc_Rpkx.m

  ○

  Calc_Rpk.m

  ○

  Calc_Rmrk1.m

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  Calc_Rmrk2.m

  ○

  Calc_Rak1.m

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  Calc_Rak2.m

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  Calc_Rak.m

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  Calc_step_p.m

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  Profile_data_process.m

• S parameter calculation: Areal roughness parameters were evaluated on that surface with the following scripts:

  ○

  Calc_Sq.m

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  Calc_Ssk.m

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  Calc_Sku.m

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  Calc_Sp.m

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  Calc_Sv.m

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  Calc_Szt.m

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  Calc_Smr.m

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  Calc_Smc.m

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  Calc_Sdq.m

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  Calc_Sdr.m

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  Calc_Sk.m

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  Calc_Svkx.m

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  Calc_Svk.m

  ○

  Calc_Spkx.m

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  Calc_Spk.m

  ○

  Calc_Smrk1.m

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  Calc_Smrk2.m

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  Calc_Sak1.m

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  Calc_Sak2.m

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  Calc_Sak.m

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  Calc_step.m

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  Data_process.m

• Data collection: Results were saved, and a report was generated as an xls file. Data collection was developed with the following scripts:

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  Write_xls.m

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  Parameter_initialize.m

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  Parameter_initialize_standard.m

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  Parameter_initialize_profile.m

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  Delete_data.m

• Plots: Plots were generated using Matlab 2024B and excel programs. The following three scripts were developed:

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  Material_ratio_curve.m

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  Iterative_graphics.m

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  Surface_graphic.m

• Comparison: Conclusions and results were studied to perform this work.

2.3. Surface Generation

A random surface was generated, and since we focused on the method, this was good enough. Triangular peak and grooves standards have been generated for analysis. A real surface could have been used instead of the random surface, but among all the standards specified, the triangular standard was shown to be the best for that.

In order to carry out the research, a surface was generated using the MySimLabs [28] tool, executed in the Matlab environment. The script used to generate the surface was ‘rsgeng2D.m’. This script gives, as an output, a matrix with the data corresponding to the Z axis and two vectors with the data of the X and Y axes. To obtain two matrixes with the X and Y axes, the Matlab meshgrid function was used. The random surface was 1400.4 µm in X and Y axes and 14.2354 µm in the Z axis. However, after the filtering operation, only 1 mm was used in both the X and Y axes.

The point cloud has been generated, leaving a distance between the points of 0.2 µm, avoiding aliasing, because the diameter of the smallest filter to be evaluated is 2 µm.

The surface was generated by five surfaces with different amplitudes and filters, and they were added together to obtain the most realistic surface possible. In Appendix A, the parameters used to generate each of the surfaces and the correction factors applied prior to the summation of all the surfaces are detailed.

It should be noted that since the morphological filter has an end effect and in the case of the surfaces there is no standard for the compensation of this defect, it has been necessary to generate a larger surface than the one to be processed with the algorithms for the calculation of the evaluation. Currently, there is only one standard for the end effect compensation, and it is the UNE-EN ISO 16610-28:2018 [29] standard, but it is only applicable to profiles, not surfaces. Therefore, the only existing solution to avoid this distortion by filtering is to use a larger data sample than needed and then discard the data affected by this effect.

Figure 2 shows the point cloud generated for the surface analysis.

2.4. Generation of the Standards

In addition to the random surface, a triangular groove standard has been used following the recommendations of the UNE-EN ISO 25178-701:2011 [30] standard; this was used in order to appreciate the influence of the program morphological filter on peaks and grooves.

Figure 3 shows the ER1 standard of the standard used.

As in the random surface, the standard surface has also been programmed with an inter-point distance of 0.2 µm. The standard has been developed with a depth (d) of 20 µm, an opening angle (α) of 130°, a rounding radius (r_f) of 1 µm, and the plane (P) is defined at Z coordinate 0 µm. These parameters have been selected so that the largest stylus or filter diameter can enter the slot to test its influence.

The standard remarks that the rounding angle must be larger than the stylus radius for the standard to be used for calibration. However, in this case, the scope of using the standard is to evaluate the evolution of morphological filtering on the surface as the filter size increases, provided that the morphological filter resembles the contacting performance for the selected particle size with the surface.

Figure 4 shows the profile of the first programmed standard.

Subsequently, the programmed standard has been inverted to have a peak upwards, and the performance of the R or S roughness parameters is checked, because for an ideal surface, as stated in these standards, the result of R or S parameters is the same. Figure 5 shows the groove (a) and peak (b) standard.

It is important to note that since the standards have ideal surfaces without roughness, only the height parameters and the hybrid parameters have been calculated. Since the functional and bearing rate parameters depend on the material ratio curve, there is no benefit from calculating them in this case. Due to the shape of the standard, the evaluated material ratio curve does not have the inverted s-shape with a single inflection point necessary to calculate these parameters.

2.5. Morphological Interaction/Filtering

In order to evaluate the influence of particles or the stylus contacting a surface, it was decided to use a closing morphological filter. For this work, the morphological filter consists of a series of mathematical operations that relate a virtual measured surface to a contact measurement system (matching the size of the filter to the size of the stylus).

Although the particles present in the lubricant can have different shapes, for this study, it is assumed that these particles are perfect spheres. Further work may consider that effect.

Whenever a measurement is made with a stylus, morphological filtering is applied automatically, as the spherical shape of the stylus cannot be in contact with every single point on the surface. In small dales, the stylus does not reach the bottom, acquiring a slightly reduced depth. However, it can measure all the peaks correctly.

The same is true for the particle as for the stylus. If the particle has a diameter larger than the width of the dale, it will not be able to penetrate completely. If there is a large difference between the filtered surface and the original surface, it means that the particle may have been left behind to block the passage of the lubricant and thus leave the lubricant unevenly distributed.

This behaviour of the sphere with the surface can be simulated for a particle in a mathematical way using an FAMCB (Filter Areal Morphological Closing Ball) filter. In fact, a morphological filter with a closing disc was selected for the profile and a closing sphere was selected for the surface filtering. In the case of the profile, the guidelines explained in the UNE-EN ISO 16610-41:2015 [31] standard have been used. There are still no published standards for morphological areal filtering.

The filters used do not present an end effect in the case of profiles or closed surfaces, but they do in the case of profiles and open surfaces. As there are no end effect compensation techniques for surfaces as in the case of profiles (UNE-EN ISO 16610-28:2018 [29]), larger surfaces and standards are generated than those that are going to be used.

It should be noted that the end effect depends on the size of the filter used; the larger the filter, the more data will be affected by it. Therefore, the larger the filter used, more data have to be discarded. In order not to have a bad influence on our evaluation by the end effect and the size of the surface used, the same amount of data have been discarded for all filter sizes. In the case of profiles, the techniques explained in the UNE-EN ISO 16610-28:2018 [29] standard could be used. In order to be consistent, the same technique has been used as for the surface.

These filters are based on two types of primary operations, namely, dilation (Figure 6) and erosion [32]. Depending on the order in which these operations are performed, the filter is either opening or closing (Figure 7). The application of the dilation operations followed by the erosion operation, using a circular disc, would highly approximate what a morphological filter would do with a closing disc.

Figure 6 and Figure 7 show graphically how a filtered profile evolves with an FPMCD filter. In Figure 6, the blue profile is the original, and the black profile is the profile resulting from the first morphological operation applied, which is dilation. In Figure 7, the dilated profile is shown in grey, and the filtered profile is shown in black. Comparing the filtered profile with the original one, it can be seen that due to the large diameter of the disc used, it does not come into contact with the bottom of the dales; therefore, in the final profile, these are reduced. It can also be seen that at the beginning and end of the filtered profile in Figure 7, there is an end effect in which there is a big difference between the original profile and the filtered profile due to the fact that there is no continuity in the data. This causes a distortion in the resulting profile; therefore, it is necessary to correct or discard these data.

