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Fractal-wavelet based classification of tribological surfaces
Wear 254 (2003) 1189–1198
Fractal-wavelet based classification of tribological surfaces P. Podsiadlo∗ , G.W. Stachowiak Tribology Laboratory, School of Mechanical Engineering, The University of Western Australia, Crawley, WA 6009, Australia
Abstract Classification of the topography of freshly machined, worn and damaged surfaces (e.g. damaged by adhesion, scoring, abrasion, pitting) is still a problem in machine failure analysis. Tribological surfaces often exhibit both a multiscale nature (i.e. different length scales of surface features) and a non-stationary nature (i.e. features which are superimposed on each other and located at different positions on a surface). The most widely used approaches to surface classification are based on the Fourier transform or statistical functions and parameters. Often these approaches are inadequate and provide incorrect classification of the tribological surfaces. The main reason is that these techniques fail to simultaneously capture the multiscale nature and the non-stationary nature of the surface data. A new method, called a hybrid fractal-wavelet method, has recently been developed for the characterization of tribological surfaces in a multiscale and non-stationary manner. In contrast to other methods, this method combines both the wavelets’ inherent ability to characterize surfaces at each individual scale and the fractals’ inherent ability to characterize surfaces in a scale-invariant manner. The application of this method to the classification of artificially generated fractal and tribological surfaces (e.g. worn surfaces) is presented in this paper. The newly developed method has been further modified to better suit tribological surface data, including a new measure of differences between initial and decoded images. The accuracy of this method in the classification of surfaces was assessed. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Fractal-wavelet analysis; Surface classification; Tribological surfaces
1. Introduction Classification of tribological surfaces and wear particles has recently been used for the studies of wear, friction and lubrication mechanisms occurring between interacting surfaces (e.g. [1–7]). Given an unclassified surface (a wear particle), its classification is either: • supervised, i.e. a classification algorithm is used to assign an unclassified surface (particle) into a specific class which is predefined (prior knowledge) according to criteria such as wear mechanism, boundary shape, surface texture, experiment conditions, etc. [5–9], or • unsupervised, i.e. a clustering algorithm is first used to find groups in a training set of unclassified surfaces (particles) in such a way that similar surfaces (particles) are clustered in the same group, while different surfaces (particles) are clustered in separate groups. For the clustering, morphological parameters such as area, aspect ratio, roundness, surface roughness, are usually used [1]. These groups are then used to produce classes of surfaces (posterior knowledge). Next, a classification algorithm is
∗ Corresponding author. E-mail address: [email protected] (P. Podsiadlo).
used to assign an unclassified surface (particle) into one of these classes [1]. One difficulty in classifying tribological surfaces is that the surface classification depends on the chosen scale. Because the topography of tribological surfaces often exhibit a non-stationary and multiscale nature, there exists no optimal “global” scale, i.e. the scale that gives the best classification for all surfaces. Consequently, a “local” scale must be individually chosen for each surface. For this purpose, classification methods based on multiscale analyses can be used, including ring-shaped filters [8], windowed Fourier filters [10], Gabor filters [11,12], 2D pseudo-Wigner transform [13] and wavelet transform [14,15]. These methods are used to decompose surface data into different frequency components (scale images) which can be individually analysed. Other methods used to overcome the difficulty in classifying surfaces are based on fractals. Fractal methods provide scale-invariant surface descriptors, e.g. fractal dimension and topothesy [16,17] or mathematical models containing information about surface data across all possible scales [5,6]. Subsequently, the fractal-based classification is scale-invariant, i.e. it does not depend on the chosen scale. Recently, a new method, called a hybrid fractal-wavelet method, has been developed for the analysis of surface topography in a multiscale manner [18]. This hybrid method
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combines wavelets and fractals in such a way that a wavelet method (a symmetric wavelet transform (SWT)) is used to decompose surface topography into ranges of different scales, while a fractal method (a partition iterated function system (PIFS)) is used to build a mathematical model for each range of scales. From initial studies conducted [18], it appears that this hybrid method would be ideal for surface classification. This is because it provides the best of two worlds, i.e. the wavelets’ inherent ability to decompose surface data into individual scales and the fractals’ inherent ability to describe surface data over many scales in a scale-invariant manner. In this paper, a new pattern recognition method, called the pattern recognition by hybrid fractal-wavelet (PR-HFW) method, has been developed and applied to computer generated images of fractal surfaces and scanning electron microscope (SEM) images of tribological surfaces. Images of fractal surfaces (that were isotropic with increasing fractal dimension) were used to evaluate the classification accuracy of the PR-HFW method, for surface topographies exhibiting different complexity. The tribological surface images were gathered in two data sets. The first set contained surfaces taken from six different tribological systems. They were used to show that the PR-HFW method is sensitive enough to classify accurately and simultaneously, tribological surfaces that exhibit similar, moderately different and significantly different topographies. The second set contained unworn surfaces and worn surfaces that were subjected to different degrees of abrasive wear. These surfaces were used to study the suitability of the PR-HFW method for monitoring the progress of wear occurring in tribological systems. The newly developed method is similar to a pattern recognition by PIFS (PR-PIFS) method [5–7]. Both methods are used to encode classified surface images into PIFSs. The PIFSs are decoded using an unclassified surface image and decoded images are obtained. The dissimilarity between each decoded image and the unclassified image is a criterion used in the surface classification. These two methods do not need feature vectors of parameters to classify surfaces and they are scale- and rotationally-invariant. The difference is that the PR-HFW method is used to classify surfaces at specific scale ranges, i.e. at scales for which differences measured between decoded and initial images take minimum values, while the PR-PIFS is used to classify surfaces at all scales. Thus, it is expected that the PR-HFW method will classify surfaces with a higher accuracy than the PR-PIFS method. This has been supported by results obtained in this study.
