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Use of descriptors based on moments from digital images for tool wear monitoring
ARTICLE IN PRESS
International Journal of Machine Tools & Manufacture 48 (2008) 1005–1013 www.elsevier.com/locate/ijmactool
Use of descriptors based on moments from digital images for tool wear monitoring J. Barreiroa,, M. Castejo´na, E. Alegrea, L.K. Herna´ndezb a
Escuela de Ingenierı´as Industrial e Informa´tica, Universidad de Leo´n, 24071 Leo´n, Spain Dpt. de Ingenierı´a Meca´nica, Industrial y Mecatro´nica, Universidad de Pamplona, Colombia
b
Received 23 July 2007; received in revised form 4 January 2008; accepted 4 January 2008 Available online 26 January 2008
Abstract The widely used criteria to determine the need for tool replacement are too much conservative. The consequence is that tools are only used for a small fraction of their possible useful life. The economical influence of tool replacement costs over total production costs demands better criteria according to current technology. The use of different moments to describe tool wear images and to classify the tool condition in wear classes has been studied. The moments used as descriptors in this paper show different behavior with regard to wear identification, concluding that Hu and Legendre descriptors provide the best performance. These descriptors have been classified using a finite mixture MCLUST model considering three wear classes (low, medium and high). The achieved results from the clustering have been checked by means of discriminant analysis techniques, linear and quadratic, using the Fowlkes–Mallow index as quality factor. The projection of image data by means of linear discriminant analysis provides useful wear maps for tool monitoring. These wear maps show us the wear classes and the frontiers among them, in such a way that wear evolution for current tool can be mapped. The quadratic discriminant analysis allows us to assign to the current tool, a probability of belonging to a wear class. This probability is used as a new wear criterion in substitution of the current conservative criteria, making possible to reduce tool replacement costs. r 2008 Elsevier Ltd. All rights reserved. Keywords: Cutting tool; Wear; Image analysis; Cluster analysis
1. Introduction Different authors [1–3] have already emphasized the impact that tool replacement operations have on costs in the metallic parts manufacturing. Costs linked to tool replacement operations comprise [4] not only the costs of cutting tools but also, and much more important, the indirect costs linked to unproductive time needed to perform the replacement. This is an issue of increasing importance for those factories which operate under high productivity constrains; hence the recent efforts to avoid superfluous replacement operations. The commonly followed criteria to identify the need of tool replacement usually consist of early stopping rules that reject the tools irrespective of their actual wear condition. In manual processes, the machinist schedules the replaceCorresponding author. Tel.: +34 987 291792; fax: +34 987 291930.
E-mail address: [email protected] (J. Barreiro). 0890-6955/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2008.01.005
ments on the basis of his own, and obviously subjective, experience. In automated lines, replacements are scheduled according to variables related to the production process, e.g. cutting time or number of parts produced, under the assumption that once a predefined limit is reached, the tool may be incapable of producing acceptable parts. Obviously, the aforementioned criteria neither ensure the optimal usage of tools nor lead to minimal replacement costs, causing in some applications important economic losses. Alternatively, a replacement policy based on tool wear measurement offers considerable benefits in costs. Apart from the derived economic implications, the International Organization for Standardization (ISO) has devoted several standards [5–7] to cutting inserts and their life. ISO’s approach already represents an important advance with regard to common practice: it establishes wear limits directly measurable on the wear land. However, this approach has its own limitations because the defined wear limits are too conservative [8] and the required
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complex measurements are too difficult to obtain without stopping the process. The former limitation demands more refined criteria, the latter clearly opens a window of opportunity for alternative measurement methods. Two main categories can be distinguished among the different approaches proposed: indirect and direct measurement methods. Indirect measurement methods provide wear estimates by monitoring observable variables related to wear, e.g. cutting forces, temperatures and vibrations [9–20]; thus avoiding unwanted machining process interruptions. In industrial environments, however, the all-pervasive noise signals severely undermine the quality of these estimates [21–24]. On the other hand, direct methods, usually based on computer vision systems, are definitely better in terms of accuracy and reliability. Nevertheless, in spite of their advantages, direct methods have not found their place in industrial practices yet, mainly due to the high cost and the poor features displayed by the early hardware available. In the last years we have witnessed a decrease of prices and the enhancement of features, so the revival of direct methods. Nowadays, computer vision can be considered a mature technology and its application to cutting tool monitoring offers a wealth of interesting possibilities [3,21,24–28]. Consequently, a brief review of the state of the art reveals a significant number of works published during the last decade on the use of digital images in wear measurement [13,14,22,24,25,28–33]. Our approach, described in the following sections, takes advantage of the information provided by a computer vision system to define a set of replacement rules according to the current state of technology.
2. Materials and methods 2.1. Materials and machining process The tests were carried out in a CNC parallel lathe. Steel cylinders ANSI SAE 4340 and 4140 (quenched and normalized) were machined. The dimensions of the parts were 250 mm length and 90 mm diameter. The tools were coated tungsten carbide inserts (CNMG 120404MF235), rhombic, high tough and low wear resistance, in order to avoid premature tool fracture and, at the same time, to increase wear rate. The machining parameters were chosen with the same objective: different values between 140 and 200 m/min were used for cutting speed, whereas feed rate and cutting depth remained constant at 0.2 mm/rev and 2 mm, respectively. The coolant was 10% diluted, as it is indicated for medium hard metals. The tool was disassembled and inspected at the end of each machining pass. A fixture was specifically made to assure that every insert was located at the very same position; therefore, the image origin coordinates were the same and all of the acquired images were comparable. The light intensity and the focal distance were also kept constant.
2.2. Image acquisition and processing The images were acquired [34] using a PULLNIX PE2015 camera with 1=300 CCD. The camera was a 752ðHÞ 582ðVÞ pixels resolution and a signal–noise ratio of 50 dB. The digitalization was carried out with a MATROX METEOR II card. The size of the image acquired was 768 576 pixels. The images were automatically cropped to 400 200 pixels in order to clearly view the wear land. The optical system was composed of an 70XL OPTEM industrial zoom, with a 1X extension tube and 0.5X/0.75X/1.5X/2.0X OPTEM lens. Lighting system selection is a critical step during the design of any vision system, because it affects the quality of captured images. Best results are obtained with a uniform illumination without shines, so that the region of interest is fully illuminated and, at the same time, with the highly possible contrast with the background. In this case, to obtain a good image of the tool corner was difficult due to the tip radius. The lighting system used was composed of a FOSTEC DCRsIII regulated light source which provided an intense cold lighting. A NER SCDI-25-F0 SCDI diffuse lighting system was used to avoid shines. This system provides diffuse lighting in the direction of the camera axis. Lighting system location was carried out by means of a FOSTEC dual bundle. Image acquisition was executed with a specifically developed application. This application includes three modules: camera setup, sequence setup and image acquisition. These modules provide capturing device information, allow user to choose the resolution, define the information storage path and save the images [25]. A set of 1383 insert flank images was acquired using this system. Previous to segmentation, several operations were carried out to improve image appearance. A low pass filter was used to soften the background and to make the segmentation easier. The filtered image was cropped to remove the inferior half of the insert, because wear only takes place at the superior half. Later on, the histogram was stretched to improve contrast [25]. Region growing was used to segment the wear area. Starting points were selected on the basis of the results of a previous threshold. Once the thresholds were obtained, a binary image was generated with the worn region set to 1 and the rest of the image set to 0. Later on, a median filter was applied to reduce the noise. If the worn region is not closed, a morphological closing is carried out. Fig. 1 shows one of these binary images. B/W
2.3. Shape description using moments Moments are descriptors obtained from point coordinates in a region and their gray level. They are applied in several areas of pattern recognition and object classification. Teh and Chin [35] studied the characteristics of the moments and discussed the correlation among them. Choksuriwong et al. [36], Kim and Yuan [37] and Alegre et al. [38] compared the ability of orthogonal moments of
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h5 ¼ ðn30 3n12 Þðn30 þ n12 Þ½ðn30 þ n12 Þ2 3ðn21 þ n03 Þ2 þ ð3n21 n03 Þðn21 þ n03 Þ½3ðn30 þ n21 Þ ðn21 þ n03 Þ2 , (5) h6 ¼ ðn20 þ n02 Þ½ðn30 þ n12 Þ2 ðn21 þ n03 Þ2 þ 4n11 ðn30 þ n12 Þ,
(6)
h7 ¼ ð3n21 n03 Þðn30 þ n12 Þ½ðn30 þ n12 Þ2 3ðn21 þ n03 Þ2 þ ð3n12 n03 Þðn21 þ n03 Þ½3ðn30 þ n12 Þ ðn21 þ n03 Þ2 , (7)
Fig. 1. Image preprocessing.