A matrix has been generated to implement the filtering, knowing that in this case the distribution between the data in the X and Y axis is constant. This matrix was used as a mobile window along the surface to apply the filter. It depends on the diameter of the filter to be applied, it simulates the shape of the sphere with trigonometric operations or Pythagorean Theorem, and its size depends on the diameter of the filter to be used. The matrix keeps the same distance between the data in the X and Y axis. The matrix used for filtering stores the distance between the diametric plane of a sphere (of the filter sphere) and the contour of the hemisphere of the sphere.

Figure 8 shows the values of the mask used for the 2 µm filter, and the groove standard for the surface morphological filter (FAMCB). The numbers in red represent the mask used for the morphological profile filter (FPMCD) of the same size.

The data represented in grey are the data corresponding to the inside or edge of the plane inside the sphere, and the values in green are the values on the outside. The grey values of the mask correspond to the heights of A (Figure 9) for all the points inside the sphere. It is important to remember that the inter-point distance in the mask must be the same as that of the surface or profile used. To ensure that the filter does not take into account the values represented in green, the parameter S_z, representing the distance between the highest peak and the deepest dale, has been calculated and multiplied with the value −2, to ensure that those values will not take in account values outside the sphere. Multiplying the surface S_z value or profile R_zt value by −2, any value outside the sphere is going to be lower than any point in the surface or profile.

In order to be able to use the mask, it is important that the advance of the data on the X and Y axes coincides with that of the mask. Another important aspect is that the size of the mask must be odd to perform the operations, since the result of the mathematical operations will be stored in the position of the data that coincides with the centre of the mask. Due to do mathematical operations, it is necessary to have a value in the centre of the mask.

To apply the dilation and closure operations, it is necessary to go through the complete point cloud with the filter. First, dilation is applied, and after, closure operation is applied.

One way to perform the dilation operation is to add the filter matrix with the matching values in the data, to calculate the maximum value between them. For the closing operation, the data to which the dilation has already been applied are subtracted from the filter values and the minimum is calculated.

2.6. Parameters Evaluated

This paper deals with a first approximation of research that aims to calculate all the standardized roughness profile and surface parameters; therefore, only some parameters have been selected among all the existing ones. Particularly, parameters that are more widely used in the industry are selected.

For the calculation of the parameters, the guidelines of the UNE-EN ISO 25178-2:2023 [6] standards have been followed for the surface parameters and UNE-EN ISO 21920-2:2023 [33] for the profile parameters.

Table 2. Parameters evaluated.

Type of Parameters	Surface Parameters	Profile Parameters
Height parameters	$\mathrm{Root}\mathrm{mean}\mathrm{square}\mathrm{height}({\mathrm{S}}_{\mathrm{q}}$)	$\mathrm{Root}\mathrm{mean}\mathrm{square}\mathrm{height}({\mathrm{R}}_{\mathrm{q}}$)
	$\mathrm{Skewness}({\mathrm{S}}_{\mathrm{s}\mathrm{k}}$)	$\mathrm{Skewness}({\mathrm{R}}_{\mathrm{s}\mathrm{k}}$)
	$\mathrm{Kurtosis}({\mathrm{S}}_{\mathrm{k}\mathrm{u}}$)	$\mathrm{Kurtosis}({\mathrm{R}}_{\mathrm{k}\mathrm{u}}$)
	$\mathrm{Maximum}\mathrm{peak}\mathrm{height}({\mathrm{S}}_{\mathrm{p}}$ *)	$\mathrm{Maximum}\mathrm{peak}\mathrm{height}({\mathrm{R}}_{\mathrm{p}}$ *)
	$\mathrm{Maximum}\mathrm{pit}\mathrm{depth}({\mathrm{S}}_{\mathrm{v}}$ *)	$\mathrm{Maximum}\mathrm{pit}\mathrm{depth}({\mathrm{R}}_{\mathrm{v}}$ *)
	$\mathrm{Maximum}\mathrm{height}({\mathrm{S}}_{\mathrm{z}}$ *)	$\mathrm{Maximum}\mathrm{height}({\mathrm{R}}_{\mathrm{z}\mathrm{t}}$ *)
	$\mathrm{Arithmetic}\mathrm{mean}\mathrm{height}({\mathrm{S}}_{\mathrm{a}}$)	$\mathrm{Arithmetic}\mathrm{mean}\mathrm{height}({\mathrm{R}}_{\mathrm{a}}$)
Hybrid parameters	$\mathrm{Root}\mathrm{mean}\mathrm{square}\mathrm{gradient}({\mathrm{S}}_{\mathrm{d}\mathrm{q}}$)	$\mathrm{Root}\mathrm{mean}\mathrm{square}\mathrm{gradient}({\mathrm{R}}_{\mathrm{d}\mathrm{q}}$)
	$\mathrm{Developed}\mathrm{interfacial}\mathrm{areal}\mathrm{ratio}({\mathrm{S}}_{\mathrm{d}\mathrm{r}}$)	$\mathrm{Developed}\mathrm{length}\mathrm{ratio}({\mathrm{R}}_{\mathrm{d}\mathrm{r}}$)
Material ratio parameters and functions	$\mathrm{Areal}\mathrm{material}\mathrm{ratio}({\mathrm{S}}_{\mathrm{m}\mathrm{r}}$)	$\mathrm{Material}\mathrm{ratio}({\mathrm{R}}_{\mathrm{m}\mathrm{c}}$)
	$\mathrm{Inverse}\mathrm{areal}\mathrm{material}\mathrm{ratio}({\mathrm{S}}_{\mathrm{m}\mathrm{c}}$)	$\mathrm{Inverse}\mathrm{material}\mathrm{ratio}({\mathrm{R}}_{\mathrm{c}\mathrm{m}}$)
	$\mathrm{Material}\mathrm{ratio}\mathrm{height}\mathrm{difference}({\mathrm{S}}_{\mathrm{d}\mathrm{c}}$)	$\mathrm{Material}\mathrm{ratio}\mathrm{height}\mathrm{difference}({\mathrm{R}}_{\mathrm{d}\mathrm{c}}$)
Areal parameters for stratified surfaces	$\mathrm{Core}\mathrm{height}({\mathrm{S}}_{\mathrm{k}}$)	$\mathrm{Core}\mathrm{height}({\mathrm{R}}_{\mathrm{k}}$)
	$\mathrm{Maximum}\mathrm{peak}\mathrm{height}({\mathrm{S}}_{\mathrm{p}\mathrm{k}\mathrm{x}}$)	$\mathrm{Maximum}\mathrm{peak}\mathrm{height}({\mathrm{R}}_{\mathrm{p}\mathrm{k}\mathrm{x}}$)
	$\mathrm{Reduced}\mathrm{peak}\mathrm{height}({\mathrm{S}}_{\mathrm{p}\mathrm{k}}$)	$\mathrm{Reduced}\mathrm{peak}\mathrm{height}({\mathrm{R}}_{\mathrm{p}\mathrm{k}}$)
	$\mathrm{Maximum}\mathrm{pit}\mathrm{depth}({\mathrm{S}}_{\mathrm{v}\mathrm{k}\mathrm{x}}$)	$\mathrm{Maximum}\mathrm{pit}\mathrm{depth}({\mathrm{R}}_{\mathrm{v}\mathrm{k}\mathrm{x}}$)
	$\mathrm{Reduced}\mathrm{pit}\mathrm{depth}({\mathrm{S}}_{\mathrm{vk}}$)	$\mathrm{Reduced}\mathrm{pit}\mathrm{depth}({\mathrm{R}}_{\mathrm{v}\mathrm{k}}$)
	$\mathrm{Material}\mathrm{ratio}\mathrm{of}\mathrm{the}\mathrm{hills}({\mathrm{S}}_{\mathrm{m}\mathrm{r}\mathrm{k}1}$)	$\mathrm{Material}\mathrm{ratio}\mathrm{of}\mathrm{the}\mathrm{hills}({\mathrm{R}}_{\mathrm{m}\mathrm{r}\mathrm{k}1}$)
	$\mathrm{Material}\mathrm{ratio}\mathrm{of}\mathrm{the}\mathrm{dales}({\mathrm{S}}_{\mathrm{m}\mathrm{r}\mathrm{k}2}$)	$\mathrm{Material}\mathrm{ratio}\mathrm{of}\mathrm{the}\mathrm{dales}({\mathrm{R}}_{\mathrm{m}\mathrm{r}\mathrm{k}2}$)
	$\mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{hills}({\mathrm{S}}_{\mathrm{a}\mathrm{k}1}$)	$\mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{hills}({\mathrm{R}}_{\mathrm{a}\mathrm{k}1}$)
	$\mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{dales}({\mathrm{S}}_{\mathrm{a}\mathrm{k}2}$)	$\mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{dales}({\mathrm{R}}_{\mathrm{a}\mathrm{k}2}$)