2. Hybrid fractal-wavelet (HFW) method The HFW method is a combination of two methods, i.e. a PIFS method [19] and a SWT method [20], used to analyse surface data both in a scale-invariant manner (similar to fractal methods) and at different scales (similar to wavelet methods).
The PIFS method is used to construct a mathematical model of a surface topography. This construction is based on ‘piecewise self-transformability’, i.e. that surface image data contain large parts which can be converted into small parts located elsewhere on the image. Each conversion is performed using an affine contractive transformation. This yields a collection of affine transformations called a PIFS. During the conversion, a geometric transformation is used to average brightness values of pixels within non-overlapping 2 × 2 pixel squares tiling the original image data and then to assign the average values to pixels of an image. As a result, an average value image equal to the half size of the original image is obtained. This image is then used for the conversion between large and small image parts. The mathematical foundation of the PIFS lies in the piecewise contractive mapping fixed-point theorem. According to this theorem, the PIFS is contractive and defines a unique ‘fixed-image’ which is the original image. This means that if the PIFS is iteratively applied to any initial image, a sequence of images eventually converging to the attractor, i.e. to the original image, is obtained. Since the attractor is unique, it is entirely specified by the PIFS. In other words, the PIFS completely describes the image, but in a numerical form rather than in a pictorial form (image) and the image can be reconstructed from the PIFS if required. The image reconstructed is not exactly same as the original image since the ‘piecewise self-transformability’ is approximately satisfied for tribological surfaces. As an example, a PIFS was constructed for the worn surface image shown in Fig. 1a. Decoded images obtained after the iterative application of the PIFS constructed to a black image (Fig. 1b) are shown in Fig. 1c–e. A two-channel filter bank is used as the wavelet transform in the SWT method to decompose surface image data into different frequency components (subimages) represented as smoothed and detail images [20]. The HFW method is performed in the following steps [18]: 1. First, the SWT method is applied to the original image data and as a result smoothed images and three detail images (in vertical, horizontal and diagonal directions) are obtained at different scales. 2. First and higher scale smoothed images obtained are resized to half of the original image size using interpolation method. These resized images together with the smoothed image obtained at scale 0, create a set of new images. 3. A PIFS is then constructed for the original image data using a scale 0 smoothed image as an average image. This yields a first PIFS. Next, a second PIFS is constructed using a resized scale 1 smoothed image as an average image. This process is repeated until all smoothed images are used. Consequently, a set of PIFSs constructed for the same image data using different smoothed images as average images, are obtained (Fig. 2).
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Fig. 1. Decoded images obtained after the iterative application of the PIFS, constructed for the worn surface shown in (a), into a black initial image shown in (b); (c) 1 iteration; (d) 4 iterations; (e) 12 iterations, respectively.
It was shown that the HFW method is more accurate than the PIFS method previously developed by the authors [18]. This is because the hybrid method uses a scale 0 smoothed image containing surface details which are either not present or are significantly distorted in an average value image used in the PIFS method. As an example, the HFW method was applied to the worn surface shown in Fig. 1a. Three smoothed images were generated by the SWT method and PIFSs were constructed using
these smoothed images as average images. Decoded images obtained for each PIFS constructed are shown in Fig. 3.
3. Pattern recognition by hybrid fractal-wavelet (PR-HFW) method A pattern recognition system based on the HFW method, called the PR-HFW method, has been developed for the
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Fig. 2. A schematic diagram of PIFSs constructed for a surface image at different scales using a hybrid fractal-wavelet method [6].
classification of tribological surfaces. The basic idea behind this method is that PIFSs constructed for classified surfaces are decoded using an unclassified surface image as the initial image. If the unclassified surface exhibits a similar morphology to the already classified surface, then the decoded image obtained just after one iteration will be ‘similar’ to the unclassified surface image. The unclassified surface is assigned to the same class as the decoded image. If the decoded image is ‘different’, then the unclassified surface image belongs to a different class and a PIFS constructed for
another surface is used. A detail description of the PR-HFW method is given below. 3.1. Surface data representation Surface data acquired by a measuring instrument (e.g. Talysurf, interferometric microscope, atomic force microscope) is represented by a 2D discrete image function z = f(x, y). This function assigns a surface height (encoded into a brightness value) z to a point (encoded into a pixel) located
Fig. 3. Decoded images obtained after the iterative application of the PIFSs constructed for the worn surface shown in Fig. 1a by the hybrid fractal-wavelet method using different smoothed images: (a) a scale 0 smoothed image; (b) a resized scale 1 smoothed image; (c) a resized scale 2 smoothed image. For the decoding of PIFSs, 12 iterations were used and the black image shown in Fig. 1b was used as an initial image.
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Fig. 4. A schematic diagram of a surface classification performed by the PR-HFW method.
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on surface at (x, y) ∈ Lx × Ly (Lx = {1, 2, . . . , M} and Ly = {1, 2, . . . , N}). M and N are the number of pixels in the x and y directions, respectively. 3.2. Dissimilarity measure A Baddeley’s distance (BD) is used to measure differences (dissimilarity) between decoded and unclassified images [21,22]. First, each image is transformed to a binary set of points in a 3D volume according to the following formula: 1 if f(x, y) = z F(x, y, z) = 0 otherwise Consequently, two binary sets are obtained, i.e. Fdec for the decoded image and Fun for the unclassified image, respectively. A BD is then calculated between these two binary sets, as 1/2 1 2 BD(Fun , Fdec ) = |dun (v) − ddec (v)| card(V) v∈V
where v = x, y, z is a voxel belonging to the volume V = Lx × Ly × Lz , card(V) is the total number of voxels in the volume V, dun (v) (ddec (v)) is the shortest distance between voxel v and the binary set Fun (Fdec ). BD takes a greater value as the differences between decoded and unclassified images increase. The calculation of dun (v) and ddec (v) distances is a lengthy process, especially for larger images. To reduce the computational time required to obtain these distances a 2D discrete chamfer distance operator d3–5 was applied [23]. This yields distances which are close approximates of dun (v) and ddec (v). The BD has been chosen in this study because it provides differences between images in both amplitude and spatial domains and it exhibits a more linear behaviour in the presence of gray-level variations, spatial shifts and shape distortions; than other dissimilarity measures [22].
