SVD ANALYSIS OF A MACHINED SURFACE IMAGE FOR THE TOOL WEAR ESTIMATION

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1 5th International DAAAM Baltic Conference "INDUSTRIAL ENGINEERING ADDING INNOVATION CAPACITY OF LABOUR FORCE AND ENTREPRENEURS" 0 April 006, Tallinn, Estonia SVD ANALYSIS OF A MACHINED SURFACE IMAGE FOR THE TOOL WEAR ESTIMATION Zawada-Tomiewicz, A., Storch, B. Abstract: Image of a machined surface is the perspective projection of a magnified surface on a converter plane. Cutting condition deterioration and tool wear influence the image of the machined surface. Change in image is distinctly noticeable from the energetic point of view. Singular value decomposition (SVD) method is a factorisation technique which effectively reduces machined surface image into a smaller portion of data. Analysis of the eigenvalues of a machined surface image maes possible their application in the estimation of tool wear. Key words: machined surface, tool wear, eigenvalue analysis, digital image.. INTRODUCTION The manufacturing industry undergoes changes due to the increased global competition. Therefore there is a need for extensive research focused on new control and monitoring methods. For the efficient and reliable operation of automated machining processes, the implementation of tool condition monitoring strategy is required. The estimation of tool wear is a complex tas because tool wear introduces very small changes in a process with a very wide dynamic range. Furthermore, it is difficult to identify whether a change in a signal is caused by tool wear or a change in the cutting conditions [ 6,7,8,0 ]. Tool condition monitoring which is defined as the ability to distinguish between a sharp, a semi-dull, or a dull tool can be successfully accomplished by evaluating CCD data of a machined surface [ 6,9 ]. In this paper it is presented research focused on the application of vision system in the tool condition monitoring system. The vision system consisted of a computer with a frame grabber card, a digital camera, lenses, a stand for a camera with movable wortable and a lighting system. Obtaining the image was the first step in an estimation process. Then there was performed image processing with feature extraction based on a calculation of image indexes with the use of statistical methods, image texture description methods, fractals and wavelets [ 9, ]. Singular value decomposition (SVD) is a multivariate statistical technique which was developed in the context of numerical analysis, but other applications have been investigated in various fields such as signal and image processing [ 4,5 ]. In a machined surface image data exhibit large spatial correlations. SVD analysis results in a more compact representation of these spatial correlations, especially with multivariate datasets and can provide insight into spatial variations exhibited in the fields of surface image data being analyzed.. SVD OF A MACHINED SURFACE IMAGE Machined surface is a combination of roughness, waviness, lay and flaws. In the machined surface image there can be distinguished periodic mappings of the cutting tool edge modulated by tool feed disturbances. These mappings change with time and tool wear (Fig. ). 7

a) b) c) d) Fig.. Machined surface image for feed 0,mm/rev, cutting speed 50m/min and various cutting time 0, 0.5, and 6 min, Tool condition changes with each turn of a worpiece.

A machined surface and a tool edge are the result of two-way interaction.

We consider a surface image point z = z( x, y), which depends on a spatial, x and y variables.

.. z( x, ym ) z( x y ) z( x y ( ) ( Z =,..., M ) ()......... z xn, y... z xn, ym ) where M and N are respectively the number of pixels in horizontal and vertical directions and it is assumed that M>=N.

Image data in their original form can be used for the visualisation but they should be processed before applying them in a monitoring system.