Hu and Zernike to recognize objects and patterns in different applications. Park and Chang [39] compared the performance of orthogonal moments of Hu and Zernike with regard to the invariants of Hu, Taubin and Flusser, analyzing the computational cost and the ability to recognition. Five different families of statistical moments—Zernike, Legendre, Hu, Taubin and Flusser—and a number of region image descriptors were used to characterize the shape of the worn region in the binary images. The first 18 moments of Zernike (up to fourth order), nine moments of Legendre, seven moments of Hu, eight moments of Taubin and six moments of Flusser and Suck were calculated for each image. These descriptors were used for their wellknown efficiency in the description and recognition of different types of shapes [36]. On the other hand, the difficulties to predict the best descriptor for a particular shape [40] are evident. Consequently, the best descriptor in the clustering and recognition process is evaluated. 2.3.1. Hu invariants Moments and functions of moments are invaluable tools in the literature for the measurement of the properties of a distribution. In the field of image analysis, their use as image descriptors was pioneered by Hu [41] when he used the 2D geometric moments for characterizing the visual patterns in images. For different images, the respective sets of moments are unique and this makes them particularly useful for the task of pattern recognition. This is further added by the advantage of being able to construct moment invariants which are insensitive to rotation, scaling and translation. Hence, geometrical moments are effective descriptors for images under different perspectives. Hu derived seven invariant moments using nonlinear combinations of normalized central moments:
where mpq npq ¼ , m00 XX mpq ¼ ðx xÞp ðy yÞq f ðx; yÞ x
(8) (9)
y
and f ðx; yÞ is the gray level of the pixel at coordinates ðx; yÞ. 2.3.2. Zernike moments Though computationally very complex compared with geometric and Legendre moments, in general, Zernike moments are better in terms of their ability to feature representation, rotation invariance, fast computation, multi-level representation for describing the shapes of patterns and low noise sensitivity. Zernike moments use a set of Zernike polynomials that is complete and orthonormal in the interior of the unit circle. The orthogonality property helps in achieving a near zero value of redundancy measure in a set of moments functions [42,40]. Let f ðr; yÞ be the gray level of the pixel at polar coordinates ðr; yÞ, ðr; yÞ being defined over the unit disc and let represent the complex conjugate operator. The Zernike moment of order m and n can be formulated as Z Z mþ1 zmn ¼ f ðr; yÞzmn ðrÞeiyn dy dr, (10) p r y where mjnj 2
zmn ðrÞ ¼
X
ðn sÞ!rn2s , ð1Þs m þ jnj m jnj s¼0 s ! s ! s! 2 2
(11)
m jnj is even, jnjpm; .
(12) (13)
rp1.
(14)
h1 ¼ n20 þ n02 ,
(1)
h2 ¼ ðn20 n02 Þ2 þ 4n211 ,
(2)
h3 ¼ ðn30 3n12 Þ2 þ ð3n21 n03 Þ2 ,
(3)
n is a positive or negative integer number related to the angular dependency or rotation. For a digital image, if f ðr; yÞ is the gray level of the pixel at polar coordinate at ðr; yÞ, then Eq. (10) can be written as m þ 1XX zmn ¼ f ðr; yÞzmn ðrÞeiyn . (15) p r y
h4 ¼ ðn30 3n12 Þ2 þ ðn21 þ n03 Þ2 ,
(4)
To calculate the Zernike moments, the image (or the region of interest) is first inscribed in the unit circle using polar
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coordinates, where the image mass center is the unit circle center and the image is included in it. Pixels out of the circle are not used for calculations. The coordinates are defined by means of the modulus of the vector from the origin to the coordinate point, r, and the angle between this vector and the horizontal axis, y, is positive if counterclockwise. The main drawback of these descriptors is that, although they are invariants to rotation, it is necessary to apply some normalization to be also invariant to scaling and translation. 2.3.3. Legendre moments The Legendre moments of ðp; qÞ order are defined by [40] lpq
ð2p þ 1Þð2q þ 1Þ ¼ 4
Z1 Z1 Pp ðxÞPq ðyÞf ðx; yÞ dx dy, (16) 1 1
where Pp and Pq are the polynomials of Legendre defined as Pi ðzÞ ¼
1 dp ðz2 1Þp , 2 p! dzp
(17)
p
where z 2 ½1; 1. In the context of digital images the expression for the moments adopts the form: ð2p þ 1Þð2q þ 1Þ l^ pq ¼ ðM 1ÞðN 1Þ
M X N X
Pp ðxi ÞPq ðyj Þf ðxi ; yj Þ,
(18)
i¼1 j¼1
where M is the number of rows and N is the number of columns in the image. The monomials basis functions increase very rapidly in range as the order increases, but they do have the advantage that simple integer data representation may be used with discrete digitized imagery. 2.3.4. Invariant moments of Taubin Taubin and Cooper [43] proposed algebraic invariant moments to describe shape features. The matching of arbitrary shaped regions is done by computing a vector of centered moments for each region. These vectors are viewpoint dependent, but the dependence on the viewpoint is algebraic and well known. They then compute invariant moments, i.e., algebraic functions of the moments that are invariant to Euclidean or affine transformations of the data set. The authors present a family of computationally efficient algorithms, based on matrix computations, for the evaluation of both Euclidean and affine algebraic invariant moments of data sets. The use of invariant moments greatly reduces the computation required for the matching and hence initial object recognition. Taubin and Cooper introduce the covariance matrix concept, which is generated using the second order central moments associated to an image: " 0 # m20 m011 M11 ¼ , (19) m011 m002
where m0pq ¼
mpq f ðx; yÞ and m00
m00 are the geometric moment of 0,0 order: XX m00 ¼ f ðx; yÞ x
y
The lower triangular matrix is obtained from M11 : " # l 11 0 L¼ l 21 l 22
(20)
in such a way that LM11 LT ¼ I and it is found using the Cholesky decomposition and the matrix inversion operation. The matrixes from which eigenvalues will be extracted are generated using the center of gravity ðx; yÞ and the L matrix, calculating the new moments Z0pq ; for p þ q42, as indicated in the following equation: Z0pq ¼
X 1 ½l 11 ðx xÞp ½l 21 ðx xÞ þ l 22 ðy yÞq . m00 f ðx; yÞ x;y (21)
Matrices for third and fourth order moments are: " # c1 Z030 Z021 c1 Z012 0 M 12 ¼ , c1 Z021 Z012 c1 Z003 2
c2 Z040
0 6 M 022 ¼ 4 c1 Z31 c2 Z022
c2 Z031 Z022 c1 Z013
c2 Z022
(22)
3
c1 Z013 7 5, 0 c2 Z04
(23)
where c1 ¼ p1ffiffi2 and c2 ¼ 12. The eight invariant moments M T ¼ ðM T1 ; . . . ; M T8 Þ which constitute the solution are the two eigenvalues of the 2 2 symmetric matrix M 12 M T12 , the three eigenvalues of the 3 3 symmetric matrix M 22 , the two eigenvalues of the 2 2 symmetric matrix M 12 M 22 M T12 and the scalar 1 0 0 4 ðZ40 þ Z04 Þ. 2.3.5. Affine invariant moments of Flusser and Suk The affine invariant moments are derived from the algebraic invariant theory. They are invariant with regard to a general affine transformation. They use the central moments relative to several orders. These moments are expressed as follows: 1 ðm m m211 Þ, m400 20 02 1 F 2 ¼ 10 ðm230 m203 6m30 m21 m12 m03 4m30 m312 þ 4m03 m321 Þ m00 F1 ¼
3
m221 m212 , m10 00
(24)
(25)
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m20 m ðm21 m03 m212 Þ 11 ðm m m21 m12 Þ 7 m00 m700 30 03 m þ 02 ðm m m212 Þ, m700 30 12 1 F 4 ¼ 11 ðm320 m303 6m220 m11 m12 m03 6m220 m02 m21 m03 Þ m00 1 þ 11 ð9m220 m02 m212 þ 12m20 m211 m21 m03 Þ m00 1 þ 11 ð6m20 m11 m02 m30 m03 18m20 m11 m02 m21 m12 Þ m00 1 þ 11 ð8m311 m30 m03 6m20 m202 m30 m12 Þ m00 1 þ 11 ð9m20 m202 m221 þ 12m211 m02 m30 m12 Þ m00 1 þ 11 ð6m11 m202 m30 m21 þ m202 m230 Þ, m00 1 F 5 ¼ 6 ðm40 m04 4m31 m12 þ 3m222 Þ, m00 1 F 6 ¼ 9 ðm40 m04 m22 m40 m212 m322 Þ m00 1 þ 9 ð2m31 m22 m12 m04 m231 Þ, m00
F3 ¼
(26)
(27) (28)
(29)
where mpq ¼
XX x
ðx xÞp ðy yÞq f ðx; yÞ.
(30)
y
Calculation of these moments can be divided in two steps: the first one uses relative complex operations and is done once, whereas the second one uses simple operations [44,45]. Drawbacks of these moments are their sensibility to partial occlusions and the requirement to do a good segmentation previously. 3. Wear analysis 3.1. Defining wear classes Once the images of the cutting tool are described by means of statistical moments, the main wear classes can be determined by analyzing the collected data. In order to improve the quality of the industrial processes, practitioners have traditionally found interesting techniques ranging from finite element models [46–48] to neural networks [49,50], and more recently data mining [51]. In a case like this, the data mining approach seems to be the most natural choice according to the dimensions and cardinality of the data set we pursue to understand. Amongst the different techniques comprised in the field of data mining, cluster analysis techniques provide the means of identifying the different wear classes. Particularly, a finite mixture model approach was adopted, in consideration of the complex nature of the data. For that
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purpose, the implementation of the MCLUST algorithm [52,53] was used running under R, a free software environment for statistical computing and graphics. Cluster analysis by means of finite mixture models requires the practitioner to set a priori the number of clusters to fit. Then, a search for the optimum values of the parameters of the different models is performed by means of the Expectation—Maximization algorithm [54]. Nevertheless, an evidence on the most appropriated number of clusters can be found by running the cluster analysis for different values of the number of clusters. The Bayesian information criterion (BIC) obtained at each configuration determines the number of clusters that provide the best fit to the given data set, thus pointing out the best choice [55]. If the BIC values are similar, as those obtained in this research, the practitioner will choose the number of clusters according to their previous experience and goals pursued. The state of the art revision performed by Sick [23] shows that the bulk of the literature tends to classify the wear level in two classes: either defective or adequate to work. In this work it has been preferred to distinguish amongst three different wear levels—low, medium and high—thus reflecting in a more faithful manner the frequently observed shape for the evolution of tool wear level (Fig. 2). This provides finer detail in the estimation of the actual wear status of the tool. Our approach corresponds with the results obtained by other authors [33]. Using the discriminant analysis techniques [56] let us test the quality of the results generated by the cluster analysis. For that purpose, the class labels obtained by the MCLUST algorithm are compared with the predictions of the linear (LDA) and quadratic (QDA) discriminant analysis. A measure of the quality of the fit can be obtained in terms of the Fowlkes—Mallows index [57]. Table 1 shows that QDA results are, as expected, better than those obtained by the LDA. On the other hand, no substantial difference can be observed amongst the different sets of descriptors as all of them provide excellent results when the QDA is considered. Nevertheless, as it will be shown soon, the LDA results will still provide us interesting insights on the nature of the data sets.