* In the UNE-EN ISO 21920-2:2023 [33] standard, the parameters ${\mathrm{R}}_{\mathrm{p}}$, ${\mathrm{R}}_{\mathrm{v}}$, and ${\mathrm{R}}_{\mathrm{z}\mathrm{t}}$ are listed in the section of elementary parameters. However, in the UNE-EN ISO 25178-2:2023 [6] standard, the parameters ${\mathrm{S}}_{\mathrm{p}}$, ${\mathrm{S}}_{\mathrm{v}}$, and ${\mathrm{S}}_{\mathrm{z}}$ are in the section on height parameters. In this paper, it has been decided to follow the structure of the areal standard, UNE-EN ISO 25178-2:2023 [6].

lists all the parameters used for the paper. These are grouped into height, hybrid, material ratio parameters, and parameters for stratified surfaces.

2.7. Collection

Results were saved in the Matlab workspace and the same program that stored data was used to create the reports with the results of all parameters.

2.8. Analysis

These data will be specified in the results section.

3. Results

This section shows the results obtained for the profile and surface roughness parameters after filtering the surface and the profile with different stylus or particle sizes.

The units recommended by UNE-EN ISO 25178-3:2013 [34] and UNE-EN ISO 21920-3:2023 [35] have been used to represent the results obtained.

As the areal material ratio and length bearing rate height parameters depend on another data besides the surface or profile. For the calculation of the ${\mathrm{S}}_{\mathrm{m}\mathrm{c}}$ and ${\mathrm{R}}_{\mathrm{c}\mathrm{m}}$ parameters, a value of c = 10 % (which is the value that UNE-EN ISO 25178-3:2013 [34] establishes for the p parameter for the calculation of ${\mathrm{S}}_{\mathrm{d}\mathrm{c}}$) has been used as input, and for the calculation of ${\mathrm{S}}_{\mathrm{m}\mathrm{r}}$ and ${\mathrm{R}}_{\mathrm{m}\mathrm{c}}$, the previously calculated value of ${\mathrm{S}}_{\mathrm{m}\mathrm{c}}$ and ${\mathrm{R}}_{\mathrm{c}\mathrm{m}}$ has been used.

In the tables in this section, some parameters are shown in red and others in orange. In red are shown parameters with large variations and in orange parameters with significant but not so large variations.

3.1. Evaluation for the Random Surface

Table 3. Results of the surface roughness of the original and filtered clouds (2–15 µm).

Parameters	Original	2 µm	4 µm	5 µm	6 µm	14 µm	15 µm
${\mathrm{S}}_{\mathrm{q}}$ [µm]	1.6825	1.6823	1.6811	1.6795	1.6773	1.6452	1.6385
${\mathrm{S}}_{\mathrm{s}\mathrm{k}}$	−0.0056	−0.0055	−0.0046	−0.0036	−0.0023	0.0261	0.0334
${\mathrm{S}}_{\mathrm{k}\mathrm{u}}$	2.9573	2.9572	2.9568	2.9564	2.9560	2.9402	2.9375
${\mathrm{S}}_{\mathrm{p}}$ [µm]	7.5584	7.5577	7.5551	7.5514	7.5462	7.4800	7.4691
${\mathrm{S}}_{\mathrm{v}}$ [µm]	6.6769	6.6776	6.6641	6.6455	6.6051	6.1742	6.1270
${\mathrm{S}}_{\mathrm{z}}$ [µm]	14.2354	14.2354	14.2191	14.1969	14.1514	13.6542	13.5960
${\mathrm{S}}_{\mathrm{a}}$ [µm]	1.3458	1.3456	1.3447	1.3434	1.3417	1.3171	1.3119
${\mathrm{S}}_{\mathrm{d}\mathrm{q}}$ [rad]	0.4304	0.4295	0.4259	0.4218	0.4170	0.3814	0.3772
${\mathrm{S}}_{\mathrm{d}\mathrm{r}}$ [%]	8.6407	8.6148	8.4735	8.3158	8.1333	6.8526	6.7066
${\mathrm{S}}_{\mathrm{m}\mathrm{r}}$ [%]	10.0000	10.0000	10.0000	10.0000	10.0000	10.0000	10.0000
${\mathrm{S}}_{\mathrm{m}\mathrm{c}}$ [µm]	2.1002	2.1006	2.1018	2.1036	2.1061	2.1372	2.1413
${\mathrm{S}}_{\mathrm{d}\mathrm{c}}$ [µm]	3.5727	3.5722	3.5697	3.5665	3.5622	3.5037	3.4922
${\mathrm{S}}_{\mathrm{k}}$ [µm]	4.3760	4.3755	4.3727	4.3689	4.3639	4.2916	4.2757
${\mathrm{S}}_{\mathrm{p}\mathrm{k}}$ [µm]	1.5694	1.5692	1.5690	1.5685	1.5677	1.5596	1.5591
${\mathrm{S}}_{\mathrm{p}\mathrm{k}\mathrm{x}}$ [µm]	5.3631	5.3627	5.3617	5.3602	5.3580	5.3351	5.3344
${\mathrm{S}}_{\mathrm{v}\mathrm{k}}$ [µm]	1.5673	1.5669	1.5647	1.5619	1.5580	1.4876	1.4717
${\mathrm{S}}_{\mathrm{v}\mathrm{k}\mathrm{x}}$ [µm]	4.4962	4.4972	4.4847	4.4677	4.4294	4.0275	3.9859
${\mathrm{S}}_{\mathrm{m}\mathrm{r}\mathrm{k}1}$ [%]	9.5198	9.5202	9.5227	9.5253	9.5293	9.6107	9.6445
${\mathrm{S}}_{\mathrm{m}\mathrm{r}\mathrm{k}2}$ [%]	90.0966	90.0973	90.1033	90.1090	90.1169	90.2672	90.3132
${\mathrm{S}}_{\mathrm{a}\mathrm{k}1}$ [µm3/mm2]	74.7000	74.6978	74.7041	74.6999	74.6941	74.9462	75.1860
${\mathrm{S}}_{\mathrm{a}\mathrm{k}2}$ [µm3/mm2]	77.6061	77.5852	77.4290	77.2451	76.9882	72.3932	71.2780

and

Table 4. Results of the surface roughness parameters of the filtered clouds (21–100 µm).