the HFW method. Next, all PIFSs constructed are decoded using their own images as initial images and BDs between the decoded and initial images are calculated. Finally, the average values of BDs obtained for unclassified (BDunclass ) and classified (BDunclass ) surface images are calculated. 3.3.3. Step 3 An unclassified surface image is decomposed into first, second and higher scale surface images by the SWT method. Each scale image is enlarged to the size of the original image using interpolation method. The scale 1 unclassified image is used as an initial image in the decoding process of PIFSs. During this process, the scale 1 PIFSs taken from the database are decoded. The scale 2 unclassified image is then used as the initial image and the scale 2 PIFSs taken from the database are decoded. The process is repeated until the highest scale is achieved. Only one iteration is allowed in the decoding processes. 3.3.4. Step 4 BDs are then calculated between images obtained after the decoding of PIFSs and the unclassified surface image at each scale. 3.3.5. Step 5 Next, a k nearest-neighbour classifier is applied to BDs calculated [24,25]. This is done in three steps: • A set of decoded surface images nearest to the unclassified surface image are first found by identifying the k smallest or largest BDs. If BDunclass is higher than BDclass then the smallest BDs are used. Otherwise, the largest BDs are used (k is a chosen odd number, e.g. 3, 5 or 7). • A majority “vote” rule is then applied to the set of decoded images. This yields an unclassified surface assigned to a class of the decoded image which has the highest number of “votes”. • If two or more decoded images have an equal highest “vote” number, then a conflict arises. For each decoded
3.3. Classification algorithm The classification of surfaces involves the following steps. 3.3.1. Step 1 First, surface images are assigned into classes according to a specific criterion, e.g. wear mechanism involved in their formation, surface texture, etc. For each surface image, a set of PIFSs is then constructed using the HFW method, i.e. PIFSs constructed for first, second and higher scale average images are obtained. This yields a multiscale database consisting of PIFSs. 3.3.2. Step 2 Subimages are randomly chosen from both the classified surface images and an unclassified (to be classified) surface image. PIFSs are then constructed for all subimages using
Fig. 5. Classification error rates calculated for isotropic fractal surfaces.
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image involved in the conflict, an average value of BDs is calculated. An unclassified surface is assigned to the class of the decoded image associated with the minimum or maximum average value, respectively. A schematic diagram of the classification algorithm is shown in Fig. 4. 4. Classification of surfaces Computer generated images of four fractal surfaces and microscopic images of 10 tribological surfaces were used
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in this study. All images were 512 × 512 pixels wide with 256 gray levels. The computer images were generated by an inverse Fourier transform technique [16], while the microscopic images were acquired with a SEM in secondary electron mode. Each image was normalized by its gray-level mean and standard deviation values. This reduces variations in pixel brightness unrelated to image texture. The mean and standard deviation were set to 128 and 40, respectively. To obtain a large database, 30 different non-overlapping 64×64 pixel subimages were randomly extracted from each 512 × 512 pixel original image. Thus, the total number of subimages used was 420. Each subimage was decomposed
Fig. 6. SEM images of tribological surfaces used in a classification test: (a, b) sandblasted Co–Cr–Mo and Ti; (c, d) PSZ ceramic and Ni–Cr–Mo steel after a sliding wear; (e, f) white cast iron abraded by silica sands and glass beads, respectively [18].
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into three smoothed images using SWT. A nearest-neighbour classifier with k equal to 7 was used. The performance of the PR-HFW method was evaluated using a leave-one-out cross-validation technique. For this purpose, a database containing images of classified surfaces and their PIFSs was used. First, a surface image and its PIFS were removed from the database. This surface image was considered as an image to be classified. Then, the PR-HFW method was used to classify it. For this classification, all PIFSs remaining in the database were decoded using the unclassified surface image as an initial image. Once the classification was completed, the surface image and its PIFS were returned to the database. Next, another surface image and its PIFS were removed from the database. This surface image was considered as an image to be classified. The PR-HFW method was used to classify it in the same manner as the first surface. Once the classification was completed, the surface image and its PIFS were returned to the database. This whole process was repeated for each individual image and its PIFS. Finally, a classification error rate was calculated as N
ec =
c 1 τi Nc
i=1
where Nc is the total number of surface images classified by the PR-HFW method and τ i the variable which takes the
value 0 if the surface image is classified correctly and 1 otherwise. This error rate indicates the percentage of surfaces that were incorrectly classified. Comparison studies between the PR-HFW method and the PR-PIFS method were also conducted. 4.1. Fractal surfaces Images of isotropic fractal surfaces with fractal dimensions D of 2.1, 2.3, 2.5 and 2.7 were used. A multiscale image database containing 120 subimages assigned to four different classes was obtained: Class 1 contained 30 subimages obtained from the fractal surface of dimension 2.1; class 2 contained 30 subimages obtained from the fractal surface of dimension 2.3; and so on. The PR-HFW method was first used to classify class 1 images. Once this classification was completed, class 2, 3 and 4 images were then classified (sequentially). The leave-one-out cross-validation technique was used to evaluate the accuracy of these classifications. Classification results for the isotropic fractal surfaces are shown in Fig. 5. As can be seen from this figure the PR-HFW method produced 0.4–2.2% lower classification error rates than the PR-PIFS method. These results indicate that the PR-HFW method provides an improved classification for fractal surfaces compared to the PR-PIFS method.