2 a) b) c) d) Fig.. Machined surface image for feed 0,mm/rev, cutting speed 50m/min and various cutting time 0, 0.5, and 6 min, Tool condition changes with each turn of a worpiece. From the time when the tool starts woring it begins wearing. Traces of wear which occur on the tool faces depend mainly on the tool geometry and material, worpiece material, cutting parameters. A machined surface and a tool edge are the result of two-way interaction. The surface image after turning reflects all the changes and interactions that co-exist during the formation of the surface [ 8,9 ]. We consider a surface image point z = z( x, y), which depends on a spatial, x and y variables. The measurement process consists of sampling the quantity z at different spatial locations, leading to a matrix of data describing the intensity of colour in each pixel, z( x, y)... z( x, ym ) z( x y ) z( x y ( ) ( Z =,..., M ) () z xn, y... z xn, ym ) where M and N are respectively the number of pixels in horizontal and vertical directions and it is assumed that M>=N. The digital images obtained from the CCD camera contain a large number of data. Image data in their original form can be used for the visualisation but they should be processed before applying them in a monitoring system. Therefore the image data are transformed into various representations suitable for image analysis. As there is a great correlation between the elements of the machined surface image, matrix Z can be decomposed as a product T Z = U S V () U and V are respectively N x M and M x M the unitary matrices T T U U = V V = I and S is an M x M diagonal matrix that consists of the eigenvalues of the matrix Z (Fig. ). The eigenvalues weights S can be positive or equal to zero values. They are sorted in descending order such that S S... S K S K. The weights S are called singular values. The size of the expansion K is equal to the ran of the image matrix Z, which in practice corresponds to the smallest value of N and M. Fig.. SVD of the machined surface image Z The orthonormality condition in matrix notation stated: T T u ul = v vl = δl (3) From () and (3) it can be shown that components u and v satisfy the eigenequations of Z : T Ax = Z Z, Ax u = S u (4) T Ay = Z Z, Ay u = S v (5) The scatter matrices A x and Ay do not necessarily have the same size, unless M=N. However, they have the same ran and their K largest eigenvalues are identical. From (4) and (5) it can be shown that SVD projects the data onto an orthonormal basis along which both spatial coherences of the signals are maximised. In spatial scatter 7

3 matrix A x, the elements are the space average of the product of the signal at different positions M ( A x ) ij = z ( xi, y ) z ( x j, y ) (6) = Equation (5) defines the space-covariance matrix of the image Z in the direction x, thus the eigenvector, corresponding to the maximum eigenvalue in (4), is the vector along which the spatial covariance is maximised. Similarly, the scatter matrix A y represents the spatial average of the product of the signals at different points such as N ( A y ) = ( i ) ( ij z x, y z x, y j ) (7) = defines the space-covariance matrix of the data matrix Z in the direction y. The eigenvector, corresponding to the maximum eigenvalue in (5), is the vector along which the covariance is maximised. SVD decomposition of the surface image shown in Fig. a is presented in Table. There can be seen a few components of the image. The first structure is the single pattern that represents the most variance in the data. In the following vectors of the matrix U there can be distinguished the averaged changes in a profile at the angle parallel to the lay direction in the image. Table. Decomposition of the surface image (Fig.a) for the first and twentieth components Image U S (=) 395 (=) 37 (=0) 7 V The structures are the elements of the eigenvectors of the variance-covariance matrix of the image. The first eigenvector points to the direction in which the data vectors jointly exhibit the most variability. A new coordinate system is created, with each axis aligned along the direction of maximum joint variability. The second structure V is the pattern that describes the second largest amount of variance, calculated the same way as the first structure. In the following vectors of the matrix there was shown the averaged profile of a machined surface digital image at the angle perpendicular to the lay direction in the image, in pitch surface generator. A very important property of the second structure is that it is completely uncorrelated with the first structure. All the structures are mutually uncorrelated. 3. PARAMETERS OF A MACHINED SURFACE IMAGE Singular values are strongly ordered and the largest one exceeds the smallest one by a few orders of magnitude. This allows the signal to be reconstructed with the use of only the most significant biorthogonal components. This ability of the SVD technique to concentrate dominant features into few spatial modes maes it well suited for the analysis of spatially extended systems. Several parameters allow to quantify the singular value distribution and the degree of compressibility of the data set. Global energy can be defined as N M E = ( Z ij ) (8) i= j= which is equal to the sum of the squared singular values K E = S (9) = The dimensionless energy S p = (0) E 73

