5th International DAAAM Baltic Conference
"INDUSTRIAL ENGINEERING – ADDING INNOVATION CAPACITY OF LABOUR
FORCE AND ENTREPRENEURS"
20–22 April 2006, Tallinn, Estonia
SVD ANALYSIS OF A MACHINED SURFACE IMAGE
FOR THE TOOL WEAR ESTIMATION
Zawada-Tomkiewicz, 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 makes possible their application in
the estimation of tool wear.
Key words: machined surface, tool wear,
eigenvalue analysis, digital image.
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 worktable 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 [9,11].
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.
1. 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 task 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 [6,7,8,10].
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].
2. 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. 1).
171
a)
b)
c)
d)
As there is a great correlation between the
elements of the machined surface image,
matrix Z can be decomposed as a product
(2)
Z  U  S V T
U and V are respectively N x M and M x
M
the
unitary
matrices
U T  U  V T  V  I and S is an M x
M diagonal matrix that consists of the
eigenvalues of the matrix Z (Fig. 2).
The eigenvalues – weights S k can be
positive or equal to zero values. They are
sorted in descending order such that
S1  S 2  ...  S K 1  S K .
Fig. 1. Machined surface image for feed
0,21mm/rev, cutting speed 250m/min and
various cutting time 0, 0.5, 1 and 6 min,
The weights S k are called singular values.
The size of the expansion K is equal to the
rank of the image matrix Z , which in
practice corresponds to the smallest value
of N and M.
Tool condition changes with each turn of a
workpiece. From the time when the tool
starts working it begins wearing. Traces of
wear which occur on the tool faces depend
mainly on the tool geometry and material,
workpiece 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  x1, y1 
 zx , y 
Z 2 1
...
 zx , y 
 N 1
...
...
...
...
z  x1, y M  
z x2 , y M  

...
z  x N , y M 
Fig. 2. SVD of the machined surface image Z
The orthonormality condition in matrix
notation stated:
(3)
ukT  ul  vkT  vl   kl
From (1) and (3) it can be shown that
components u k and v k satisfy the
eigenequations of Z :
Ax  Z T  Z ,  Ax  uk  Sk2uk (4)
Ay  Z  Z T ,  Ay  uk  S k2 vk (5)
The scatter matrices Ax and A y do not
necessarily have the same size, unless
M=N. However, they have the same rank
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
(1)
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.
172
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.
matrix Ax , the elements are the space
average of the product of the signal at
different positions
M
 Ax ij   z xi , yk  z x j , yk (6)
k 1
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   z xk , yi  z xk , y j (7)
ij
k 1
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. 1a is presented in Table 1.
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.
 


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 makes 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
Table 1. Decomposition of the surface image
(Fig.1a) for the first and twentieth components
Image
Uk
Sk
Vk
E
(k=1)
  Zij 2
N M
(8)
i 1 j 1
395
which is equal to the sum of the squared
singular values
K
E   S k2
(9)
k 1
The dimensionless energy
S k2
(10)
pk 
(k=2)
37
(k=20)
7
E
173
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 1. 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 1.
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
681x582 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 checked various
portions of data for their usability in tool
flank 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 1a.
Fig. 3. Normalised cumulative energy of the
first 40 main components of images presented
in table 2 (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 20
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.
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 flank wear, which mainly influences
machined surface, is usually described by
the VB parameter. It is the width of
abrasion on the minor flank face.
174
There was measured VB parameter of tool
flank 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.
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 network 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
rank of each input neuron determined.
Neurons from input layer, which weights
were frequently pruned, were rejected.
At the beginning, the network was
composed of twenty neurons in an input
layer, five neurons in hidden layer and one
neuron in an output layer (20–5–1). There
were carried out fifty pruning experiments.
During each experiment the network 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 network was
reduced to the structure 10–5–1.
Fig. 5. The VB parameter of the tool flank
wear.
normalized feature weight
Accordingly to the experiment there were
machined surface digital images acquired
with the frequency of 1 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 20 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 network estimator.
Neural network 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 network 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
network structure optimization [1,2,11].
There was used feed-forward neural
network composed of three layers. It was
trained with the use of LavenbergMarquard 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
0,1
0,08
0,06
0,04
0,02
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
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
175
set increased and for the testing set
consequently declined.
Further reduction of the network 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 network 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
0.1 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 flank
wear to apply them in monitoring.
4. Salgado D.R., Alonso F.J., Tool wear
detection in turning operations using
singular spectrum analysis, Journal of
Materials Processing Technology 171,
451–458, 2006
5. Shuxin Gu, Jun Ni, Jingxia Yuan, Nonstationary signal analysis and transient
machining
process
condition
monitoring, International Journal of
Machine Tools & Manufacture 42 41–
51, 2002
6. Sick B., Review. On-line and indirect
tool wear monitoring in turning with
artificial neural networks. W review of
a more than a decade of research.
Mechanical Systems and Signal
Processing, 16 (4), 487-546, 2002
7. Storch B.: The principals of machining
(in Polish), Technical University of
Koszalin Academic Press, Koszalin,
2001
8. Storch B.: Influence between the corner
of the edge and machined material (in
Polish), Monograph 8, Technical
University of Wrocław, 1994.
9. Zawada–Tomkiewicz A., Application
of digital image features of machined
surfaces for tool wear monitoring in
machining (in Polish), PhD thesis,
Technical University of Koszalin,
Koszalin, 2002
10. Zawada–Tomkiewicz A., Storch B.:
Classifying the wear of turning tools
with neural networks, Journal of
Materials Processing Technology 109,
pp. 300–304, 2001.
11. Zawada-Tomkiewicz A., Tomkiewicz
D., The Application of Optimal Brain
Surgeon Method for optimization of
tool
wear
estimator
structure,
Polioptymalizacja i Komputerowe
Wspomaganie Projektowania, Vol. III,
WNT, Warszawa, 218-225, 2004
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
flank wear estimation there have been
applied neural networks. The results of
estimation confirm the usability of a SVD
in a digital machined surface image
description.
5. REFERENCES
1. Hassibi B., Stork D., Second Order
Derivatives for Network Pruning: Optimal Brain Surgeon, Neural Information
Processing Systems–92, 1992
2. Ljung L., System identification. Theory
for the user. Prentice-Hall, Inc. 1987
3. Otto, T. Kurik, L., A digital measuring
module for tool wear estimation. 13th
DAAAM International Symposium
23-26th October 2002
Corresponding author:
Anna Zawada-Tomkiewicz, PhD, TU
Koszalin, Poland Phone: +4894
3478451,Fax: +4894 3426753, E-mail:
[email protected]
176