Parameters	21 µm	25 µm	38 µm	50 µm	70 µm	100 µm
${\mathrm{S}}_{\mathrm{q}}$ [µm]	1.5848	1.5426	1.4127	1.3209	1.2125	1.1109
${\mathrm{S}}_{\mathrm{s}\mathrm{k}}$	0.0873	0.1227	0.2088	0.2491	0.2841	0.3102
${\mathrm{S}}_{\mathrm{k}\mathrm{u}}$	2.9364	2.9522	3.0300	3.1063	3.2002	3.2732
${\mathrm{S}}_{\mathrm{p}}$ [µm]	7.3869	7.3200	7.0813	6.8738	6.5819	6.2478
${\mathrm{S}}_{\mathrm{v}}$ [µm]	5.6975	5.5356	4.7164	4.4972	4.0934	3.6758
${\mathrm{S}}_{\mathrm{z}}$ [µm]	13.0845	12.8556	11.7977	11.3709	10.6753	9.9236
${\mathrm{S}}_{\mathrm{a}}$ [µm]	1.2695	1.2353	1.1284	1.0526	0.9629	0.8794
${\mathrm{S}}_{\mathrm{d}\mathrm{q}}$ [rad]	0.3498	0.3315	0.2800	0.2453	0.2057	0.1690
${\mathrm{S}}_{\mathrm{d}\mathrm{r}}$ [%]	5.8130	5.2451	3.7930	2.9366	2.0815	1.4130
${\mathrm{S}}_{\mathrm{m}\mathrm{r}}$ [%]	10.0000	10.0000	10.0000	10.0000	10.0000	10.0000
${\mathrm{S}}_{\mathrm{m}\mathrm{c}}$ [µm]	2.1660	2.1850	2.2663	2.3573	2.5106	2.7095
${\mathrm{S}}_{\mathrm{d}\mathrm{c}}$ [µm]	3.3961	3.3148	3.0459	2.8475	2.6087	2.3816
${\mathrm{S}}_{\mathrm{k}}$ [µm]	4.1399	4.0225	3.6501	3.3878	3.0714	2.7794
${\mathrm{S}}_{\mathrm{p}\mathrm{k}}$ [µm]	1.5568	1.5528	1.5110	1.4573	1.3774	1.2890
${\mathrm{S}}_{\mathrm{p}\mathrm{k}\mathrm{x}}$ [µm]	5.3351	5.3362	5.3020	5.2312	5.1007	4.9107
${\mathrm{S}}_{\mathrm{v}\mathrm{k}}$ [µm]	1.3599	1.2932	1.1251	1.0439	0.9626	0.8856
${\mathrm{S}}_{\mathrm{v}\mathrm{k}\mathrm{x}}$ [µm]	3.6095	3.4969	2.8456	2.7519	2.5032	2.2335
${\mathrm{S}}_{\mathrm{m}\mathrm{r}\mathrm{k}1}$ [%]	9.9076	10.1207	10.6837	10.9739	11.3506	11.6327
${\mathrm{S}}_{\mathrm{m}\mathrm{r}\mathrm{k}2}$ [%]	90.6360	90.8333	91.1788	91.3833	91.3464	91.1754
${\mathrm{S}}_{\mathrm{a}\mathrm{k}1}$ [µm3/mm2]	77.1222	78.5763	80.7170	79.9638	78.1706	74.9701
${\mathrm{S}}_{\mathrm{a}\mathrm{k}2}$ [µm3/mm2]	63.6721	59.2715	49.6228	44.9760	41.6490	39.0753

show the results for the areal roughness parameters for particle sizes from 2 µm to 100 µm.

Looking at the results in the tables above, it is evident that the parameter most affected by the increasing particle size is the S_dr parameter. This parameter exhibits the greatest variation on the unfiltered surface, a value of 8.6407%, which decreases to 1.41301% on the 100 µm filter. The S_dr parameter is the ratio between the interfacial surface and the projected surface; therefore, it makes sense that when a filtering operation is applied, it is considerably reduced, since the variation between peaks and dales will be reduced. Therefore, the interfacial surface will be reduced as well.

Another parameter that has a large variation is S_dq, which on the unfiltered surface is 0.4304 rad, and for the 100 µm filter, it is 0.169 rad. The parameter calculation is based on the gradient of the surface when enlarging the filter size.

The S_sk parameter varies from −0.0056 to 0.3102. This parameter evaluates the level of asymmetry in the surface height distribution.

The parameters S_vk, S_vkx, and S_ak2 also have a remarkable variation. S_vkx ranges from 4.4962 µm to 2.2335 µm, and S_v is reduced to almost half of its initial value.

Some parameters increase with an increasing particle size, and others are reduced. The S_ku parameter is one of the parameters that increases with the size of the filter, but the most curious thing is that it exceeds the limit of 3. When S_ku has a value of 3, the distribution of the surface in height is normal; when it has a value below 3, the distribution is uniform, and above 3, it is sharp [36].

The graphs in Figure 10 show two point clouds; the one on the left is the original one and the one on the right is the one resulting from applying a 70 µm FAMCB filter. A clear reduction in the depth of the pits is observed.

The material ratio curve describes the surface texture. It is a graphical representation of the surface or profile material ratio of the scale-limited surface or profile as a function of an intersection level. This curve represents the percentage of points (horizontal axis) above a given height (vertical axis). The left side of the curve corresponds to the highest points of the surface and the right side to the lowest points. Therefore, it makes sense that the right part of the curve is more affected with an increasing filter or particle size.

Figure 11 shows the areal material ratio curve for the created surface. It is shown that the secant of the curve becomes smaller with an increasing filter size. The reduction in this curve results in a reduction in the S_k parameter. The curve shows a reduction in the size of the dales at around 2 µm.

In the original curve for a height of 0 µm, there is 49.1694% of the surface data above this intersection level. But once the morphological filter of 100 µm is applied, the dales are reduced and 87.7874 % of the data are above that height. The S_mc and S_mr functions are the ones that relate the height to the material ratio curves.

In S_mr, a height is given as a datum and gives the material ratio as a result, and in the case of S_mc, a probability is given as a parameter and calculates the height to which it corresponds.

In Figure 12, the evolution of each areal roughness parameter is observed with the filter size increasing.

As seen in Table 3 and Table 4, there are significant variations in the parameters S_sk, S_dr, S_dq, S_v, S_vk, S_vkx, and S_ak2. Due to the change in the sign of the parameter S_sk, it seems that the parameter has a larger change than it actually has.

3.2. Evaluation for Profile

For the evaluation of the profile roughness parameters, the first profile after removing all data affected by the end effect was used. This profile corresponds to 1002 of the original random surface. This evaluation could have been performed with any of the profiles. For the evaluation of the profile parameters, a disc of the same diameter as the particle has been used instead of a sphere, since we are working in two dimensions and not in three.

Table 5. Results of the profile roughness of the original and filtered clouds (2–15 µm).

Parameters	Original	2 µm	4 µm	5 µm	6 µm	14 µm	15 µm
$R_q$ [µm]	1.5478	1.5477	1.5474	1.5469	1.5460	1.5349	1.5320
$R_{sk}$	0.2952	0.2952	0.2949	0.2951	0.2957	0.3023	0.3061
$R_{ku}$	3.4020	3.4019	3.4014	3.4002	3.3986	3.4001	3.4026
$R_p$ [µm]	5.3538	5.3535	5.3521	5.3506	5.3485	5.3216	5.3172
$R_v$ [µm]	4.3783	4.3787	4.3742	4.3515	4.3204	4.2297	4.2092
$R_{zt}$ [µm]	9.7321	9.7321	9.7264	9.7022	9.6689	9.5512	9.5264
$R_a$ [µm]	1.2068	1.2067	1.2064	1.2060	1.2053	1.1963	1.1943
$R_{dq}$ [rad]	0.2986	0.2981	0.2960	0.2940	0.2915	0.2726	0.2703
$R_{dr}$ [%]	4.2369	4.2268	4.1656	4.1101	4.0412	3.5374	3.4805
$R_{mc}$ [%]	10.0000	10.0000	10.0000	10.0000	10.0000	10.0000	10.0000
$R_{cm}$ [µm]	1.8687	1.8687	1.8711	1.8687	1.8687	1.8841	1.8826
$R_{dc}$ [µm]	3.2085	3.2085	3.2090	3.2051	3.2027	3.1947	3.1871
$R_k$ [µm]	3.7311	3.7306	3.7479	3.7461	3.7425	3.6595	3.6510
$R_{pk}$ [µm]	1.8802	1.8794	1.8743	1.8772	1.8788	1.8690	1.8717
$R_{pkx}$ [µm]	3.5332	3.5331	3.5218	3.5211	3.5204	3.5450	3.5460
$R_{vk}$ [µm]	1.4236	1.4230	1.4160	1.4161	1.4192	1.4188	1.4083
$R_{vkx}$ [µm]	2.4678	2.4684	2.4566	2.4349	2.4060	2.3467	2.3295
$R_{mrk1}$ [%]	11.8097	11.8149	11.7124	11.6901	11.6812	12.1646	12.1694
$R_{mrk2}$ [%]	90.1687	90.1639	90.2265	90.2338	90.2566	90.2128	90.2299
$R_{ak1}$ [µm2/mm]	111.0213	111.0229	109.7636	109.7209	109.7343	113.6792	113.8854
$R_{ak2}$ [µm2/mm]	69.9784	69.9824	69.1953	69.1502	69.1399	69.4281	68.7978

and

Table 6. Results of the profile roughness parameters of the filtered clouds (21–100 µm).