Fig. 2. Evolution of cutting tool wear.
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LDA QDA
MCLUST
assignments vs. LDA and QDA 4
Zernike
Legendre
Hu
Taubin
Flusser
0.935 1.000
0.852 0.997
0.917 0.966
0.750 0.993
0.784 0.971
The Fowlkes—Mallows indexes reflect that the data have structures easy to reproduce by distributional models, thus validating the quality of the clustering results in terms of suitability of the models to the data. However, as the wear itself has not been considered to perform the clustering, the clusters obtained might not necessarily display significant differences on the wear. This will depend on the amount of information related to wear contained by each family of descriptors. The following section elaborates on this issue. 3.2. Insights from the discriminant analysis Even though the QDA showed better performance, LDA has proved to be more adequate to show the correlation of the descriptors to the tool wear. LDA is a simplification of the QDA under the assumption of homocedasticity over the different groups, that is to say, the different clusters share a common variance–covariance matrix and they only differ on their location vectors. Considering this truth provides a simplification of the QDA formulas where the quadratic terms disappear and the resulting (LDA) formulas can be seen as a linear combination of the original vector space, or equivalently, a linear projection of the data onto a new vector space. Thus, QDA provides the possibility of capturing finer details due to the quadratic terms and usually provides higher accuracy at the numerical results, but no graphical representation can be directly obtain by the interpretation of its results. The LDA analysis, despite having an accuracy limited by the homocedasticity assumption, can be interpreted from the point of view of the change of basis matrix of its linear projection. Thus, QDA and LDA should not be seen as competitors but as complementary methods specialized in different tasks: QDA in providing higher accuracy for heteroscedastic structures, LDA providing graphical insights of these structures. The data, once projected onto the LDA map, can show the location of the different wear classes and, more importantly, the track of a cutting tool while the machining process proceeds. Fig. 3 shows such LDA projection of the data set as described by the Hu moments. The figure shows the areas where the data set concentrates and the borders amongst classes. The first class, the one located at the left, corresponds with the low wear level class; the medium wear level class lays in the center of the image; and the high level class is located at the right.
2
LD2
Table 1 Fowlkes–Mallows index of the predictions
0
-2
-4
-8
-6
-4
-2
0
2
4
LD1
Fig. 3. LDA projection of the images data as described by the Hu moments.
4
2
LD2
1010
0
-2
-4
-8
-6
-4
-2 LD1
0
2
4
Fig. 4. The evolution of the wear of a particular cutting edge as described by the Hu moments.
Fig. 4 shows the trace of a particular single cutting edge while evolves starting at the left in the low wear level class, passes through the medium wear level class and enters into the high wear level class at the right. There is a connection amongst the wear level and the information described by the Hu moments, since the cutting edge crosses the borders of the different classes during progress of the machining process. On the other hand, drawing the LDA map corresponding to the same data set as described by the Zernike
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4
4 2 0 -2
LD2
moments, shows us (Fig. 5) that the three classes found are quite separated, not showing the behavior of a continuous process that evolves through the different classes. A look into the evolution of each cutting edge confirms this fact, as it can be seen for example in Fig. 6, where a particular single cutting edge remains in one of the classes during the whole machining process until its replacement. Table 2 shows the percentage of cutting edges that evolved through the three different classes during the machining process. As it can be seen, the information described by the Hu and Legendre moments of the wear region is directly related with the wear level while the rest of the moments considered seem to pay attention to other features than those related with the wear itself. The features with greater relevance in the discrimination process can be known with a closer look at the LDA results obtained with the Hu and Legendre moments; that is to say, which ones are more important in describing the wear level. This is due to the fact that the LDA can be seen as projection, as it has been shown, of the original data into a new vector space where the clusters are as compact and separated from each other as possible. The change of basis matrix that defines this linear mapping contains information about the impact of each descriptor on the direction of the new axes. Thus, the contribution of each descriptor to the first components of the LDA basis can be used as an indicator of the relationship between the information described by the descriptor and the wear level. Inspecting the results obtained by the LDA, it can be seen that in the case of the Legendre moments (Table 4) the information about the wear level is distributed amongst a
1011
-4 -6 -8 -10
-6h
-4h
-2h
0 LD1
2h
4h
6h
Fig. 6. The evolution of the wear of a particular cutting edge as described by the Zernike moments.
Table 2 Percentage of cutting edges that evolve through the detected clusters as an indicator of the relationship amongst the information provided by the descriptors and the wear level Hu
Legendre
Taubin
Flusser
Zernike
86
75
1
0
0
Table 3 Contribution (in %) of the HU descriptors to the first discriminant function h1
h2
h3
h4
h5
h6
h7
99.88
0.08
0.02
0.02
0.00
0.00
0.00
2
Table 4 Contribution (in %) of the Legendre descriptors to the first discriminant function
0
LD2
-2 -4
l^ 02
l^ 12
l^ 11
l^ 00
l^ 20
l^ 10
l^ 22
l^ 21
l^ 01
23.77
16.86
16.01
12.86
10.82
8.02
7.78
3.02
0.87
-6 -8 -10 -6
-4
-2
0 LD1
2
4
6
Fig. 5. LDA projection of the images data as described by the Zernike moments.
large number of descriptors; while in the case of the Hu moments (Table 3), it is remarkable that the first descriptor ðh1 ¼ n20 þ n02 Þ is by far the feature that contains the larger proportion ð99:88%Þ of information about the wear level. h1 being a combination of the second order statistical moments that describe the vertical and horizontal variance of the region, it is clear now that the discrimination amongst wear levels is mainly performed according to the changes in n20 and n02 .
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3.3. Monitoring wear level The former analysis allowed to gain a better understanding of the wear process in terms of the main factors that have impact over the discrimination amongst different wear levels. Those insights will now be applied to the machining process itself by determining a cutting tool replacement criterion. In fact, most of the work has already been done and it is only necessary to consider the former results from a different perspective or view. So as to prevent the machining process to be performed with a defective cutting tool, it should be replaced just before it enters into the high wear level class. Fortunately, it is possible to take advantage of the fine details the QDA provides. Being a Bayes’ theorem based technique, the QDA provides the probability to belong to each of the classes for a given cutting edge. Thus, following a procedure similar to that described by Castejo´n et al. [58], the evolution of the cutting tool can be traced as the value of the probability to belong to each class. The cutting tool will be replaced as soon as it is detected that the probability to belong to the medium wear level class diminishes and the probability to belong to the high wear level class increases over a defined threshold. 4. Conclusions An analysis of the tool wear process and the most adequate manners of describing it stands to reason according to the impact of the cutting tool replacement on the production costs. As a result of this research, it has been shown that not all of the analyzed descriptors based on moments are adequate to describe the wear level of cutting tools. Amongst them, the Hu descriptors [41] seem to be the better for that purpose. It is remarkable that the first Hu descriptor, h1 , is responsible of 99:88% of the discrimination amongst wear levels. A new tool life criterion is proposed that is more efficient than traditional approaches. Following the methodology described, the status of the cutting edge can be efficiently monitored, it is possible to prolong its useful life and the production costs are reduced significatively. The values for the maximum and average width on the flank wear land (VBmax and VBavg) were always higher when using our descriptors based methodology than those proposed by ISO3685 standard. The tool life criteria defined in ISO3685, that indicates if a tool is useful or non-useful, maps in our approach at values belonging to the medium wear class. In the case of VBmax, ISO standard postulates a limit value of 600 mm, whereas in our approach the limit value is 300 mm higher, in the case of simple region descriptors, and 730 mm higher, in the case of Legendre descriptors. Similarly, the VBavg is also higher using our approach. In this case, the limit value for VBavg is established at 300 mm in the ISO standard, whereas with our approach the value is 139 mm higher for Hu
descriptors, 280 mm higher for simple descriptors, and 376 mm higher for Legendre descriptors. Acknowledgments We gratefully acknowledge the financial support provided by the Spanish Ministerio de Educacio´n y Ciencia through grants DPI2006-14784 and DPI2006-02550, the Junta de Castilla y Leo´n through Grant no. LE018B06 and the Universidad of Leo´n through Grant no. ULE2005-01. References [1] S. Kurada, C. Bradley, A review of machine vision sensors for tool condition monitoring, Computers in Industry 34 (1997) 55–72. [2] R. Teti, Machining of composite materials, Annals of the CIRP 51 (2) (2002) 611–634. [3] A. Weckenmann, K. Nalbantic, Precision measurement of cutting tools with two matched optical 3d-sensors, Annals of CIRP 52 (1) (2003) 443–446. [4] J. Kopac, S. Sali, Tool wear monitoring during the turning process, Journal of Materials Processing Technology 113 (2001) 312–316. [5] ISO 3685:1993, Tool-life testing with single-point turning tools. [6] ISO 8688-1:1989, Tool life testing in milling—Part 1: face milling. [7] ISO 8688-2:1989, Tool life testing in milling–Part 2: end milling. [8] G. Lim, Tool-wear monitoring in machine turning, Journal of Materials Processing Technology 51 (1995) 25–36. [9] F. Al—Sulaiman, M. Baseer, A. Sheikh, Use of electrical power for online monitoring of tool condition, Journal of Materials Processing Technology 166 (2005) 364–371. [10] D. Axinte, N. Gindy, K. Fox, I. Unanue, Process monitoring to assist the workpiece surface quality in machining, International Journal of Machine Tools and Manufacture 44 (2004) 1091–1108. [11] M. Cakir, Y. Isik, Detecting tool breakage in turning AISI 1050 steel using coated and uncoated cutting tools, Journal of Materials Processing Technology 159 (2005) 191–198. [12] S. Choudhury, P. Srinivas, Tool wear prediction in turning, Journal of Materials Processing Technology 153–154 (2004) 276–280. [13] J. Gajewski, J. Jonak, Utilisation of neural networks to identify the status of the cutting tool point, Tunnelling and Underground Space Technology 21 (2006) 180–184. [14] N. Ghosh, Y. Ravi, A. Patra, S. Mukhopadhyay, S. Paul, A. Mohanty, A. Chattopadhyay, Estimation of tool wear during CNC milling using neural network-based sensor fusion, Mechanical Systems and Signal Processing 21 (1) (2007) 466–479. [15] W. Haili, S. Hua, C. Ming, H. Dejin, On-line breakage monitoring in turning, Journal of Materials Processing Technology 139 (2003) 237–242. [16] E. Kuljanic, M. Sortino, Twem, a method based on cutting forces— monitoring tool wear in face milling, International Journal of Machine Tools and Manufacture 45 (2005) 29–34. [17] M. Ritou, S. Garnier, B. Furet, J. Hascoet, A new versatile in-process monitoring system for milling, International Journal of Machine Tools and Manufacture 46 (15) (2006) 2026–2035. [18] D. Salgado, F. Alonso, Tool wear detection in turning operations using singular spectrum analysis, Journal of Materials Processing Technology 171 (2006) 451–458. [19] C. Sceffer, P. Heyns, An industrial tool wear monitoring system for interrupted turning, Mechanical Systems and Signal Processing 18 (2004) 1219–1242. [20] H. Shao, H. Wang, X. Zhao, A cutting power model for tool wear monitoring in milling, International Journal of Machine Tools and Manufacture 44 (2004) 1503–1509.