4 measures the relative amount of energy, which is stored in each component and it can be a useful parameter for comparing different image data. The variance of the nth main component is the nth eigenvalue. Therefore, the total variation exhibited by the image data is equal to the sum of all eigenvalues. Eigenvalues are normalized such that the sum of all eigenvalues equals to. A normalized eigenvalue will indicate the percentage of total image variance explained by its corresponding structure. Structures have also been normalized so that the root mean square equals to. Fig. 3. Normalised cumulative energy of the first 40 main components of images presented in table (full line for the cutting time equal to 6 min and dashed line for the cutting time equal to 0 min) The first few eigenvectors point in directions where the data jointly exhibit large variation. For this reason, it is possible to capture most of the variation by considering only the first few eigenvectors. As can be seen in Fig. 3 most of the energy of the machined surface image is focused in a few first components. About 96% of the image variation is focused only in the first component and 99% of it is focused in the first 0 components. The remaining eigenvectors, along with their corresponding main components were truncated. The ability of SVD to eliminate a large portion of the data was a primary reason for its use in tool wear estimation. Comparisons between original image data and truncated ones reveal a great similarity. Original images contain more details, more high frequencies but the general character of the data remains the same. A single image which was acquired with the use of a vision system consisted of 68x58 pixels and could be described by almost 40 thousands of twenty-four bit numbers. The input data were of great redundancy that was reduced by the conversion an RGB image into a grey scale one. Accordingly to the previous research [ 9 ] where there had been checed various portions of data for their usability in tool flan wear estimation, there was applied only a part of data from the central part of the image. But this subimage was chosen large enough to preserve sufficient information about the whole image. To fulfil this assumption there ware conducted experiments and was chosen subimage equal to 64 x 64 pixels. Such an image contained enough energy and was treated as sufficiently representative. In Fig 4 there were presented spatial structures for such subimage from the central part of the image shown in a figure a. Fig. 4. Spatial components U*S for the matrix of the size 64 x 64 from the central part of the image 3. TOOL WEAR ESTIMATION BASED ON A MACHINED SURFACE IMAGE Tool flan wear, which mainly influences machined surface, is usually described by the VB parameter. It is the width of abrasion on the minor flan face. 74

Accordingly to the experiment there were machined surface digital images acquired with the frequency of minute. These images constitute the base of a monitoring system.

5 There was measured VB parameter of tool flan wear for the uncoated sintered carbides inserts with the use of an optical system (Fig. 5). All the measurements were experimentally designed where the cutting parameters were chosen to be appropriate for precise and medium precise turning. Fig. 5. The VB parameter of the tool flan wear. Accordingly to the experiment there were machined surface digital images acquired with the frequency of minute. These images constitute the base of a monitoring system. As was stated earlier most of the information of the machined surface is contained in singular values of SVD decomposition. These first 0 singular values calculated on a base of data from the central part of the image were chosen as features describing machined surface image. These values were applied as the input of the neural networ estimator. Neural networ was applied to fulfil two goals. The main objective was the estimation of a tool wear VB parameter. Additional goal was to eliminate redundant features from the input set. The problem of feature selection in case of applying the neural networ became reduced to the problem of optimization of the input layer i.e. selection of such inputs, which changed, produce the biggest output error during training and eliminate those input, which produce small change of output error. There have been applied the Optimal Brain Surgeon (OBS) Method for neural networ structure optimization [,, ]. There was used feed-forward neural networ composed of three layers. It was trained with the use of Lavenberg- Marquard algorithm. The number of neurons in an input layer was equal to number of image features. The neurons in the hidden layer had hyperbolic tangent activation function; the single neuron in the output layer had a linear activation function. The data were divided into training and testing sets. The number of examples in a training set was two times bigger then in a testing set. The sets came from two independent experiments. During the optimization procedure the neural networ was trained several times and after the process of learning the input layer was pruned. Each time the pruned weights were counted again. At the end of the optimization procedure there were a ran of each input neuron determined. Neurons from input layer, which weights were frequently pruned, were rejected. At the beginning, the networ was composed of twenty neurons in an input layer, five neurons in hidden layer and one neuron in an output layer (0 5 ). There were carried out fifty pruning experiments. During each experiment the networ was trained and after training process the weights were pruned. The results of these experiments are presented in Fig. 6. Each bar represents the normalized sum of connections that were not pruned during fifty experiments. The higher the bar the more important feature was for the tool wear estimation. The neurons which output signals were frequently eliminated by pruning were removed. The networ was reduced to the structure 0 5. normalized feature weight 0, 0,08 0,06 0,04 0, feature number Fig. 6. Normalized feature weight (normalized sum of neuron connections which were not prunned) Limiting Absolute Error was applied as a measure of the estimator structure quality. During the optimization procedure when the number of input features was reduced the Limiting Absolute Error for the training 75