Parameters	21 µm	25 µm	38 µm	50 µm	70 µm	100 µm
$R_q$ [µm]	1.5141	1.5029	1.4405	1.3987	1.3546	1.3046
$R_{sk}$	0.3169	0.3174	0.4353	0.4805	0.5002	0.5417
$R_{ku}$	3.4372	3.4523	3.3889	3.4063	3.4154	3.3557
$R_p$ [µm]	5.2845	5.2592	5.1560	5.0634	4.9296	4.7585
$R_v$ [µm]	4.1236	4.0864	3.3434	2.9977	2.7497	2.5255
$R_{zt}$ [µm]	9.4081	9.3456	8.4994	8.0611	7.6793	7.2840
$R_a$ [µm]	1.1780	1.1680	1.1270	1.0966	1.0601	1.0350
$R_{dq}$ [rad]	0.2556	0.2462	0.2154	0.1936	0.1677	0.1406
$R_{dr}$ [%]	3.1284	2.9103	2.2543	1.8329	1.3845	0.9790
$R_{mr}$ [%]	10.0000	10.0000	10.0000	10.0000	10.0000	10.0000
$R_{mc}$ [µm]	1.8904	1.9053	1.9619	2.0264	2.1060	2.2676
$R_{dc}$ [µm]	3.1473	3.1347	3.0525	2.9907	2.9098	2.8986
$R_k$ [µm]	3.5738	3.5772	3.4146	3.4346	3.2031	3.2932
$R_{pk}$ [µm]	1.8679	1.8349	1.8230	1.8694	1.8005	1.6701
$R_{pkx}$ [µm]	3.5418	3.5276	3.5023	3.4137	3.3975	3.1945
$R_{vk}$ [µm]	1.4826	1.4364	1.1759	0.9848	0.9873	0.6862
$R_{vkx}$ [µm]	2.2925	2.2409	1.5825	1.2128	1.0787	0.7963
$R_{mrk1}$ [%]	12.2189	12.3297	12.4866	11.4784	12.5013	11.8171
$R_{mrk2}$ [%]	90.5259	90.6584	90.3033	90.7477	90.0200	91.2505
$R_{ak1}$ [µm2/mm]	114.1195	113.1207	113.8143	107.2857	112.5437	98.6763
$R_{ak2}$ [µm2/mm]	70.2334	67.0920	57.0128	45.5567	49.2659	30.0181

contain the results of the calculation of the profile roughness parameters of the original and filtered profiles.

The parameter with the greatest variation is R_dr, which starts at 4.2369% and ends with a value of 0.9790%. The R_kvx parameter also shows significant evolution, changing from 2.4678 µm to 0.7963 µm. Rak2 starts at 69.9784 µm²/mm and ends at 30.0181 µm²/mm; this may seem like a large change, but it is due to the unit of this parameter. R_vk also has considerable evolution, exceeding 50%, starting at 1.4236 µm and ending at 0.6862 µm. R_dq also varies significantly, from 0.2986 rad to 0.1406 rad.

The results obtained from the profile filtering of the original point cloud profile 1002 (which is the first profile after removing the end effect of the biggest FAMCB filter used in the paper) and the calculation of roughness parameters for the profile are shown below. Figure 13 shows the selected and fitted profile with all the selected stylus sizes. Figure 14 shows a part of this profile, which has been trimmed so that the behaviour of the FPMCD filter for the peaks and dales of the profile can be observed.

Figure 14 shows the profile expanded to better appreciate the differences between different particle sizes.

Figure 15 shows a detailed enlargement in the 340–440 um range of the X order.

Figure 16 shows the evolution of the material ratio curve with an increasing stylus size and the comparison with the original profile. With the increase in the filter size, a clear reduction in the profile dales can be seen on the right side of the curve, reducing the depth of the dales by about 2 µm for the 100 µm filter.

Comparing this curve with the one in Figure 11, it can be seen that it has a more irregular shape; this is due to the fact that as it is only a profile, and not the complete surface, many fewer points are used to generate it and it has some small undulations.

Reviewing the filtering of the dales, it can be seen that up to 25 µm, it only filters the small dales, but from 38 µm onwards, it starts to reduce larger dales.

A notable reduction in the dales is observed in the bearing capacity curve, as the dales decrease by at least 2 µm. A reduction in the minimum secant of the curve is also observed, which could lead to an increase in the R_mrk2 parameter and a reduction in the core (R_k) of the profile.

The reduction in the minimum secant with an increasing particle size is smaller than that at the surface.

In the original profile for a height of 0 µm, a bearing rate of 46.8106% is observed, and in the filtered profile curve with a particle size of 100 µm, a bearing rate of 63.6873% is observed.

The difference between the two material rate curves for the surface is greater than in the case of the profile.

Figure 17 shows the evolution of each profile roughness parameter as a percentage for each of the filter sizes analysed.

This graph presents a similar aspect to that of Figure 12, although in this case the R_sk parameter does not present such a severe variation graphically, as occurs with S_sk. This is due to the fact that in the case of the random surface, the S_sk parameter changes sign, and graphically, it seems that the change is much greater.

The previous graph shows a great evolution in the results of the parameters R_dr, R_vkx, R_vk, R_ak2, and R_dq. The variation in R_dr is almost 80%.

3.3. Evaluation for Groove and Peak Standard

As explained above, in addition to the random surface, groove and peak standards have also been generated in order to clearly appreciate the effect of the filtering on the peaks and dales.

3.3.1. Comparison of Results of Standard with α = 130°

In this section, the variation in the surface roughness parameter results for the peak and groove standards is analysed.

Table 7. Results of the areal roughness parameters of the original and filtered groove standard (2–15 µm).

Parameters	Original	2 µm	4 µm	5 µm	6 µm	14 µm	15 µm
Sq [µm]	3.2957	3.2957	3.2951	3.2948	3.2943	3.2875	3.2863
Ssk	−4.0838	−4.0838	−4.0831	−4.0826	−4.0819	−4.0733	−4.0718
Sku	19.1490	19.1490	19.1406	19.1344	19.1270	19.0269	19.0100
Sp [µm]	0.8665	0.8665	0.8664	0.8663	0.8663	0.8651	0.8648
Sv [µm]	19.1335	19.1335	19.0309	18.9794	18.9280	18.5148	18.4633
Sz [µm]	20.0000	20.0000	19.8973	19.8458	19.7943	19.3799	19.3281
Sa [µm]	1.5868	1.5868	1.5866	1.5865	1.5864	1.5842	1.5838
Sdq [rad]	0.1363	0.1363	0.1358	0.1356	0.1354	0.1335	0.1333
Sdr [%]	0.8842	0.8842	0.8784	0.8754	0.8724	0.8484	0.8454

and

Table 8. Results of the areal roughness parameters of the filtered groove standard (21–100 µm).