ARTICLE IN PRESS J. Barreiro et al. / International Journal of Machine Tools & Manufacture 48 (2008) 1005–1013 [21] J. Jurkovic, M. Korosec, J. Kopac, New approach in tool wear measuring technique using ccd vision system, International Journal of Machine Tools and Manufacture 45 (9) (2005) 1023–1030. [22] S. Kurada, C. Bradley, A machine vision system for tool wear assessment, Tribology International 30 (4) (1997) 295–304. [23] B. Sick, On-line and indirect tool wear monitoring in turning with artificial neural networks: a review of more than a decade of research, Mechanical Systems and Signal Processing 16 (4) (2002) 487–546. [24] W. Wang, Y. Wong, G. Hong, Flank wear measurement by successive image analysis, Computers in Industry 56 (2005) 816–830. [25] E. Alegre, J. Barreiro, H. Ca´ceres, L. Herna´ndez, R. Ferna´ndez, M. Castejo´n, Design of a computer vision system to estimate tool wearing, Materials Science Forum 526 (2006) 61–66. [26] M. Lanzetta, A new flexible high-resolution vision sensor for tool condition monitoring, Journal of Materials Processing Technology 119 (1–3) (2001) 73–82. [27] M. Sortino, Application of statistical filtering for optical detection of tool wear, International Journal of Machine Tools and Manufacture 43 (2003) 493–497. [28] W. Wang, G. Hong, Y. Wong, Flank wear measurement by a threshold independent method with sub-pixel accuracy, International Journal of Machine Tools and Manufacture 46 (2) (2006) 199–207. [29] T. Obikawa, J. Shinozuka, Monitoring of flank wear of coated tools in high speed machining with a neural network art 2, International Journal Machine Tools and Manufacture 44 (2004) 1311–1318. [30] D. Brezak, T. Udiljak, K. Mihoci, D. Majetic, B. Novakovic, J. Kasac, Tool wear monitoring using radial basis function neural network, Neural Networks 3 (2004) 25–29. [31] G. Littlefair, M. Javed, G. Smith, Fusion of integrated multisensor data for tool wear monitoring, Neural Networks 2 (1995) 734–737. [32] V. Sunil, C. Yung, Design and implementation of tool wear monitoring with radial basis function neural networks, American Control Conference 3 (1995) 1722–1726. [33] H. Guo, G. Xiong, F. Xi, J. Zheng, X. Ping, Detection and compensation of cutting tool wear based on Lipschitz exponent, in: Proceedings of the 2005 IEEE Conference on Control Applications, 2005. [34] L. Herna´ndez, H. Ca´ceres, J. Barreiro, E. Alegre, M. Castejo´n, R. A. Ferna´ndez, Monitorizacio´n del desgaste de plaquitas de corte usando visio´n artificial, Congreso Iberoamericano de Ingenierı´ a Meca´nica (2005). [35] C. Teh, R.T. Chin, On image analysis by the methods of moments, IEEE Transactions on Pattern Analysis and Machine Intelligence 10 (4) (1988) 498–514. [36] A. Choksuriwong, H. Laurent, B. Emil, Comparison of invariant descriptors for object recognition, in: IEEE International Conference on Image Processing, 2005, pp. 377–380. [37] W. Kim, P. Yuan, A practical pattern recognition system for translation, scale and rotation invariance, IEEE (1994) 391–396. [38] E. Alegre, L. Sa´nchez, R. Alaiz, J. C. Rodrı´ guez, Utilizacio´n de momentos estadı´ sticos y redes neuronales en la clasificacio´n de cabezas de espermatozoides de verraco, XXV Jornada de Automatica (2004).
1013
[39] J.S. Park, D. Chang, 2D invariant descriptors for shape-based images retrieval, Pattern Recognition Letters (2001). [40] R.J. Prokop, A.P. Reeves, A survey of moment-based techniques for unoccluded object representation and recognition, CCVGIP: Graphical Models and Image Processing 54 (5) (1992) 438–460. [41] M. Hu, Visual pattern recognition by moment invariants, IRE Transactions on Information Theory (1962) 179–182. [42] A. Khotanzad, Y.H. Hong, Invariant image recognition by Zernike moments, IEEE Transaction on Pattern Analysis and Machine Intelligence 22 (1) (1990) 489–497. [43] G. Taubin, D. Cooper, Object recognition based on moment (or algebraic) invariants, Geometric Invariance in Computer Vision (1992) 375–397. [44] J. Flusser, T. Suk, On the calculation of image moments, Research Report no. 1946. [45] J. Flusser, B. Zitovi, T. Suk, Invariant–based registration of rotated and blurred images, IEEE (1999) 1262–1265. [46] C. Mene´ndez, J.B. Ordieres, F. Ortega, Importance of information pre-processing in the improvement of neural network results, Expert Systems 13 (2) (1996) 95–102. [47] C. Gonzalez Nicieza, J.B. Ordieres Mere, A. Menendez Diaz, J. Taboada Castro, Optimizing of stripping in a slate bank, International Journal of Surface Mining and Reclamation 8 (1) (1994) 17–22. [48] A. Bello Garcı´ a, A. Mene´ndez Dı´ az, J.B. Ordieres Mere´, C. Gonza´lez Nicieza, Generalization of the influence function method in mining subsidence, International Journal of Surface Mining, Reclamation and Environment 10 (4) (1996) 195–202. [49] A.V. Pernı´ a-Espinoza, J.B. Ordieres-Mere´, F.J. Martı´ nez-de-Piso´n, A. Gonza´lez-Marcos, TAO–robust backpropagation learning algorithm, Neural Networks 18 (2) (2005) 191–204. [50] J.B.O. Mere´, A.G. Marcos, J.A. Gonza´lez, V.L. Rubio, Estimation of mechanical properties of steel strip in hot dip galvanising lines, Ironmaking and Steelmaking 31 (1) (2004) 43–50. [51] T. Hastie, R. Tibshirani, J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Springer, Berlin, 2001. [52] J. Banfield, A. Raftery, Model-based Gaussian and non-Gaussian Clustering, Biometrics 49 (1993) 803–821. [53] C. Fraley, A.E. Raftery, Enhanced software for model–based clustering, discriminant analysis and density estimation: Mclust, Journal of Classification 20 (2003) 263–286. [54] G. McLachlan, T. Krishnan, The EM Algorithm and Extensions, Wiley, New York, 1997. [55] G. McLachlan, D. Peel, Finite Mixture Models, Wiley, New York, 2000. [56] G. McLachlan, Dis criminant Analysis and Statistical Pattern Recognition, Wiley, New York, 2004. [57] M. Meila, Comparing clusterings, Technical Report, University of Washington (2002). [58] M. Castejo´n, E. Alegre, J. Barreiro, L. Herna´ndez, On-line tool wear monitoring using geometric descriptors from digital images, International Journal of Machine Tools and Manufacture 47 (2007) 1847–1853.