Fig. 7. SEM images of polished mild steel surfaces used in a classification test: (a) unworn surface; (b–d) slightly, moderately and severely worn surfaces after three-body abrasive wear, respectively.
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Fig. 8. Classification error rates calculated for tribological surfaces shown in Fig. 6: (1, 2) sandblasted Co–Cr–Mo and Ti; (3, 4) PSZ ceramic and Ni–Cr–Mo steel after a sliding wear; (5, 6) white cast iron abraded by silica sands and glass beads.
4.2. Tribological surfaces The PR-HFW method was also used to classify tribological surfaces. This includes various surfaces exhibiting different textures (first group) taken from the previously published work [6] and polished mild steel (0.2% C, 130 HV) surfaces subject to a three-body abrasive wear using 10 wt.% quartz slurry with particles varying in size from 250 to 300 m (second group). These surfaces were obtained from wear tests described in [26]. First group surfaces are shown in Fig. 6; two sandblasted surfaces made of Cr–Co–Mo (Fig. 6a) and Ti (Fig. 6b), a PSZ ceramic surface obtained by dry sliding at elevated temperature (Fig. 6c), a Ni–Cr–Mo steel surface obtained by sliding under boundary lubrication conditions (Fig. 6d), and white cast iron surfaces abraded by silica sands (Fig. 6e) and glass beads (Fig. 6f). Second group surfaces are shown in Fig. 7; unworn (Fig. 7a), slightly (Fig. 7b), moderately (Fig. 7c) and severely (Fig. 7d) worn polished mild steel surfaces. This figure shows the progress of damage made by abrasive wear to an unworn polished surface (Fig. 7a). The slightly worn surface exhibits a small number of indents and scratches and large areas of unworn polished mild steel (Fig. 7b). The moderately worn surface exhibits a number of indents and scratches and small areas of unworn polished mild steel (Fig. 7c), while the severely worn surface exhibits a mixture of indents and scratches and no areas of unworn polished mild steel (Fig. 7d). Classification results obtained for these surfaces are shown in Figs. 8 and 9. Fig. 8 shows that the PR-PIFS and PR-HFW methods produced same classification error rates for the white cast iron surface abraded by silica sands and the sandblasted Ti surface, while the PR-HFW method produced 0.8 and 0.7% lower classification error rates for the remaining surfaces. Fig. 9 shows that the PR-HFW method
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Fig. 9. Classification error rates calculated for polished mild steel surfaces shown in Fig. 7: (1) unworn surface; (2) slightly; (3) moderately; (4) severely worn surfaces.
produced 1.3 and 0.3% lower classification error rates for the polished steel surfaces than the PR-PIFS method, except for the severely worn polished surface shown in Fig. 6c for which they were equal. These results indicate that the PR-HFW method both accurately classified the tribological surfaces used in this study and gave better classification results than the PR-PIFS method.
5. Conclusions From the work conducted the following conclusions can be drawn: • A pattern recognition by hybrid fractal-wavelet (PR-HFW) method has been developed and then used to classify computer generated images of fractal surfaces and SEM images of tribological surfaces. This method is based on a hybrid fractal-wavelet method and a pattern recognition by partition iterated function system (PR-PIFS) method previously developed by the authors. • The PR-HFW method allows a multiscale surface classification. Initially, the wavelet transform was used to decompose the image to be classified into first, second and higher scale surface images. These scale images were then used as initial images to decode a multiscale reference database, that contains PIFSs constructed for classified scale surfaces. As a result, decoded images were obtained at each scale. Next, Baddeley’s distances were calculated between decoded and initial images. Finally, a k nearest-neighbour classifier was applied to these distances and the image was classified. • Results obtained showed that the PR-HFW method gives significantly better classification results as compared to the PR-PIFS method previously developed by the authors.
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These results showed that the PR-HFW has low classification error rates, i.e. <6% for the fractal surfaces and <4% for tribological surfaces tested. • Future work will focus on the development of an automated pattern recognition system for tribological surfaces, based on the PR-HFW method, that can be employed in condition monitoring and failure analysis. Acknowledgements The authors would like to thank Australian Research Council for sponsoring this project and the School of Mechanical Engineering, University of Western Australia for its support during the writing of this paper. References [1] U. Cho, J.A. Tichy, Quantitative correlation of wear debris morphology: grouping and classification, Tribol. Int. 33 (2000) 461– 467. [2] X. Kun, A.R. Luxmoore, F. Deravi, Comparison of shape features for the classification of wear particles, Eng. Appl. Artif. Intell. 10 (1997) 485–493. [3] N.K. Myshkin, O.K. Kwon, A.Ya. Grigoriev, H.-S. Ahn, H. Kong, Classification of wear debris using a neural network, Wear 203–204 (1997) 658–662. [4] B.J. Roylance, I.A. Albidewi, A.L. Price, A.R. Luxmoore, The development of a computer-aided systematic particle analysis procedure—CASPA, Lubr. Eng. 48 (1992) 940–946. [5] P. Podsiadlo, G.W. Stachowiak, Scale-invariant analysis of tribological surfaces, Thinning Films And Tribological Interfaces, in: D. Dowson, et al. (Eds.), Tribology Series 38, Elsevier, Amsterdam, 2000, pp. 546–557. [6] P. Podsiadlo, G.W. Stachowiak, Rotationally- and scale-invariant pattern recognition of tribological surfaces, From Model Experiment To Industrial Problem, in: D. Dowson, et al. (Eds.), Tribology Series 39, Elsevier, Amsterdam, 2001, pp. 697–708. [7] G.W. Stachowiak, P. Podsiadlo, Characterization and classification of wear particles and surfaces, Wear 249 (2001) 194–200. [8] D.-M. Tsai, C.-F. Tseng, Surface roughness classification for castings, Pattern Recognit. 32 (1999) 389–405.