Parameters	21 µm	25 µm	38 µm	50 µm	70 µm	100 µm
Sq [µm]	3.2773	3.2699	3.2372	3.1965	3.1077	2.9300
Ssk	−4.0614	−4.0531	−4.0202	−3.9841	−3.9170	−3.8120
Sku	18.8919	18.7994	18.4419	18.0627	17.3877	16.3956
Sp [µm]	0.8632	0.8618	0.8557	0.8478	0.8298	0.7916
Sv [µm]	18.1548	17.9497	17.2835	16.6712	15.6553	14.1429
Sz [µm]	19.0180	18.8115	18.1392	17.5190	16.4852	14.9345
Sa [µm]	1.5808	1.5783	1.5670	1.5525	1.5195	1.4494
Sdq [rad]	0.1318	0.1308	0.1276	0.1246	0.1194	0.1111
Sdr [%]	0.8274	0.8155	0.7765	0.7405	0.6806	0.5906

contain the results of the calculation of the surface roughness parameters of the original and filtered groove standard.

There is a large difference in the results obtained for the S_dr parameter, ranging from 0.8842 µm to 0.5906 µm. the S_z parameter also has a considerable difference in its value, which is originally 20 µm, and ends up being 14.9345 µm. The S_v parameter has a variation similar to that of S_z, starting at 19.1335 µm and ending at 14.1429 µm.

Figure 18 shows the evolution of the groove with an increasing particle size. A reduction in the dale size of around 25% is observed for a filter size of 100 µm.

Figure 19 shows the evolution of each surface roughness parameter as a percentage for each of the filter sizes analysed for the groove standard.

It is curious that, being a groove standard, all the parameters are reduced with the increase in the particle size.

Table 9. Results of the areal roughness parameters of the original and filtered peak standard (2–15 µm).

Parameters	Original	2 µm	4 µm	5 µm	6 µm	14 µm	15 µm
Sq [µm]	3.2957	3.2956	3.2956	3.2956	3.2956	3.2956	3.2956
Ssk	4.0838	4.0838	4.0838	4.0838	4.0838	4.0838	4.0838
Sku	19.1490	19.1491	19.1491	19.1491	19.1491	19.1491	19.1491
Sp [µm]	19.1335	19.1335	19.1335	19.1334	19.1334	19.1331	19.1331
Sv [µm]	0.8665	0.8665	0.8665	0.8666	0.8666	0.8669	0.8669
Sz [µm]	20.0000	20.0000	20.0000	20.0000	20.0000	20.0000	20.0000
Sa [µm]	1.5868	1.5868	1.5868	1.5868	1.5868	1.5867	1.5867
Sdq [rad]	0.1363	0.1362	0.1361	0.1360	0.1360	0.1355	0.1355
Sdr [%]	0.8842	0.8828	0.8817	0.8808	0.8806	0.8745	0.8742

and

Table 10. Results of the areal roughness parameters of the filtered peak standard (21–100 µm).

Parameters	21 µm	25 µm	38 µm	50 µm	70 µm	100 µm
Sq [µm]	3.2955	3.2954	3.2952	3.2951	3.2951	3.2960
Ssk	4.0837	4.0836	4.0829	4.0818	4.0780	4.0673
Sku	19.1488	19.1483	19.1450	19.1389	19.1174	19.0532
Sp [µm]	19.1327	19.1323	19.1308	19.1289	19.1245	19.1158
Sv [µm]	0.8673	0.8677	0.8692	0.8711	0.8755	0.8842
Sz [µm]	20.0000	20.0000	20.0000	20.0000	20.0000	20.0000
Sa [µm]	1.5867	1.5866	1.5869	1.5876	1.5906	1.5988
Sdq [rad]	0.1352	0.1349	0.1342	0.1335	0.1323	0.1306
Sdr [%]	0.8697	0.8666	0.8575	0.8491	0.8349	0.8146

contain the results of the calculation of the surface roughness parameters of the original and filtered peak standard.

There is a large difference in the results obtained for the S_dr parameter, ranging from 0.8842 µm to 0.8146 µm.

Figure 20 shows the graphical comparison of the shape of the original and filtered peak standard. The variations are only seen at the lower rounding radii. As this is a morphological closing filter, as is the case with the random surface and the profile, the peak is not affected.

Figure 21 shows an expansion of Figure 20, where the increase in the roundness radius with the particle size increase can be seen.

Figure 22 shows the evolution of each surface roughness parameter as a percentage for each of the filter sizes analysed for the peak standard.

An important evolution is observed in the results of the parameters S_dr and S_dq.

3.3.2. Comparison of Results with Variations of ±5° of α

This section shows the evolution of the results for an alpha variation of ±5° of the groove and peak standards. It has been decided to consider ±5°, because it is a total variation of 10° and it is considered to be a large variation.

The results are presented for particle or stylus sizes of 5 µm, 25 µm, 50 µm, and 100 µm. It is considered that these four sizes are sufficient to be able to appreciate an evolution in the calculated parameters. Parameters have been calculated for all particle sizes considered in the article and are presented in Appendix B.

Table 11. Results of applying a 5 µm filter and a ±5° variation in α for a groove standard.

Parameters	α = 125°	α = 130°	α = 135°
Sq [µm]	3.1345	3.2948	3.4757
Ssk	−4.3480	−4.0826	−3.8094
Sku	21.5168	19.1344	16.8381
Sp [µm]	0.7778	0.8663	0.9733
Sv [µm]	19.0317	18.9794	18.9036
Sz [µm]	19.8095	19.8458	19.8769
Sa [µm]	1.4377	1.5865	1.7625
Sdq [rad]	0.1431	0.1356	0.1280
Sdr [%]	0.9636	0.8754	0.7874

and Figure 23 show the results of areal parameters for the variation in the groove standard for the 5 µm particle size.

Table 12. Results of applying a 25 µm filter and a ±5° variation in α for a groove standard.

Parameters	α = 125°	α = 130°	α = 135°
Sq [µm]	3.0992	3.2699	3.4587
Ssk	−4.3038	−4.0531	−3.7905
Sku	20.9899	18.7994	16.6354
Sp [µm]	0.7717	0.8618	0.9701
Sv [µm]	17.7634	17.9497	18.0824
Sz [µm]	18.5352	18.8115	19.0525
Sa [µm]	1.4264	1.5783	1.7567
Sdq [rad]	0.1368	0.1308	0.1244
Sdr [%]	0.8824	0.8155	0.7443

and Figure 24 show the results of areal parameters for the variation in the groove standard for the 25 µm particle size.

Table 13. Results of applying a 50 µm filter and a ±5° variation in α for a groove standard.

Parameters	α = 125°	α = 130°	α = 135°
Sq [µm]	2.9966	3.1965	3.4081
Ssk	−4.2064	−3.9841	−3.7441
Sku	19.9092	18.0627	16.1611
Sp [µm]	0.7527	0.8478	0.9600
Sv [µm]	16.1902	16.6712	17.0626
Sz [µm]	16.9429	17.5190	18.0226
Sa [µm]	1.3912	1.5525	1.7383
Sdq [rad]	0.1286	0.1246	0.1198
Sdr [%]	0.7809	0.7405	0.6906

and Figure 25 show the results of areal parameters for the variation in the groove standard for the 50 µm particle size.

Table 14. Results of applying a 100 µm filter and a ±5° variation in α for a groove standard.

Parameters	α = 125°	α = 130°	α = 135°
Sq [µm]	2.6293	2.9300	3.2214
Ssk	−3.9862	−3.8120	−3.6179
Sku	17.7271	16.3956	14.9752
Sp [µm]	0.6766	0.7916	0.9197
Sv [µm]	13.0818	14.1429	15.0431
Sz [µm]	13.7584	14.9345	15.9628
Sa [µm]	1.2503	1.4494	1.6651
Sdq [rad]	0.1104	0.1111	0.1100
Sdr [%]	0.5779	0.5906	0.5830

and Figure 26 show the results of areal parameters for the variation in the groove standard for the 100 µm particle size.