6 set increased and for the testing set consequently declined. Further reduction of the networ structure caused significant increase of the error. This reasoning made it possible to come to the conclusion that ten features had the greatest influence on the output error. There have been tested a neural networ as an estimator of a VB tool wear parameter from a digital image of a machined surface. Limiting Absolute Error of estimation for uncoated sintered carbides was less then 0. mm. The results of the estimation authorize one to draw a conclusion that digital images of a machined surfaces contain enough information of a tool flan wear to apply them in monitoring. 4. FINAL CONCLUSIONS An innovative methodology for tool wear estimation have been presented. There is enormously difficult to obtain information of a tool wear in a cutting zone. There was used optical method for the measurement of a machined surface. There was applied vision system to acquire machined surface digital image. This image was decomposed with the use of singular value decomposition. The eigenvectors were used as features describing the image. For the tool flan wear estimation there have been applied neural networs. The results of estimation confirm the usability of a SVD in a digital machined surface image description. 5. REFERENCES. Hassibi B., Stor D., Second Order Derivatives for Networ Pruning: Optimal Brain Surgeon, Neural Information Processing Systems 9, 99. Ljung L., System identification. Theory for the user. Prentice-Hall, Inc Otto, T. Kuri, L., A digital measuring module for tool wear estimation. 3th DAAAM International Symposium 3-6th October Salgado D.R., Alonso F.J., Tool wear detection in turning operations using singular spectrum analysis, Journal of Materials Processing Technology 7, , Shuxin Gu, Jun Ni, Jingxia Yuan, Nonstationary signal analysis and transient machining process condition monitoring, International Journal of Machine Tools & Manufacture 4 4 5, Sic B., Review. On-line and indirect tool wear monitoring in turning with artificial neural networs. W review of a more than a decade of research. Mechanical Systems and Signal Processing, 6 (4), , Storch B.: The principals of machining (in Polish), Technical University of Koszalin Academic Press, Koszalin, Storch B.: Influence between the corner of the edge and machined material (in Polish), Monograph 8, Technical University of Wrocław, Zawada Tomiewicz A., Application of digital image features of machined surfaces for tool wear monitoring in machining (in Polish), PhD thesis, Technical University of Koszalin, Koszalin, Zawada Tomiewicz A., Storch B.: Classifying the wear of turning tools with neural networs, Journal of Materials Processing Technology 09, pp , 00.. Zawada-Tomiewicz A., Tomiewicz D., The Application of Optimal Brain Surgeon Method for optimization of tool wear estimator structure, Polioptymalizacja i Komputerowe Wspomaganie Projetowania, Vol. III, WNT, Warszawa, 8-5, 004 Corresponding author: Anna Zawada-Tomiewicz, PhD, TU Koszalin, Poland Phone: ,Fax: , annaz@tu.oszalin.pl 76

Image Compression with Singular Value Decomposition & Correlation: a Graphical Analysis

ISSN -7X Volume, Issue June 7 Image Compression with Singular Value Decomposition & Correlation: a Graphical Analysis Tamojay Deb, Anjan K Ghosh, Anjan Mukherjee Tripura University (A Central University),