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Fractal-wavelet based classification of tribological surfaces P. Podsiadlo∗ , G.W. Stachowiak Tribology Laboratory, School of Mechanical Engineering, The University of Western Australia, Crawley, WA 6009, Australia
Abstract Classification of the topography of freshly machined, worn and damaged surfaces (e.g. damaged by adhesion, scoring, abrasion, pitting) is still a problem in machine failure analysis. Tribological surfaces often exhibit both a multiscale nature (i.e. different length scales of surface features) and a non-stationary nature (i.e. features which are superimposed on each other and located at different positions on a surface). The most widely used approaches to surface classification are based on the Fourier transform or statistical functions and parameters. Often these approaches are inadequate and provide incorrect classification of the tribological surfaces. The main reason is that these techniques fail to simultaneously capture the multiscale nature and the non-stationary nature of the surface data. A new method, called a hybrid fractal-wavelet method, has recently been developed for the characterization of tribological surfaces in a multiscale and non-stationary manner. In contrast to other methods, this method combines both the wavelets’ inherent ability to characterize surfaces at each individual scale and the fractals’ inherent ability to characterize surfaces in a scale-invariant manner. The application of this method to the classification of artificially generated fractal and tribological surfaces (e.g. worn surfaces) is presented in this paper. The newly developed method has been further modified to better suit tribological surface data, including a new measure of differences between initial and decoded images. The accuracy of this method in the classification of surfaces was assessed. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Fractal-wavelet analysis; Surface classification; Tribological surfaces
1. Introduction Classification of tribological surfaces and wear particles has recently been used for the studies of wear, friction and lubrication mechanisms occurring between interacting surfaces (e.g. [1–7]). Given an unclassified surface (a wear particle), its classification is either: • supervised, i.e. a classification algorithm is used to assign an unclassified surface (particle) into a specific class which is predefined (prior knowledge) according to criteria such as wear mechanism, boundary shape, surface texture, experiment conditions, etc. [5–9], or • unsupervised, i.e. a clustering algorithm is first used to find groups in a training set of unclassified surfaces (particles) in such a way that similar surfaces (particles) are clustered in the same group, while different surfaces (particles) are clustered in separate groups. For the clustering, morphological parameters such as area, aspect ratio, roundness, surface roughness, are usually used [1]. These groups are then used to produce classes of surfaces (posterior knowledge). Next, a classification algorithm is
∗ Corresponding author. E-mail address: [email protected] (P. Podsiadlo).
used to assign an unclassified surface (particle) into one of these classes [1]. One difficulty in classifying tribological surfaces is that the surface classification depends on the chosen scale. Because the topography of tribological surfaces often exhibit a non-stationary and multiscale nature, there exists no optimal “global” scale, i.e. the scale that gives the best classification for all surfaces. Consequently, a “local” scale must be individually chosen for each surface. For this purpose, classification methods based on multiscale analyses can be used, including ring-shaped filters [8], windowed Fourier filters [10], Gabor filters [11,12], 2D pseudo-Wigner transform [13] and wavelet transform [14,15]. These methods are used to decompose surface data into different frequency components (scale images) which can be individually analysed. Other methods used to overcome the difficulty in classifying surfaces are based on fractals. Fractal methods provide scale-invariant surface descriptors, e.g. fractal dimension and topothesy [16,17] or mathematical models containing information about surface data across all possible scales [5,6]. Subsequently, the fractal-based classification is scale-invariant, i.e. it does not depend on the chosen scale. Recently, a new method, called a hybrid fractal-wavelet method, has been developed for the analysis of surface topography in a multiscale manner [18]. This hybrid method
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combines wavelets and fractals in such a way that a wavelet method (a symmetric wavelet transform (SWT)) is used to decompose surface topography into ranges of different scales, while a fractal method (a partition iterated function system (PIFS)) is used to build a mathematical model for each range of scales. From initial studies conducted [18], it appears that this hybrid method would be ideal for surface classification. This is because it provides the best of two worlds, i.e. the wavelets’ inherent ability to decompose surface data into individual scales and the fractals’ inherent ability to describe surface data over many scales in a scale-invariant manner. In this paper, a new pattern recognition method, called the pattern recognition by hybrid fractal-wavelet (PR-HFW) method, has been developed and applied to computer generated images of fractal surfaces and scanning electron microscope (SEM) images of tribological surfaces. Images of fractal surfaces (that were isotropic with increasing fractal dimension) were used to evaluate the classification accuracy of the PR-HFW method, for surface topographies exhibiting different complexity. The tribological surface images were gathered in two data sets. The first set contained surfaces taken from six different tribological systems. They were used to show that the PR-HFW method is sensitive enough to classify accurately and simultaneously, tribological surfaces that exhibit similar, moderately different and significantly different topographies. The second set contained unworn surfaces and worn surfaces that were subjected to different degrees of abrasive wear. These surfaces were used to study the suitability of the PR-HFW method for monitoring the progress of wear occurring in tribological systems. The newly developed method is similar to a pattern recognition by PIFS (PR-PIFS) method [5–7]. Both methods are used to encode classified surface images into PIFSs. The PIFSs are decoded using an unclassified surface image and decoded images are obtained. The dissimilarity between each decoded image and the unclassified image is a criterion used in the surface classification. These two methods do not need feature vectors of parameters to classify surfaces and they are scale- and rotationally-invariant. The difference is that the PR-HFW method is used to classify surfaces at specific scale ranges, i.e. at scales for which differences measured between decoded and initial images take minimum values, while the PR-PIFS is used to classify surfaces at all scales. Thus, it is expected that the PR-HFW method will classify surfaces with a higher accuracy than the PR-PIFS method. This has been supported by results obtained in this study.