International Journal of Machine Tools & Manufacture 48 (2008) 1005–1013 www.elsevier.com/locate/ijmactool
Use of descriptors based on moments from digital images for tool wear monitoring J. Barreiroa,, M. Castejo´na, E. Alegrea, L.K. Herna´ndezb a
Escuela de Ingenierı´as Industrial e Informa´tica, Universidad de Leo´n, 24071 Leo´n, Spain Dpt. de Ingenierı´a Meca´nica, Industrial y Mecatro´nica, Universidad de Pamplona, Colombia
b
Received 23 July 2007; received in revised form 4 January 2008; accepted 4 January 2008 Available online 26 January 2008
Abstract The widely used criteria to determine the need for tool replacement are too much conservative. The consequence is that tools are only used for a small fraction of their possible useful life. The economical influence of tool replacement costs over total production costs demands better criteria according to current technology. The use of different moments to describe tool wear images and to classify the tool condition in wear classes has been studied. The moments used as descriptors in this paper show different behavior with regard to wear identification, concluding that Hu and Legendre descriptors provide the best performance. These descriptors have been classified using a finite mixture MCLUST model considering three wear classes (low, medium and high). The achieved results from the clustering have been checked by means of discriminant analysis techniques, linear and quadratic, using the Fowlkes–Mallow index as quality factor. The projection of image data by means of linear discriminant analysis provides useful wear maps for tool monitoring. These wear maps show us the wear classes and the frontiers among them, in such a way that wear evolution for current tool can be mapped. The quadratic discriminant analysis allows us to assign to the current tool, a probability of belonging to a wear class. This probability is used as a new wear criterion in substitution of the current conservative criteria, making possible to reduce tool replacement costs. r 2008 Elsevier Ltd. All rights reserved. Keywords: Cutting tool; Wear; Image analysis; Cluster analysis
1. Introduction Different authors [1–3] have already emphasized the impact that tool replacement operations have on costs in the metallic parts manufacturing. Costs linked to tool replacement operations comprise [4] not only the costs of cutting tools but also, and much more important, the indirect costs linked to unproductive time needed to perform the replacement. This is an issue of increasing importance for those factories which operate under high productivity constrains; hence the recent efforts to avoid superfluous replacement operations. The commonly followed criteria to identify the need of tool replacement usually consist of early stopping rules that reject the tools irrespective of their actual wear condition. In manual processes, the machinist schedules the replaceCorresponding author. Tel.: +34 987 291792; fax: +34 987 291930.
E-mail address: [email protected] (J. Barreiro). 0890-6955/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2008.01.005
ments on the basis of his own, and obviously subjective, experience. In automated lines, replacements are scheduled according to variables related to the production process, e.g. cutting time or number of parts produced, under the assumption that once a predefined limit is reached, the tool may be incapable of producing acceptable parts. Obviously, the aforementioned criteria neither ensure the optimal usage of tools nor lead to minimal replacement costs, causing in some applications important economic losses. Alternatively, a replacement policy based on tool wear measurement offers considerable benefits in costs. Apart from the derived economic implications, the International Organization for Standardization (ISO) has devoted several standards [5–7] to cutting inserts and their life. ISO’s approach already represents an important advance with regard to common practice: it establishes wear limits directly measurable on the wear land. However, this approach has its own limitations because the defined wear limits are too conservative [8] and the required
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complex measurements are too difficult to obtain without stopping the process. The former limitation demands more refined criteria, the latter clearly opens a window of opportunity for alternative measurement methods. Two main categories can be distinguished among the different approaches proposed: indirect and direct measurement methods. Indirect measurement methods provide wear estimates by monitoring observable variables related to wear, e.g. cutting forces, temperatures and vibrations [9–20]; thus avoiding unwanted machining process interruptions. In industrial environments, however, the all-pervasive noise signals severely undermine the quality of these estimates [21–24]. On the other hand, direct methods, usually based on computer vision systems, are definitely better in terms of accuracy and reliability. Nevertheless, in spite of their advantages, direct methods have not found their place in industrial practices yet, mainly due to the high cost and the poor features displayed by the early hardware available. In the last years we have witnessed a decrease of prices and the enhancement of features, so the revival of direct methods. Nowadays, computer vision can be considered a mature technology and its application to cutting tool monitoring offers a wealth of interesting possibilities [3,21,24–28]. Consequently, a brief review of the state of the art reveals a significant number of works published during the last decade on the use of digital images in wear measurement [13,14,22,24,25,28–33]. Our approach, described in the following sections, takes advantage of the information provided by a computer vision system to define a set of replacement rules according to the current state of technology.
2. Materials and methods 2.1. Materials and machining process The tests were carried out in a CNC parallel lathe. Steel cylinders ANSI SAE 4340 and 4140 (quenched and normalized) were machined. The dimensions of the parts were 250 mm length and 90 mm diameter. The tools were coated tungsten carbide inserts (CNMG 120404MF235), rhombic, high tough and low wear resistance, in order to avoid premature tool fracture and, at the same time, to increase wear rate. The machining parameters were chosen with the same objective: different values between 140 and 200 m/min were used for cutting speed, whereas feed rate and cutting depth remained constant at 0.2 mm/rev and 2 mm, respectively. The coolant was 10% diluted, as it is indicated for medium hard metals. The tool was disassembled and inspected at the end of each machining pass. A fixture was specifically made to assure that every insert was located at the very same position; therefore, the image origin coordinates were the same and all of the acquired images were comparable. The light intensity and the focal distance were also kept constant.
2.2. Image acquisition and processing The images were acquired [34] using a PULLNIX PE2015 camera with 1=300 CCD. The camera was a 752ðHÞ 582ðVÞ pixels resolution and a signal–noise ratio of 50 dB. The digitalization was carried out with a MATROX METEOR II card. The size of the image acquired was 768 576 pixels. The images were automatically cropped to 400 200 pixels in order to clearly view the wear land. The optical system was composed of an 70XL OPTEM industrial zoom, with a 1X extension tube and 0.5X/0.75X/1.5X/2.0X OPTEM lens. Lighting system selection is a critical step during the design of any vision system, because it affects the quality of captured images. Best results are obtained with a uniform illumination without shines, so that the region of interest is fully illuminated and, at the same time, with the highly possible contrast with the background. In this case, to obtain a good image of the tool corner was difficult due to the tip radius. The lighting system used was composed of a FOSTEC DCRsIII regulated light source which provided an intense cold lighting. A NER SCDI-25-F0 SCDI diffuse lighting system was used to avoid shines. This system provides diffuse lighting in the direction of the camera axis. Lighting system location was carried out by means of a FOSTEC dual bundle. Image acquisition was executed with a specifically developed application. This application includes three modules: camera setup, sequence setup and image acquisition. These modules provide capturing device information, allow user to choose the resolution, define the information storage path and save the images [25]. A set of 1383 insert flank images was acquired using this system. Previous to segmentation, several operations were carried out to improve image appearance. A low pass filter was used to soften the background and to make the segmentation easier. The filtered image was cropped to remove the inferior half of the insert, because wear only takes place at the superior half. Later on, the histogram was stretched to improve contrast [25]. Region growing was used to segment the wear area. Starting points were selected on the basis of the results of a previous threshold. Once the thresholds were obtained, a binary image was generated with the worn region set to 1 and the rest of the image set to 0. Later on, a median filter was applied to reduce the noise. If the worn region is not closed, a morphological closing is carried out. Fig. 1 shows one of these binary images. B/W
2.3. Shape description using moments Moments are descriptors obtained from point coordinates in a region and their gray level. They are applied in several areas of pattern recognition and object classification. Teh and Chin [35] studied the characteristics of the moments and discussed the correlation among them. Choksuriwong et al. [36], Kim and Yuan [37] and Alegre et al. [38] compared the ability of orthogonal moments of
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h5 ¼ ðn30 3n12 Þðn30 þ n12 Þ½ðn30 þ n12 Þ2 3ðn21 þ n03 Þ2 þ ð3n21 n03 Þðn21 þ n03 Þ½3ðn30 þ n21 Þ ðn21 þ n03 Þ2 , (5) h6 ¼ ðn20 þ n02 Þ½ðn30 þ n12 Þ2 ðn21 þ n03 Þ2 þ 4n11 ðn30 þ n12 Þ,
(6)
h7 ¼ ð3n21 n03 Þðn30 þ n12 Þ½ðn30 þ n12 Þ2 3ðn21 þ n03 Þ2 þ ð3n12 n03 Þðn21 þ n03 Þ½3ðn30 þ n12 Þ ðn21 þ n03 Þ2 , (7)
Fig. 1. Image preprocessing.