Comparing the four graphs, the larger the particle size, the less dependent the S_dr and S_dq parameter are on the angle of the standard.

Table 15. Results of applying a 5 µm filter and a ±5° variation in α for a peak standard.

Parameters	α = 125°	α = 130°	α = 135°
Sq [µm]	3.1358	3.2956	3.4763
Ssk	4.3499	4.0838	3.8101
Sku	21.5412	19.1491	16.8466
Sp [µm]	19.2219	19.1334	19.0265
Sv [µm]	0.7781	0.8666	0.9735
Sz [µm]	20.0000	20.0000	20.0000
Sa [µm]	1.4380	1.5868	1.7627
Sdq [rad]	0.1436	0.1360	0.1283
Sdr [%]	0.9708	0.8808	0.7916

and Figure 27 show the results of areal parameters for the variation in the peak standard for the 5 µm particle size.

The parameters with the highest variability are S_ku, S_v, S_dr, and S_a, all of which are around 20%.

Table 16. Results of applying a 25 µm filter and a ±5° variation in α for a peak standard.

Parameters	α = 125°	α = 130°	α = 135°
Sq [µm]	3.1356	3.2954	3.4761
Ssk	4.3494	4.0836	3.8101
Sku	21.5388	19.1483	16.8465
Sp [µm]	19.2204	19.1323	19.0257
Sv [µm]	0.7796	0.8677	0.9743
Sz [µm]	20.0000	20.0000	20.0000
Sa [µm]	1.4380	1.5866	1.7626
Sdq [rad]	0.1422	0.1349	0.1275
Sdr [%]	0.9520	0.8666	0.7813

and Figure 28 show the results of areal parameters for the variation in the peak standard for the 25 µm particle size.

Table 17. Results of applying a 50 µm filter and a ±5° variation in α for a peak standard.

Parameters	α = 125°	α = 130°	α = 135°
Sq [µm]	3.1354	3.2951	3.4757
Ssk	4.3459	4.0818	3.8092
Sku	21.5183	19.1389	16.8428
Sp [µm]	19.2160	19.1289	19.0232
Sv [µm]	0.7840	0.8711	0.9768
Sz [µm]	20.0000	20.0000	20.0000
Sa [µm]	1.4406	1.5876	1.7626
Sdq [rad]	0.1404	0.1335	0.1264
Sdr [%]	0.9292	0.8491	0.7682

and Figure 29 show the results of areal parameters for the variation in the peak standard for the 50 µm particle size.

Table 18. Results of applying a 100 µm filter and a ±5° variation in α for a peak standard.

Parameters	α = 125°	α = 130°	α = 135°
Sq [µm]	3.1386	3.2960	3.4755
Ssk	4.3178	4.0673	3.8020
Sku	21.3328	19.0532	16.8051
Sp [µm]	19.1978	19.1158	19.0135
Sv [µm]	0.8022	0.8842	0.9865
Sz [µm]	20.0000	20.0000	20.0000
Sa [µm]	1.4599	1.5988	1.7682
Sdq [rad]	0.1365	0.1306	0.1242
Sdr [%]	0.8812	0.8146	0.7428

and Figure 30 show the results of areal parameters for the variation in the peak standard for the 100 µm particle size.

A variation of up to 20% of the value is observed for all particle sizes. It is reasonable that it remains similar because the change is in the angle of the standard and not in the particle size.

The parameter S_ku seems to be the one that varies the most in the numbers, but it has a variability of approximately 20% in all cases.

In the case of the peak standard, the variation between the parameters seems to be more constant and not so dependent on the particle size. In the case of the groove standard, the difference between values increases with increasing particle size, i.e., at the 5 µm size there is less variability than at the 100 µm size.

3.3.3. Comparison of Results for ±5% Variations in Stylus Size

The size of the stylus varies by 5% for the groove and peak standards with α = 130° and the following results are obtained. It has been decided to consider ±5% because it is a total variation of 10% and when a stylus shows a wear of 5%, it is discarded.

In this section, as in the previous one, the results are presented for particle or stylus sizes of 5 µm, 25 µm, 50 µm, and 100 µm. It is considered that these four sizes are sufficient to be able to appreciate an evolution in the calculated parameters. Parameters have been calculated for all particle sizes considered in the article and are presented in Appendix B.

Table 19. Results of applying a stylus size variation of ±5% to the 5 µm stylus for the groove standard with α = 130°.

Parameters	4.75 µm	5 µm	5.25 µm
Sq [µm]	3.2948	3.2948	3.2947
Ssk	−4.0827	−4.0826	−4.0824
Sku	19.1359	19.1344	19.1329
Sp [µm]	0.8664	0.8663	0.8663
Sv [µm]	18.9915	18.9794	18.9678
Sz [µm]	19.8579	19.8458	19.8342
Sa [µm]	1.5866	1.5865	1.5865
Sdq [rad]	0.1357	0.1356	0.1356
Sdr [%]	0.8761	0.8754	0.8748

and Figure 31 show the results of areal parameters for the 5 µm particle size variation for the groove standard.

The parameters have a minimum variation of only 0.15% of the maximum value.

Table 20. Results of applying a stylus size variation of ±5% to the 25 µm stylus for the groove standard with α = 130°.

Parameters	23.75 µm	25 µm	26.25 µm
Sq [µm]	3.2723	3.2699	3.2672
Ssk	−4.0558	−4.0531	−4.0503
Sku	18.8292	18.7994	18.7684
Sp [µm]	0.8623	0.8618	0.8614
Sv [µm]	18.0135	17.9497	17.8853
Sz [µm]	18.8758	18.8115	18.7467
Sa [µm]	1.5791	1.5783	1.5774
Sdq [rad]	0.1311	0.1308	0.1305
Sdr [%]	0.8192	0.8155	0.8117

and Figure 32 show the results of areal parameters for the 25 µm particle size variation for the groove standard.

Table 21. Results of applying a stylus size variation of ±5% to the 50 µm stylus for the groove standard with α = 130°.

Parameters	47.5 µm	50 µm	52.5 µm
Sq [µm]	3.2058	3.1965	3.1869
Ssk	−3.9920	−3.9841	−3.9761
Sku	18.1443	18.0627	17.9804
Sp [µm]	0.8496	0.8478	0.8459
Sv [µm]	16.7986	16.6712	16.5441
Sz [µm]	17.6482	17.5190	17.3899
Sa [µm]	1.5558	1.5525	1.5490
Sdq [rad]	0.1253	0.1246	0.1240
Sdr [%]	0.7480	0.7405	0.7330

and Figure 33 show the results of areal parameters for the 50 µm particle size variation for the groove standard.

Table 22. Results of applying a stylus size variation of ±5% to the 100 µm stylus for the groove standard with α = 130°.

Parameters	95 µm	100 µm	105 µm
Sq [µm]	2.9631	2.9300	2.8956
Ssk	−3.8293	−3.8120	−3.7950
Sku	16.5542	16.3956	16.2410
Sp [µm]	0.7989	0.7916	0.7839
Sv [µm]	14.3941	14.1429	13.8921
Sz [µm]	15.1930	14.9345	14.6760
Sa [µm]	1.4628	1.4494	1.4353
Sdq [rad]	0.1126	0.1111	0.1097
Sdr [%]	0.6056	0.5906	0.5756

and Figure 34 show the results of areal parameters for the 100 µm particle size variation for the groove standard.

Table 23. Results of applying a stylus size variation of ±5% to the 5 µm stylus for the peak standard with α = 130°.