2. Hybrid fractal-wavelet (HFW) method The HFW method is a combination of two methods, i.e. a PIFS method [19] and a SWT method [20], used to analyse surface data both in a scale-invariant manner (similar to fractal methods) and at different scales (similar to wavelet methods).
The PIFS method is used to construct a mathematical model of a surface topography. This construction is based on ‘piecewise self-transformability’, i.e. that surface image data contain large parts which can be converted into small parts located elsewhere on the image. Each conversion is performed using an affine contractive transformation. This yields a collection of affine transformations called a PIFS. During the conversion, a geometric transformation is used to average brightness values of pixels within non-overlapping 2 × 2 pixel squares tiling the original image data and then to assign the average values to pixels of an image. As a result, an average value image equal to the half size of the original image is obtained. This image is then used for the conversion between large and small image parts. The mathematical foundation of the PIFS lies in the piecewise contractive mapping fixed-point theorem. According to this theorem, the PIFS is contractive and defines a unique ‘fixed-image’ which is the original image. This means that if the PIFS is iteratively applied to any initial image, a sequence of images eventually converging to the attractor, i.e. to the original image, is obtained. Since the attractor is unique, it is entirely specified by the PIFS. In other words, the PIFS completely describes the image, but in a numerical form rather than in a pictorial form (image) and the image can be reconstructed from the PIFS if required. The image reconstructed is not exactly same as the original image since the ‘piecewise self-transformability’ is approximately satisfied for tribological surfaces. As an example, a PIFS was constructed for the worn surface image shown in Fig. 1a. Decoded images obtained after the iterative application of the PIFS constructed to a black image (Fig. 1b) are shown in Fig. 1c–e. A two-channel filter bank is used as the wavelet transform in the SWT method to decompose surface image data into different frequency components (subimages) represented as smoothed and detail images [20]. The HFW method is performed in the following steps [18]: 1. First, the SWT method is applied to the original image data and as a result smoothed images and three detail images (in vertical, horizontal and diagonal directions) are obtained at different scales. 2. First and higher scale smoothed images obtained are resized to half of the original image size using interpolation method. These resized images together with the smoothed image obtained at scale 0, create a set of new images. 3. A PIFS is then constructed for the original image data using a scale 0 smoothed image as an average image. This yields a first PIFS. Next, a second PIFS is constructed using a resized scale 1 smoothed image as an average image. This process is repeated until all smoothed images are used. Consequently, a set of PIFSs constructed for the same image data using different smoothed images as average images, are obtained (Fig. 2).
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Fig. 1. Decoded images obtained after the iterative application of the PIFS, constructed for the worn surface shown in (a), into a black initial image shown in (b); (c) 1 iteration; (d) 4 iterations; (e) 12 iterations, respectively.
It was shown that the HFW method is more accurate than the PIFS method previously developed by the authors [18]. This is because the hybrid method uses a scale 0 smoothed image containing surface details which are either not present or are significantly distorted in an average value image used in the PIFS method. As an example, the HFW method was applied to the worn surface shown in Fig. 1a. Three smoothed images were generated by the SWT method and PIFSs were constructed using
these smoothed images as average images. Decoded images obtained for each PIFS constructed are shown in Fig. 3.
3. Pattern recognition by hybrid fractal-wavelet (PR-HFW) method A pattern recognition system based on the HFW method, called the PR-HFW method, has been developed for the
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Fig. 2. A schematic diagram of PIFSs constructed for a surface image at different scales using a hybrid fractal-wavelet method [6].
classification of tribological surfaces. The basic idea behind this method is that PIFSs constructed for classified surfaces are decoded using an unclassified surface image as the initial image. If the unclassified surface exhibits a similar morphology to the already classified surface, then the decoded image obtained just after one iteration will be ‘similar’ to the unclassified surface image. The unclassified surface is assigned to the same class as the decoded image. If the decoded image is ‘different’, then the unclassified surface image belongs to a different class and a PIFS constructed for
another surface is used. A detail description of the PR-HFW method is given below. 3.1. Surface data representation Surface data acquired by a measuring instrument (e.g. Talysurf, interferometric microscope, atomic force microscope) is represented by a 2D discrete image function z = f(x, y). This function assigns a surface height (encoded into a brightness value) z to a point (encoded into a pixel) located
Fig. 3. Decoded images obtained after the iterative application of the PIFSs constructed for the worn surface shown in Fig. 1a by the hybrid fractal-wavelet method using different smoothed images: (a) a scale 0 smoothed image; (b) a resized scale 1 smoothed image; (c) a resized scale 2 smoothed image. For the decoding of PIFSs, 12 iterations were used and the black image shown in Fig. 1b was used as an initial image.
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Fig. 4. A schematic diagram of a surface classification performed by the PR-HFW method.
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on surface at (x, y) ∈ Lx × Ly (Lx = {1, 2, . . . , M} and Ly = {1, 2, . . . , N}). M and N are the number of pixels in the x and y directions, respectively. 3.2. Dissimilarity measure A Baddeley’s distance (BD) is used to measure differences (dissimilarity) between decoded and unclassified images [21,22]. First, each image is transformed to a binary set of points in a 3D volume according to the following formula: 1 if f(x, y) = z F(x, y, z) = 0 otherwise Consequently, two binary sets are obtained, i.e. Fdec for the decoded image and Fun for the unclassified image, respectively. A BD is then calculated between these two binary sets, as 1/2 1 2 BD(Fun , Fdec ) = |dun (v) − ddec (v)| card(V) v∈V
where v = x, y, z is a voxel belonging to the volume V = Lx × Ly × Lz , card(V) is the total number of voxels in the volume V, dun (v) (ddec (v)) is the shortest distance between voxel v and the binary set Fun (Fdec ). BD takes a greater value as the differences between decoded and unclassified images increase. The calculation of dun (v) and ddec (v) distances is a lengthy process, especially for larger images. To reduce the computational time required to obtain these distances a 2D discrete chamfer distance operator d3–5 was applied [23]. This yields distances which are close approximates of dun (v) and ddec (v). The BD has been chosen in this study because it provides differences between images in both amplitude and spatial domains and it exhibits a more linear behaviour in the presence of gray-level variations, spatial shifts and shape distortions; than other dissimilarity measures [22].