Hu and Zernike to recognize objects and patterns in different applications. Park and Chang [39] compared the performance of orthogonal moments of Hu and Zernike with regard to the invariants of Hu, Taubin and Flusser, analyzing the computational cost and the ability to recognition. Five different families of statistical moments—Zernike, Legendre, Hu, Taubin and Flusser—and a number of region image descriptors were used to characterize the shape of the worn region in the binary images. The first 18 moments of Zernike (up to fourth order), nine moments of Legendre, seven moments of Hu, eight moments of Taubin and six moments of Flusser and Suck were calculated for each image. These descriptors were used for their wellknown efficiency in the description and recognition of different types of shapes [36]. On the other hand, the difficulties to predict the best descriptor for a particular shape [40] are evident. Consequently, the best descriptor in the clustering and recognition process is evaluated. 2.3.1. Hu invariants Moments and functions of moments are invaluable tools in the literature for the measurement of the properties of a distribution. In the field of image analysis, their use as image descriptors was pioneered by Hu [41] when he used the 2D geometric moments for characterizing the visual patterns in images. For different images, the respective sets of moments are unique and this makes them particularly useful for the task of pattern recognition. This is further added by the advantage of being able to construct moment invariants which are insensitive to rotation, scaling and translation. Hence, geometrical moments are effective descriptors for images under different perspectives. Hu derived seven invariant moments using nonlinear combinations of normalized central moments:
where mpq npq ¼ , m00 XX mpq ¼ ðx xÞp ðy yÞq f ðx; yÞ x
(8) (9)
y
and f ðx; yÞ is the gray level of the pixel at coordinates ðx; yÞ. 2.3.2. Zernike moments Though computationally very complex compared with geometric and Legendre moments, in general, Zernike moments are better in terms of their ability to feature representation, rotation invariance, fast computation, multi-level representation for describing the shapes of patterns and low noise sensitivity. Zernike moments use a set of Zernike polynomials that is complete and orthonormal in the interior of the unit circle. The orthogonality property helps in achieving a near zero value of redundancy measure in a set of moments functions [42,40]. Let f ðr; yÞ be the gray level of the pixel at polar coordinates ðr; yÞ, ðr; yÞ being defined over the unit disc and let represent the complex conjugate operator. The Zernike moment of order m and n can be formulated as Z Z mþ1 zmn ¼ f ðr; yÞzmn ðrÞeiyn dy dr, (10) p r y where mjnj 2
zmn ðrÞ ¼
X
ðn sÞ!rn2s , ð1Þs m þ jnj m jnj s¼0 s ! s ! s! 2 2
(11)
m jnj is even, jnjpm; .
(12) (13)
rp1.
(14)
h1 ¼ n20 þ n02 ,
(1)
h2 ¼ ðn20 n02 Þ2 þ 4n211 ,
(2)
h3 ¼ ðn30 3n12 Þ2 þ ð3n21 n03 Þ2 ,
(3)
n is a positive or negative integer number related to the angular dependency or rotation. For a digital image, if f ðr; yÞ is the gray level of the pixel at polar coordinate at ðr; yÞ, then Eq. (10) can be written as m þ 1XX zmn ¼ f ðr; yÞzmn ðrÞeiyn . (15) p r y
h4 ¼ ðn30 3n12 Þ2 þ ðn21 þ n03 Þ2 ,
(4)
To calculate the Zernike moments, the image (or the region of interest) is first inscribed in the unit circle using polar
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coordinates, where the image mass center is the unit circle center and the image is included in it. Pixels out of the circle are not used for calculations. The coordinates are defined by means of the modulus of the vector from the origin to the coordinate point, r, and the angle between this vector and the horizontal axis, y, is positive if counterclockwise. The main drawback of these descriptors is that, although they are invariants to rotation, it is necessary to apply some normalization to be also invariant to scaling and translation. 2.3.3. Legendre moments The Legendre moments of ðp; qÞ order are defined by [40] lpq
ð2p þ 1Þð2q þ 1Þ ¼ 4
Z1 Z1 Pp ðxÞPq ðyÞf ðx; yÞ dx dy, (16) 1 1
where Pp and Pq are the polynomials of Legendre defined as Pi ðzÞ ¼
1 dp ðz2 1Þp , 2 p! dzp
(17)
p
where z 2 ½1; 1. In the context of digital images the expression for the moments adopts the form: ð2p þ 1Þð2q þ 1Þ l^ pq ¼ ðM 1ÞðN 1Þ
M X N X
Pp ðxi ÞPq ðyj Þf ðxi ; yj Þ,
(18)
i¼1 j¼1
where M is the number of rows and N is the number of columns in the image. The monomials basis functions increase very rapidly in range as the order increases, but they do have the advantage that simple integer data representation may be used with discrete digitized imagery. 2.3.4. Invariant moments of Taubin Taubin and Cooper [43] proposed algebraic invariant moments to describe shape features. The matching of arbitrary shaped regions is done by computing a vector of centered moments for each region. These vectors are viewpoint dependent, but the dependence on the viewpoint is algebraic and well known. They then compute invariant moments, i.e., algebraic functions of the moments that are invariant to Euclidean or affine transformations of the data set. The authors present a family of computationally efficient algorithms, based on matrix computations, for the evaluation of both Euclidean and affine algebraic invariant moments of data sets. The use of invariant moments greatly reduces the computation required for the matching and hence initial object recognition. Taubin and Cooper introduce the covariance matrix concept, which is generated using the second order central moments associated to an image: " 0 # m20 m011 M11 ¼ , (19) m011 m002
where m0pq ¼
mpq f ðx; yÞ and m00
m00 are the geometric moment of 0,0 order: XX m00 ¼ f ðx; yÞ x
y
The lower triangular matrix is obtained from M11 : " # l 11 0 L¼ l 21 l 22
(20)
in such a way that LM11 LT ¼ I and it is found using the Cholesky decomposition and the matrix inversion operation. The matrixes from which eigenvalues will be extracted are generated using the center of gravity ðx; yÞ and the L matrix, calculating the new moments Z0pq ; for p þ q42, as indicated in the following equation: Z0pq ¼
X 1 ½l 11 ðx xÞp ½l 21 ðx xÞ þ l 22 ðy yÞq . m00 f ðx; yÞ x;y (21)
Matrices for third and fourth order moments are: " # c1 Z030 Z021 c1 Z012 0 M 12 ¼ , c1 Z021 Z012 c1 Z003 2
c2 Z040
0 6 M 022 ¼ 4 c1 Z31 c2 Z022
c2 Z031 Z022 c1 Z013
c2 Z022
(22)
3
c1 Z013 7 5, 0 c2 Z04
(23)
where c1 ¼ p1ffiffi2 and c2 ¼ 12. The eight invariant moments M T ¼ ðM T1 ; . . . ; M T8 Þ which constitute the solution are the two eigenvalues of the 2 2 symmetric matrix M 12 M T12 , the three eigenvalues of the 3 3 symmetric matrix M 22 , the two eigenvalues of the 2 2 symmetric matrix M 12 M 22 M T12 and the scalar 1 0 0 4 ðZ40 þ Z04 Þ. 2.3.5. Affine invariant moments of Flusser and Suk The affine invariant moments are derived from the algebraic invariant theory. They are invariant with regard to a general affine transformation. They use the central moments relative to several orders. These moments are expressed as follows: 1 ðm m m211 Þ, m400 20 02 1 F 2 ¼ 10 ðm230 m203 6m30 m21 m12 m03 4m30 m312 þ 4m03 m321 Þ m00 F1 ¼
3
m221 m212 , m10 00
(24)
(25)
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m20 m ðm21 m03 m212 Þ 11 ðm m m21 m12 Þ 7 m00 m700 30 03 m þ 02 ðm m m212 Þ, m700 30 12 1 F 4 ¼ 11 ðm320 m303 6m220 m11 m12 m03 6m220 m02 m21 m03 Þ m00 1 þ 11 ð9m220 m02 m212 þ 12m20 m211 m21 m03 Þ m00 1 þ 11 ð6m20 m11 m02 m30 m03 18m20 m11 m02 m21 m12 Þ m00 1 þ 11 ð8m311 m30 m03 6m20 m202 m30 m12 Þ m00 1 þ 11 ð9m20 m202 m221 þ 12m211 m02 m30 m12 Þ m00 1 þ 11 ð6m11 m202 m30 m21 þ m202 m230 Þ, m00 1 F 5 ¼ 6 ðm40 m04 4m31 m12 þ 3m222 Þ, m00 1 F 6 ¼ 9 ðm40 m04 m22 m40 m212 m322 Þ m00 1 þ 9 ð2m31 m22 m12 m04 m231 Þ, m00
F3 ¼
(26)
(27) (28)
(29)
where mpq ¼
XX x
ðx xÞp ðy yÞq f ðx; yÞ.
(30)
y
Calculation of these moments can be divided in two steps: the first one uses relative complex operations and is done once, whereas the second one uses simple operations [44,45]. Drawbacks of these moments are their sensibility to partial occlusions and the requirement to do a good segmentation previously. 3. Wear analysis 3.1. Defining wear classes Once the images of the cutting tool are described by means of statistical moments, the main wear classes can be determined by analyzing the collected data. In order to improve the quality of the industrial processes, practitioners have traditionally found interesting techniques ranging from finite element models [46–48] to neural networks [49,50], and more recently data mining [51]. In a case like this, the data mining approach seems to be the most natural choice according to the dimensions and cardinality of the data set we pursue to understand. Amongst the different techniques comprised in the field of data mining, cluster analysis techniques provide the means of identifying the different wear classes. Particularly, a finite mixture model approach was adopted, in consideration of the complex nature of the data. For that
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purpose, the implementation of the MCLUST algorithm [52,53] was used running under R, a free software environment for statistical computing and graphics. Cluster analysis by means of finite mixture models requires the practitioner to set a priori the number of clusters to fit. Then, a search for the optimum values of the parameters of the different models is performed by means of the Expectation—Maximization algorithm [54]. Nevertheless, an evidence on the most appropriated number of clusters can be found by running the cluster analysis for different values of the number of clusters. The Bayesian information criterion (BIC) obtained at each configuration determines the number of clusters that provide the best fit to the given data set, thus pointing out the best choice [55]. If the BIC values are similar, as those obtained in this research, the practitioner will choose the number of clusters according to their previous experience and goals pursued. The state of the art revision performed by Sick [23] shows that the bulk of the literature tends to classify the wear level in two classes: either defective or adequate to work. In this work it has been preferred to distinguish amongst three different wear levels—low, medium and high—thus reflecting in a more faithful manner the frequently observed shape for the evolution of tool wear level (Fig. 2). This provides finer detail in the estimation of the actual wear status of the tool. Our approach corresponds with the results obtained by other authors [33]. Using the discriminant analysis techniques [56] let us test the quality of the results generated by the cluster analysis. For that purpose, the class labels obtained by the MCLUST algorithm are compared with the predictions of the linear (LDA) and quadratic (QDA) discriminant analysis. A measure of the quality of the fit can be obtained in terms of the Fowlkes—Mallows index [57]. Table 1 shows that QDA results are, as expected, better than those obtained by the LDA. On the other hand, no substantial difference can be observed amongst the different sets of descriptors as all of them provide excellent results when the QDA is considered. Nevertheless, as it will be shown soon, the LDA results will still provide us interesting insights on the nature of the data sets.