Parameters	4.75 µm	5 µm	5.25 µm
Sq [µm]	3.2956	3.2956	3.2956
Ssk	4.0838	4.0838	4.0838
Sku	19.1491	19.1491	19.1491
Sp [µm]	19.1335	19.1334	19.1334
Sv [µm]	0.8665	0.8666	0.8666
Sz [µm]	20.0000	20.0000	20.0000
Sa [µm]	1.5868	1.5868	1.5868
Sdq [rad]	0.1361	0.1360	0.1360
Sdr [%]	0.8812	0.8808	0.8806

and Figure 35 show the results of areal parameters for the 5 µm particle size variation for the peak standard.

Table 24. Results of applying a stylus size variation of ±5% to the 25 µm stylus for the peak standard with α = 130°.

Parameters	23.75 µm	25 µm	26.25 µm
Sq [µm]	3.2954	3.2954	3.2954
Ssk	4.0836	4.0836	4.0836
Sku	19.1484	19.1483	19.1483
Sp [µm]	19.1324	19.1323	19.1322
Sv [µm]	0.8676	0.8677	0.8678
Sz [µm]	20.0000	20.0000	20.0000
Sa [µm]	1.5867	1.5866	1.5866
Sdq [rad]	0.1350	0.1349	0.1349
Sdr [%]	0.8677	0.8666	0.8663

and Figure 36 show the results of areal parameters for the 25 µm particle size variation for the peak standard.

Table 25. Results of applying a stylus size variation of ±5% to the 50 µm stylus for the peak standard with α = 130°.

Parameters	47.5 µm	50 µm	52.5 µm
Sq [µm]	3.2951	3.2951	3.2951
Ssk	4.0821	4.0818	4.0813
Sku	19.1405	19.1389	19.1365
Sp [µm]	19.1294	19.1289	19.1284
Sv [µm]	0.8706	0.8711	0.8716
Sz [µm]	20.0000	20.0000	20.0000
Sa [µm]	1.5875	1.5876	1.5880
Sdq [rad]	0.1337	0.1335	0.1333
Sdr [%]	0.8513	0.8491	0.8473

and Figure 37 show the results of areal parameters for the 100 µm particle size variation for the peak standard.

Table 26. Results of applying a stylus size variation of ±5% to the 100 µm stylus for the peak standard with α = 130°.

Parameters	95 µm	100 µm	105 µm
Sq [µm]	3.2958	3.2960	3.2965
Ssk	4.0694	4.0673	4.0641
Sku	19.0663	19.0532	19.0331
Sp [µm]	19.1173	19.1158	19.1137
Sv [µm]	0.8827	0.8842	0.8863
Sz [µm]	20.0000	20.0000	20.0000
Sa [µm]	1.5972	1.5988	1.6011
Sdq [rad]	0.1309	0.1306	0.1303
Sdr [%]	0.8175	0.8146	0.8104

and Figure 38 show the results of areal parameters for the 100 µm particle size variation for the peak standard.

There is a very small variability in the surface roughness parameters with the variation of ±5% of the stylus size. This variation does not reach 1% in any case, although it seems that the variation increases for larger particle sizes. The parameter with the greatest variability is S_dr, which exceeds 0.8% for the 100 µm particle.

4. Discussion

Comparing the point clouds and the material ratio curves in Figure 11 and Figure 16 show a significant reduction in the dales, which are represented on the right side of the curve. With the increase in the filter size, the dale of the curve is reduced and with it, the minimum secant of the curve is reduced as well. The change in the slope of the curve is more noticeable on the surface than in the profile. This reduction in the slope causes the height of the surface core to be reduced (S_k).

The parameters S_mc, R_cm, S_mr, and R_mc have a very slight variation, because they receive another input parameter in addition to the surface or profile. As explained above, in the case of S_mc and R_cm, a constant (10%) has been used, and in the case of S_mr and R_mc, the result of S_mc and R_cm has been used. A value of 10% for the calculation on the left-hand side of the curve has been used.

It can be seen that in the case of the surface, the S_sk parameter changes from negative to positive as the filter size increases. The S_sk parameter shows the deviation between the data and the median line or plane. If the parameter is negative, its distribution is above the median line or plane. It is normal that when using a filtering technique that only affects the dales, this parameter may change sign.

The kurtosis (S_ku and R_ku) parameter increases with the filter size for the surface but decreases for the profile. This makes this parameter unsuitable for the evaluation of the critical particle size.

As for the peak standard, it is observed that there is no variation in the S_z parameter as expected, since, when using a morphological closing filter, this should only affect the dales and, being a triangle upwards, this standard does not present any dale. There is an increase in the S_v parameter and a reduction in the S_p parameter as the filter size increases. Although the S_z parameter is not affected, the surface changes on the sides of the triangle increasing the rounding radius; this causes the arithmetic mean of the plane to increase slightly. As it is the one used to define a peak and a dale, it causes a slight variation in the S_p and S_v parameters, but not in the sum of both.

A high sensitivity of the S_dr and S_dq parameters is observed in all cases, but mainly, they seem to be more sensitive to the variation in the stylus size.

In addition to the variations in S_dr and S_dq in the point cloud and in the profile, a remarkable variation in the parameters R_v, R_vk, R_vkx, and R_ak2 is observed. All these parameters are relative to the dales, so this variation is to be expected.

The parameter Sa is less sensitive to changes in the point clouds due to filtering than other parameters.

5. Conclusions

Because of the results analysed in this paper, the following conclusions were drawn:

• Special attention must be paid to the wear of the stylus used to measure roughness if the S_dr and S_dq parameters are to be used.
• S_dr and R_dr are the best parameters to evaluate the damage that can be caused by the particles in the contacting surfaces or pieces.
• The parameters S_vkx, S_vk, S_ak2, R_vkx, R_vk, and R_ak2 are also to be taken into account in this analysis. They are not as sensitive as R_dr or S_dr, but a variation in their value of more than 40% of their initial value with an increasing particle size has been observed.
• The parameters S_dq and R_dq also appear to be good indicators of the problems of poor lubricant distribution that particles can cause.
• The parameters R_ku and S_ku are not reliable for the evaluation of the critical particle size.
• Figure 11 shows that with larger particle sizes, larger dales can be clogged, and the lubricant is not evenly distributed over the entire surface. This generates more friction between the surfaces being examined, in addition to the friction caused by the particles themselves.
• The areas that can be left without oil due to the presence of particles of considerable size can lead to poorer heat dissipation from the machine.
• In this work, it has been observed that morphological filtering has a high computational cost. Before implementing it in a commonly used surface inspection system, it is advisable to have an estimate of the time the system may need to perform the filtering and whether it is really necessary to perform it; consideration should be given to whether another technique can be used, or whether the profiles instead of the entire surface can be inspected.
• Looking at Figure 15, the original and filtered profiles greatly expanded; it is estimated that the limiting particle size is around 38 µm, as it can be seen that the larger valleys of the profile also start to be affected.
• Surface parameters are more effective in assessing particle size, as they have greater variation, in addition to having more information on the entire surface.

Future Lines

Future lines of research include the following:

• Finish the implementation of calculation algorithms for the rest of the parameters of the UNE-EN ISO 25178-2:2023 [6] and UNE-EN ISO 21920-2:2023 [33] standards.
• Implement the F operation and apply S and L filters [6], which are not being applied now.
• Implement other filters of the UNE-EN ISO 16610 standard and compare the results and processing times of each one.
• Compare results of the software created with other libraries or calculation software.
• Test morphological filters with different spacings between data; this would be interesting, because an advantage of morphological filters is that they allow irregular spacing between data.
• Generate a mathematical model that can estimate the processing time that a large morphological filter can have on a given computer, knowing the processing time of two small filters on that computer.
• Try to optimize the filter to minimize the processing time required.
• Compare the results obtained from several profiles of a surface on which end effect compensation techniques [29] have been applied with the use of a larger surface with data discarding.
• Compare a surface filtered by profiles with an area filter to estimate the possible uncertainty that may be added by profile processing compared to area processing, given the shorter processing time it has.
• Consider the surface energy of the lubricant.
• Take into account the surface material and the possible wear of the surface.