the HFW method. Next, all PIFSs constructed are decoded using their own images as initial images and BDs between the decoded and initial images are calculated. Finally, the average values of BDs obtained for unclassified (BDunclass ) and classified (BDunclass ) surface images are calculated. 3.3.3. Step 3 An unclassified surface image is decomposed into first, second and higher scale surface images by the SWT method. Each scale image is enlarged to the size of the original image using interpolation method. The scale 1 unclassified image is used as an initial image in the decoding process of PIFSs. During this process, the scale 1 PIFSs taken from the database are decoded. The scale 2 unclassified image is then used as the initial image and the scale 2 PIFSs taken from the database are decoded. The process is repeated until the highest scale is achieved. Only one iteration is allowed in the decoding processes. 3.3.4. Step 4 BDs are then calculated between images obtained after the decoding of PIFSs and the unclassified surface image at each scale. 3.3.5. Step 5 Next, a k nearest-neighbour classifier is applied to BDs calculated [24,25]. This is done in three steps: • A set of decoded surface images nearest to the unclassified surface image are first found by identifying the k smallest or largest BDs. If BDunclass is higher than BDclass then the smallest BDs are used. Otherwise, the largest BDs are used (k is a chosen odd number, e.g. 3, 5 or 7). • A majority “vote” rule is then applied to the set of decoded images. This yields an unclassified surface assigned to a class of the decoded image which has the highest number of “votes”. • If two or more decoded images have an equal highest “vote” number, then a conflict arises. For each decoded
3.3. Classification algorithm The classification of surfaces involves the following steps. 3.3.1. Step 1 First, surface images are assigned into classes according to a specific criterion, e.g. wear mechanism involved in their formation, surface texture, etc. For each surface image, a set of PIFSs is then constructed using the HFW method, i.e. PIFSs constructed for first, second and higher scale average images are obtained. This yields a multiscale database consisting of PIFSs. 3.3.2. Step 2 Subimages are randomly chosen from both the classified surface images and an unclassified (to be classified) surface image. PIFSs are then constructed for all subimages using
Fig. 5. Classification error rates calculated for isotropic fractal surfaces.
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image involved in the conflict, an average value of BDs is calculated. An unclassified surface is assigned to the class of the decoded image associated with the minimum or maximum average value, respectively. A schematic diagram of the classification algorithm is shown in Fig. 4. 4. Classification of surfaces Computer generated images of four fractal surfaces and microscopic images of 10 tribological surfaces were used
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in this study. All images were 512 × 512 pixels wide with 256 gray levels. The computer images were generated by an inverse Fourier transform technique [16], while the microscopic images were acquired with a SEM in secondary electron mode. Each image was normalized by its gray-level mean and standard deviation values. This reduces variations in pixel brightness unrelated to image texture. The mean and standard deviation were set to 128 and 40, respectively. To obtain a large database, 30 different non-overlapping 64×64 pixel subimages were randomly extracted from each 512 × 512 pixel original image. Thus, the total number of subimages used was 420. Each subimage was decomposed
Fig. 6. SEM images of tribological surfaces used in a classification test: (a, b) sandblasted Co–Cr–Mo and Ti; (c, d) PSZ ceramic and Ni–Cr–Mo steel after a sliding wear; (e, f) white cast iron abraded by silica sands and glass beads, respectively [18].
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into three smoothed images using SWT. A nearest-neighbour classifier with k equal to 7 was used. The performance of the PR-HFW method was evaluated using a leave-one-out cross-validation technique. For this purpose, a database containing images of classified surfaces and their PIFSs was used. First, a surface image and its PIFS were removed from the database. This surface image was considered as an image to be classified. Then, the PR-HFW method was used to classify it. For this classification, all PIFSs remaining in the database were decoded using the unclassified surface image as an initial image. Once the classification was completed, the surface image and its PIFS were returned to the database. Next, another surface image and its PIFS were removed from the database. This surface image was considered as an image to be classified. The PR-HFW method was used to classify it in the same manner as the first surface. Once the classification was completed, the surface image and its PIFS were returned to the database. This whole process was repeated for each individual image and its PIFS. Finally, a classification error rate was calculated as N
ec =
c 1 τi Nc
i=1
where Nc is the total number of surface images classified by the PR-HFW method and τ i the variable which takes the
value 0 if the surface image is classified correctly and 1 otherwise. This error rate indicates the percentage of surfaces that were incorrectly classified. Comparison studies between the PR-HFW method and the PR-PIFS method were also conducted. 4.1. Fractal surfaces Images of isotropic fractal surfaces with fractal dimensions D of 2.1, 2.3, 2.5 and 2.7 were used. A multiscale image database containing 120 subimages assigned to four different classes was obtained: Class 1 contained 30 subimages obtained from the fractal surface of dimension 2.1; class 2 contained 30 subimages obtained from the fractal surface of dimension 2.3; and so on. The PR-HFW method was first used to classify class 1 images. Once this classification was completed, class 2, 3 and 4 images were then classified (sequentially). The leave-one-out cross-validation technique was used to evaluate the accuracy of these classifications. Classification results for the isotropic fractal surfaces are shown in Fig. 5. As can be seen from this figure the PR-HFW method produced 0.4–2.2% lower classification error rates than the PR-PIFS method. These results indicate that the PR-HFW method provides an improved classification for fractal surfaces compared to the PR-PIFS method.