Fig. 2. Evolution of cutting tool wear.
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LDA QDA
MCLUST
assignments vs. LDA and QDA 4
Zernike
Legendre
Hu
Taubin
Flusser
0.935 1.000
0.852 0.997
0.917 0.966
0.750 0.993
0.784 0.971
The Fowlkes—Mallows indexes reflect that the data have structures easy to reproduce by distributional models, thus validating the quality of the clustering results in terms of suitability of the models to the data. However, as the wear itself has not been considered to perform the clustering, the clusters obtained might not necessarily display significant differences on the wear. This will depend on the amount of information related to wear contained by each family of descriptors. The following section elaborates on this issue. 3.2. Insights from the discriminant analysis Even though the QDA showed better performance, LDA has proved to be more adequate to show the correlation of the descriptors to the tool wear. LDA is a simplification of the QDA under the assumption of homocedasticity over the different groups, that is to say, the different clusters share a common variance–covariance matrix and they only differ on their location vectors. Considering this truth provides a simplification of the QDA formulas where the quadratic terms disappear and the resulting (LDA) formulas can be seen as a linear combination of the original vector space, or equivalently, a linear projection of the data onto a new vector space. Thus, QDA provides the possibility of capturing finer details due to the quadratic terms and usually provides higher accuracy at the numerical results, but no graphical representation can be directly obtain by the interpretation of its results. The LDA analysis, despite having an accuracy limited by the homocedasticity assumption, can be interpreted from the point of view of the change of basis matrix of its linear projection. Thus, QDA and LDA should not be seen as competitors but as complementary methods specialized in different tasks: QDA in providing higher accuracy for heteroscedastic structures, LDA providing graphical insights of these structures. The data, once projected onto the LDA map, can show the location of the different wear classes and, more importantly, the track of a cutting tool while the machining process proceeds. Fig. 3 shows such LDA projection of the data set as described by the Hu moments. The figure shows the areas where the data set concentrates and the borders amongst classes. The first class, the one located at the left, corresponds with the low wear level class; the medium wear level class lays in the center of the image; and the high level class is located at the right.
2
LD2
Table 1 Fowlkes–Mallows index of the predictions
0
-2
-4
-8
-6
-4
-2
0
2
4
LD1
Fig. 3. LDA projection of the images data as described by the Hu moments.
4
2
LD2
1010
0
-2
-4
-8
-6
-4
-2 LD1
0
2
4
Fig. 4. The evolution of the wear of a particular cutting edge as described by the Hu moments.
Fig. 4 shows the trace of a particular single cutting edge while evolves starting at the left in the low wear level class, passes through the medium wear level class and enters into the high wear level class at the right. There is a connection amongst the wear level and the information described by the Hu moments, since the cutting edge crosses the borders of the different classes during progress of the machining process. On the other hand, drawing the LDA map corresponding to the same data set as described by the Zernike
ARTICLE IN PRESS J. Barreiro et al. / International Journal of Machine Tools & Manufacture 48 (2008) 1005–1013
4
4 2 0 -2
LD2
moments, shows us (Fig. 5) that the three classes found are quite separated, not showing the behavior of a continuous process that evolves through the different classes. A look into the evolution of each cutting edge confirms this fact, as it can be seen for example in Fig. 6, where a particular single cutting edge remains in one of the classes during the whole machining process until its replacement. Table 2 shows the percentage of cutting edges that evolved through the three different classes during the machining process. As it can be seen, the information described by the Hu and Legendre moments of the wear region is directly related with the wear level while the rest of the moments considered seem to pay attention to other features than those related with the wear itself. The features with greater relevance in the discrimination process can be known with a closer look at the LDA results obtained with the Hu and Legendre moments; that is to say, which ones are more important in describing the wear level. This is due to the fact that the LDA can be seen as projection, as it has been shown, of the original data into a new vector space where the clusters are as compact and separated from each other as possible. The change of basis matrix that defines this linear mapping contains information about the impact of each descriptor on the direction of the new axes. Thus, the contribution of each descriptor to the first components of the LDA basis can be used as an indicator of the relationship between the information described by the descriptor and the wear level. Inspecting the results obtained by the LDA, it can be seen that in the case of the Legendre moments (Table 4) the information about the wear level is distributed amongst a
1011
-4 -6 -8 -10
-6h
-4h
-2h
0 LD1
2h
4h
6h
Fig. 6. The evolution of the wear of a particular cutting edge as described by the Zernike moments.
Table 2 Percentage of cutting edges that evolve through the detected clusters as an indicator of the relationship amongst the information provided by the descriptors and the wear level Hu
Legendre
Taubin
Flusser
Zernike
86
75
1
0
0
Table 3 Contribution (in %) of the HU descriptors to the first discriminant function h1
h2
h3
h4
h5
h6
h7
99.88
0.08
0.02
0.02
0.00
0.00
0.00
2
Table 4 Contribution (in %) of the Legendre descriptors to the first discriminant function
0
LD2
-2 -4
l^ 02
l^ 12
l^ 11
l^ 00
l^ 20
l^ 10
l^ 22
l^ 21
l^ 01
23.77
16.86
16.01
12.86
10.82
8.02
7.78
3.02
0.87
-6 -8 -10 -6
-4
-2
0 LD1
2
4
6
Fig. 5. LDA projection of the images data as described by the Zernike moments.
large number of descriptors; while in the case of the Hu moments (Table 3), it is remarkable that the first descriptor ðh1 ¼ n20 þ n02 Þ is by far the feature that contains the larger proportion ð99:88%Þ of information about the wear level. h1 being a combination of the second order statistical moments that describe the vertical and horizontal variance of the region, it is clear now that the discrimination amongst wear levels is mainly performed according to the changes in n20 and n02 .
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3.3. Monitoring wear level The former analysis allowed to gain a better understanding of the wear process in terms of the main factors that have impact over the discrimination amongst different wear levels. Those insights will now be applied to the machining process itself by determining a cutting tool replacement criterion. In fact, most of the work has already been done and it is only necessary to consider the former results from a different perspective or view. So as to prevent the machining process to be performed with a defective cutting tool, it should be replaced just before it enters into the high wear level class. Fortunately, it is possible to take advantage of the fine details the QDA provides. Being a Bayes’ theorem based technique, the QDA provides the probability to belong to each of the classes for a given cutting edge. Thus, following a procedure similar to that described by Castejo´n et al. [58], the evolution of the cutting tool can be traced as the value of the probability to belong to each class. The cutting tool will be replaced as soon as it is detected that the probability to belong to the medium wear level class diminishes and the probability to belong to the high wear level class increases over a defined threshold. 4. Conclusions An analysis of the tool wear process and the most adequate manners of describing it stands to reason according to the impact of the cutting tool replacement on the production costs. As a result of this research, it has been shown that not all of the analyzed descriptors based on moments are adequate to describe the wear level of cutting tools. Amongst them, the Hu descriptors [41] seem to be the better for that purpose. It is remarkable that the first Hu descriptor, h1 , is responsible of 99:88% of the discrimination amongst wear levels. A new tool life criterion is proposed that is more efficient than traditional approaches. Following the methodology described, the status of the cutting edge can be efficiently monitored, it is possible to prolong its useful life and the production costs are reduced significatively. The values for the maximum and average width on the flank wear land (VBmax and VBavg) were always higher when using our descriptors based methodology than those proposed by ISO3685 standard. The tool life criteria defined in ISO3685, that indicates if a tool is useful or non-useful, maps in our approach at values belonging to the medium wear class. In the case of VBmax, ISO standard postulates a limit value of 600 mm, whereas in our approach the limit value is 300 mm higher, in the case of simple region descriptors, and 730 mm higher, in the case of Legendre descriptors. Similarly, the VBavg is also higher using our approach. In this case, the limit value for VBavg is established at 300 mm in the ISO standard, whereas with our approach the value is 139 mm higher for Hu
descriptors, 280 mm higher for simple descriptors, and 376 mm higher for Legendre descriptors. Acknowledgments We gratefully acknowledge the financial support provided by the Spanish Ministerio de Educacio´n y Ciencia through grants DPI2006-14784 and DPI2006-02550, the Junta de Castilla y Leo´n through Grant no. LE018B06 and the Universidad of Leo´n through Grant no. ULE2005-01. References [1] S. Kurada, C. Bradley, A review of machine vision sensors for tool condition monitoring, Computers in Industry 34 (1997) 55–72. [2] R. Teti, Machining of composite materials, Annals of the CIRP 51 (2) (2002) 611–634. [3] A. Weckenmann, K. Nalbantic, Precision measurement of cutting tools with two matched optical 3d-sensors, Annals of CIRP 52 (1) (2003) 443–446. [4] J. Kopac, S. Sali, Tool wear monitoring during the turning process, Journal of Materials Processing Technology 113 (2001) 312–316. [5] ISO 3685:1993, Tool-life testing with single-point turning tools. [6] ISO 8688-1:1989, Tool life testing in milling—Part 1: face milling. [7] ISO 8688-2:1989, Tool life testing in milling–Part 2: end milling. [8] G. Lim, Tool-wear monitoring in machine turning, Journal of Materials Processing Technology 51 (1995) 25–36. [9] F. Al—Sulaiman, M. Baseer, A. Sheikh, Use of electrical power for online monitoring of tool condition, Journal of Materials Processing Technology 166 (2005) 364–371. [10] D. Axinte, N. Gindy, K. Fox, I. Unanue, Process monitoring to assist the workpiece surface quality in machining, International Journal of Machine Tools and Manufacture 44 (2004) 1091–1108. [11] M. Cakir, Y. Isik, Detecting tool breakage in turning AISI 1050 steel using coated and uncoated cutting tools, Journal of Materials Processing Technology 159 (2005) 191–198. [12] S. Choudhury, P. Srinivas, Tool wear prediction in turning, Journal of Materials Processing Technology 153–154 (2004) 276–280. [13] J. Gajewski, J. Jonak, Utilisation of neural networks to identify the status of the cutting tool point, Tunnelling and Underground Space Technology 21 (2006) 180–184. [14] N. Ghosh, Y. Ravi, A. Patra, S. Mukhopadhyay, S. Paul, A. Mohanty, A. Chattopadhyay, Estimation of tool wear during CNC milling using neural network-based sensor fusion, Mechanical Systems and Signal Processing 21 (1) (2007) 466–479. [15] W. Haili, S. Hua, C. Ming, H. Dejin, On-line breakage monitoring in turning, Journal of Materials Processing Technology 139 (2003) 237–242. [16] E. Kuljanic, M. Sortino, Twem, a method based on cutting forces— monitoring tool wear in face milling, International Journal of Machine Tools and Manufacture 45 (2005) 29–34. [17] M. Ritou, S. Garnier, B. Furet, J. Hascoet, A new versatile in-process monitoring system for milling, International Journal of Machine Tools and Manufacture 46 (15) (2006) 2026–2035. [18] D. Salgado, F. Alonso, Tool wear detection in turning operations using singular spectrum analysis, Journal of Materials Processing Technology 171 (2006) 451–458. [19] C. Sceffer, P. Heyns, An industrial tool wear monitoring system for interrupted turning, Mechanical Systems and Signal Processing 18 (2004) 1219–1242. [20] H. Shao, H. Wang, X. Zhao, A cutting power model for tool wear monitoring in milling, International Journal of Machine Tools and Manufacture 44 (2004) 1503–1509.
ARTICLE IN PRESS J. Barreiro et al. / International Journal of Machine Tools & Manufacture 48 (2008) 1005–1013 [21] J. Jurkovic, M. Korosec, J. Kopac, New approach in tool wear measuring technique using ccd vision system, International Journal of Machine Tools and Manufacture 45 (9) (2005) 1023–1030. [22] S. Kurada, C. Bradley, A machine vision system for tool wear assessment, Tribology International 30 (4) (1997) 295–304. [23] B. Sick, On-line and indirect tool wear monitoring in turning with artificial neural networks: a review of more than a decade of research, Mechanical Systems and Signal Processing 16 (4) (2002) 487–546. [24] W. Wang, Y. Wong, G. Hong, Flank wear measurement by successive image analysis, Computers in Industry 56 (2005) 816–830. [25] E. Alegre, J. Barreiro, H. Ca´ceres, L. Herna´ndez, R. Ferna´ndez, M. Castejo´n, Design of a computer vision system to estimate tool wearing, Materials Science Forum 526 (2006) 61–66. [26] M. Lanzetta, A new flexible high-resolution vision sensor for tool condition monitoring, Journal of Materials Processing Technology 119 (1–3) (2001) 73–82. [27] M. Sortino, Application of statistical filtering for optical detection of tool wear, International Journal of Machine Tools and Manufacture 43 (2003) 493–497. [28] W. Wang, G. Hong, Y. Wong, Flank wear measurement by a threshold independent method with sub-pixel accuracy, International Journal of Machine Tools and Manufacture 46 (2) (2006) 199–207. [29] T. Obikawa, J. Shinozuka, Monitoring of flank wear of coated tools in high speed machining with a neural network art 2, International Journal Machine Tools and Manufacture 44 (2004) 1311–1318. [30] D. Brezak, T. Udiljak, K. Mihoci, D. Majetic, B. Novakovic, J. Kasac, Tool wear monitoring using radial basis function neural network, Neural Networks 3 (2004) 25–29. [31] G. Littlefair, M. Javed, G. Smith, Fusion of integrated multisensor data for tool wear monitoring, Neural Networks 2 (1995) 734–737. [32] V. Sunil, C. Yung, Design and implementation of tool wear monitoring with radial basis function neural networks, American Control Conference 3 (1995) 1722–1726. [33] H. Guo, G. Xiong, F. Xi, J. Zheng, X. Ping, Detection and compensation of cutting tool wear based on Lipschitz exponent, in: Proceedings of the 2005 IEEE Conference on Control Applications, 2005. [34] L. Herna´ndez, H. Ca´ceres, J. Barreiro, E. Alegre, M. Castejo´n, R. A. Ferna´ndez, Monitorizacio´n del desgaste de plaquitas de corte usando visio´n artificial, Congreso Iberoamericano de Ingenierı´ a Meca´nica (2005). [35] C. Teh, R.T. Chin, On image analysis by the methods of moments, IEEE Transactions on Pattern Analysis and Machine Intelligence 10 (4) (1988) 498–514. [36] A. Choksuriwong, H. Laurent, B. Emil, Comparison of invariant descriptors for object recognition, in: IEEE International Conference on Image Processing, 2005, pp. 377–380. [37] W. Kim, P. Yuan, A practical pattern recognition system for translation, scale and rotation invariance, IEEE (1994) 391–396. [38] E. Alegre, L. Sa´nchez, R. Alaiz, J. C. Rodrı´ guez, Utilizacio´n de momentos estadı´ sticos y redes neuronales en la clasificacio´n de cabezas de espermatozoides de verraco, XXV Jornada de Automatica (2004).
1013
[39] J.S. Park, D. Chang, 2D invariant descriptors for shape-based images retrieval, Pattern Recognition Letters (2001). [40] R.J. Prokop, A.P. Reeves, A survey of moment-based techniques for unoccluded object representation and recognition, CCVGIP: Graphical Models and Image Processing 54 (5) (1992) 438–460. [41] M. Hu, Visual pattern recognition by moment invariants, IRE Transactions on Information Theory (1962) 179–182. [42] A. Khotanzad, Y.H. Hong, Invariant image recognition by Zernike moments, IEEE Transaction on Pattern Analysis and Machine Intelligence 22 (1) (1990) 489–497. [43] G. Taubin, D. Cooper, Object recognition based on moment (or algebraic) invariants, Geometric Invariance in Computer Vision (1992) 375–397. [44] J. Flusser, T. Suk, On the calculation of image moments, Research Report no. 1946. [45] J. Flusser, B. Zitovi, T. Suk, Invariant–based registration of rotated and blurred images, IEEE (1999) 1262–1265. [46] C. Mene´ndez, J.B. Ordieres, F. Ortega, Importance of information pre-processing in the improvement of neural network results, Expert Systems 13 (2) (1996) 95–102. [47] C. Gonzalez Nicieza, J.B. Ordieres Mere, A. Menendez Diaz, J. Taboada Castro, Optimizing of stripping in a slate bank, International Journal of Surface Mining and Reclamation 8 (1) (1994) 17–22. [48] A. Bello Garcı´ a, A. Mene´ndez Dı´ az, J.B. Ordieres Mere´, C. Gonza´lez Nicieza, Generalization of the influence function method in mining subsidence, International Journal of Surface Mining, Reclamation and Environment 10 (4) (1996) 195–202. [49] A.V. Pernı´ a-Espinoza, J.B. Ordieres-Mere´, F.J. Martı´ nez-de-Piso´n, A. Gonza´lez-Marcos, TAO–robust backpropagation learning algorithm, Neural Networks 18 (2) (2005) 191–204. [50] J.B.O. Mere´, A.G. Marcos, J.A. Gonza´lez, V.L. Rubio, Estimation of mechanical properties of steel strip in hot dip galvanising lines, Ironmaking and Steelmaking 31 (1) (2004) 43–50. [51] T. Hastie, R. Tibshirani, J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Springer, Berlin, 2001. [52] J. Banfield, A. Raftery, Model-based Gaussian and non-Gaussian Clustering, Biometrics 49 (1993) 803–821. [53] C. Fraley, A.E. Raftery, Enhanced software for model–based clustering, discriminant analysis and density estimation: Mclust, Journal of Classification 20 (2003) 263–286. [54] G. McLachlan, T. Krishnan, The EM Algorithm and Extensions, Wiley, New York, 1997. [55] G. McLachlan, D. Peel, Finite Mixture Models, Wiley, New York, 2000. [56] G. McLachlan, Dis criminant Analysis and Statistical Pattern Recognition, Wiley, New York, 2004. [57] M. Meila, Comparing clusterings, Technical Report, University of Washington (2002). [58] M. Castejo´n, E. Alegre, J. Barreiro, L. Herna´ndez, On-line tool wear monitoring using geometric descriptors from digital images, International Journal of Machine Tools and Manufacture 47 (2007) 1847–1853.