Fig. 7. SEM images of polished mild steel surfaces used in a classification test: (a) unworn surface; (b–d) slightly, moderately and severely worn surfaces after three-body abrasive wear, respectively.
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Fig. 8. Classification error rates calculated for tribological surfaces shown in Fig. 6: (1, 2) sandblasted Co–Cr–Mo and Ti; (3, 4) PSZ ceramic and Ni–Cr–Mo steel after a sliding wear; (5, 6) white cast iron abraded by silica sands and glass beads.
4.2. Tribological surfaces The PR-HFW method was also used to classify tribological surfaces. This includes various surfaces exhibiting different textures (first group) taken from the previously published work [6] and polished mild steel (0.2% C, 130 HV) surfaces subject to a three-body abrasive wear using 10 wt.% quartz slurry with particles varying in size from 250 to 300 m (second group). These surfaces were obtained from wear tests described in [26]. First group surfaces are shown in Fig. 6; two sandblasted surfaces made of Cr–Co–Mo (Fig. 6a) and Ti (Fig. 6b), a PSZ ceramic surface obtained by dry sliding at elevated temperature (Fig. 6c), a Ni–Cr–Mo steel surface obtained by sliding under boundary lubrication conditions (Fig. 6d), and white cast iron surfaces abraded by silica sands (Fig. 6e) and glass beads (Fig. 6f). Second group surfaces are shown in Fig. 7; unworn (Fig. 7a), slightly (Fig. 7b), moderately (Fig. 7c) and severely (Fig. 7d) worn polished mild steel surfaces. This figure shows the progress of damage made by abrasive wear to an unworn polished surface (Fig. 7a). The slightly worn surface exhibits a small number of indents and scratches and large areas of unworn polished mild steel (Fig. 7b). The moderately worn surface exhibits a number of indents and scratches and small areas of unworn polished mild steel (Fig. 7c), while the severely worn surface exhibits a mixture of indents and scratches and no areas of unworn polished mild steel (Fig. 7d). Classification results obtained for these surfaces are shown in Figs. 8 and 9. Fig. 8 shows that the PR-PIFS and PR-HFW methods produced same classification error rates for the white cast iron surface abraded by silica sands and the sandblasted Ti surface, while the PR-HFW method produced 0.8 and 0.7% lower classification error rates for the remaining surfaces. Fig. 9 shows that the PR-HFW method
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Fig. 9. Classification error rates calculated for polished mild steel surfaces shown in Fig. 7: (1) unworn surface; (2) slightly; (3) moderately; (4) severely worn surfaces.
produced 1.3 and 0.3% lower classification error rates for the polished steel surfaces than the PR-PIFS method, except for the severely worn polished surface shown in Fig. 6c for which they were equal. These results indicate that the PR-HFW method both accurately classified the tribological surfaces used in this study and gave better classification results than the PR-PIFS method.
5. Conclusions From the work conducted the following conclusions can be drawn: • A pattern recognition by hybrid fractal-wavelet (PR-HFW) method has been developed and then used to classify computer generated images of fractal surfaces and SEM images of tribological surfaces. This method is based on a hybrid fractal-wavelet method and a pattern recognition by partition iterated function system (PR-PIFS) method previously developed by the authors. • The PR-HFW method allows a multiscale surface classification. Initially, the wavelet transform was used to decompose the image to be classified into first, second and higher scale surface images. These scale images were then used as initial images to decode a multiscale reference database, that contains PIFSs constructed for classified scale surfaces. As a result, decoded images were obtained at each scale. Next, Baddeley’s distances were calculated between decoded and initial images. Finally, a k nearest-neighbour classifier was applied to these distances and the image was classified. • Results obtained showed that the PR-HFW method gives significantly better classification results as compared to the PR-PIFS method previously developed by the authors.
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These results showed that the PR-HFW has low classification error rates, i.e. <6% for the fractal surfaces and <4% for tribological surfaces tested. • Future work will focus on the development of an automated pattern recognition system for tribological surfaces, based on the PR-HFW method, that can be employed in condition monitoring and failure analysis. Acknowledgements The authors would like to thank Australian Research Council for sponsoring this project and the School of Mechanical Engineering, University of Western Australia for its support during the writing of this paper. References [1] U. Cho, J.A. Tichy, Quantitative correlation of wear debris morphology: grouping and classification, Tribol. Int. 33 (2000) 461– 467. [2] X. Kun, A.R. Luxmoore, F. Deravi, Comparison of shape features for the classification of wear particles, Eng. Appl. Artif. Intell. 10 (1997) 485–493. [3] N.K. Myshkin, O.K. Kwon, A.Ya. Grigoriev, H.-S. Ahn, H. Kong, Classification of wear debris using a neural network, Wear 203–204 (1997) 658–662. [4] B.J. Roylance, I.A. Albidewi, A.L. Price, A.R. Luxmoore, The development of a computer-aided systematic particle analysis procedure—CASPA, Lubr. Eng. 48 (1992) 940–946. [5] P. Podsiadlo, G.W. Stachowiak, Scale-invariant analysis of tribological surfaces, Thinning Films And Tribological Interfaces, in: D. Dowson, et al. (Eds.), Tribology Series 38, Elsevier, Amsterdam, 2000, pp. 546–557. [6] P. Podsiadlo, G.W. Stachowiak, Rotationally- and scale-invariant pattern recognition of tribological surfaces, From Model Experiment To Industrial Problem, in: D. Dowson, et al. (Eds.), Tribology Series 39, Elsevier, Amsterdam, 2001, pp. 697–708. [7] G.W. Stachowiak, P. Podsiadlo, Characterization and classification of wear particles and surfaces, Wear 249 (2001) 194–200. [8] D.-M. Tsai, C.-F. Tseng, Surface roughness classification for castings, Pattern Recognit. 32 (1999) 389–405.
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