We propose a process control and analysis technique for two-dimensional, irregular- shaped parts whose digitized images are transformed of a discrete wavelet ...
WAVELET TRANSFORMS FOR QUALITY ENGINEERING APPLICATIONS Liang Jin and Shing I Chang Industrial and Manufacturing System Engineering Kansas State University 237 Durland Hall, Manhattan KS 66502,USA Phone:(785) 532-5606 Fax (785) 532-7810
ABSTRACT We propose a process control and analysis technique for two-dimensional, irregularshaped parts whose digitized images are transformed of a discrete wavelet transform (DWT) algorithm. Image processing methods are applied to obtain two-dimensional part profiles and changed to one-dimensional boundary discrete signal in order to simplify computation complexity. Wavelet transform is employed to extract “signature” of the signal. Wavelet coefficients, which are arranged in layers, are chosen for the signal compact representation. Simulation tests are made to study the relationship between sample size and wavelet coefficient statistics that are useful to distinguish dimensional variations among parts due to production inperfection. Initial results show that this technique is very promising for automatic inspection and quality control in modern production environments. Keywords: SPC , Discrete Wavelet Transform(DWT), Multiresolution Analysis 1. INTRODUCTION Part shape recognition and inspection is one of basic problems in the pattern recognition of image processing field and has wide applications in industrial. Many vision-based techniques have been applied to achieve the goal and are continuously evolving. Persoon (1977) developed Fourier Descriptor to discriminate shapes. Ballard (1981) used Hough transform to detect arbitrary shapes. Mokhtarian (1986) applied varying levels of details for the planar curves and two-dimensional shapes analyses. Sekita (1992) proposed Complex Autoregressive (CAR) model to recognize shapes by describing a boundary of the pattern. Chen and Ventura (1992) developed a machine vision algorithm for the automated part inspections on two dimension regular shapes. Grinde (1994) dealt with the inspection of spherical and elliptical profiles parts. Nakamura's (1994) algorithm can learn two dimension shapes by using wavelet local extrema. Tieng (1997) recognized two-dimensional object contour using wavelet transform’s zero-crossing representatio. Lu (1998) used part pixel difference as an inspection index. In this study, we investigate possible uses of a discrete wavelet transform (DWT) technique for inspection and process control on two-dimentional part profiles. Specifically, statistics analyses are done based on the transformed coefficients rather than
1
on the part dimensions. That is the measurement and analyses are now changed from spatial to frequency domain to capture the small dimensional variation which is not easily measured directly in spatial domain. A simulated ellipse will be used as an example although the approaches developed can be adopted in other two-dimensional industrial parts for inspection and process control. In order to reduce the computation complexity to have more efficiency, we do not apply two-dimensional part image wavelet transform directly but first change a twodimensional part image to a one-dimensional signal in a specific format and then apply wavelet transform. In this way ,we can not only uniquely represent two-dimensional part profiles more efficiently but also achieve calculation simplicity. 2. Transforming Two-dimensional Part profile to One-dimensional Signal By using edge detection and tracking (Freeman,1981), we can reduce the dimensionality and get an ordered boundary coordinates of two dimension image parts. Radial Function (Chetverikov, 1992 ) can represent such a boundary by an ordered sampling sequence as angularly equipspaced radius vector and its angle: (r j ,θ j ), (θ j = 2πj / N )
Where each radius vector is the length of each of N radius vectors between the boundary coordinates and boundary centroid. If any of the radial vector intersects the object boundary more than once, radius vector will be multivalued. To overcome this restriction, the Modified Radial Function(MRF)( Tieng. Q, 1997 ) is used to approximate the boundary by radius vectors between boundary coordinates and its centroid as the function of another parameter which varies from zero to the length of the perimeter of the part. So in this way, any two dimension part profile can be transformed to one dimension boundary signal. 3. One Dimension DWT Wavelets are building blocks of wavelet transform, just as trigonometric functions of different frequencies are building blocks used in Fourier transforms (Palavajjhala, 1994). They are generated by the dilation and translation of a single prototype:
ψ
a ,b
(t ) = 1 / a *ψ ((t − b) / a )
Where a is scaling factor and b is translation factor respectively In case of discrete wavelet transformation(DWT),a, b should take only discrete values.
a = a0
b = nb0 a 0
m
m
Therefore:
ψ
m ,n
(t ) = a0
− m/ 2
−m
ψ (a0 t − nb0 ) 2
A particular interesting class of wavelets are orthonormal wavelets which are linearly independent, complete and orthogonal. This means that there is no “redundant” data to represent the signal. Daubechies (1988) developed conditions under which form the orthonormal bases. Thus the Discrete Wavelet Coefficients are the inner products of the signal and wavelet function. That is: f (t ) =
∑ψ
m ,n∈ Z
m, n
(t ) < f (t ),ψ
m, n
(t ) >
If b 0 = 1 a 0 = 2 The reconstruction can use: < f (t ),ψ
− m/2 m .n (t ) >= 2
∞
∫f (t )ψ
(2 − m t − n)dt
−∞
There are several wavelets with good properties(smooth, fast decay, compactly) such as Haar ,Meyer , Battle-Lemarie , and the most commonly used Daubechies. In practice, any signal is measured at a finite resolution, and therefore, the value m should be ranged [0,L], in which m=0 corresponds to the finest scale while m=L to the coarsest scale. This so called multiresolution analysis allow us to look a signal at different scale by “zooming in “ or “zooming out”, that is, an approximation of a given signal at low resolution can go to an approximation at immediate higher resolution just by adding some “details” information. We can see clearly this from following multiresolution decomposition by representing reconstruction formula as: f (t ) = φL,n (t ) < f (t ),φL ,n (t ) > +
∑ψ
m ,n m∈ [1, L ], n∈ Z
(t ) < f (t ),ψ
m ,n
(t ) >
Mallat (1989) developed a fast wavelet algorithm based on the pyramid algorithm of Burt and Adelson (Burt, 1983 ). The basic components in each stage of the pyramid are two analysis filters: a low-pass filter H and high-pass filter G and a decimation by two operation. Fig.1 shows the block diagram of such decomposition: x
Hx
HHx
Gx
HGx
HHGx HHHx
3
Fig. 1 Mutiresolution Analysis by Decomposition of a Signal The output of the algorithm is the set of detail coefficient(D) and approximation coefficients(A) at each level as following: D j = (d j ,1 , d j , 2 ..., d j ,n j ) AJ = (a J ,1 , a J ,2 ..., a J ,nJ )
Where j = 1,2,..., J
nj = n / 2 j ,
4. STUDIES OF WAVELET COEFFICIENT We use ellipse(10 x 4) as our example two dimension parts because of its simple and easily controllable parameters. Boundary points are sampled from X0 at every 6 degree and 60 points are obtained as shown Fig.2. ( r j ,θ j ) 4 X0 10
Fig. 2 RF Representation of Ellipse Using above mentioned RF method, one dimension radius vector signal is shown as Fig.3.
55
49
43
37
31
25
19
13
7
15 10 5 0 1
Radius Vector(r)
1D Radius Vector Signal
Angle(theta)
Fig. 3 two dimension Radius Vector Signal Zeros are padded of the signal to the length of 64 points before dyadic wavelet transform is implemented. Daubechies 8 wavelet is chosen for our experiment. Usually the finer levels always contain undesired noise so the coefficients from those levels are discarded and only coefficients from the rest levels are used as the feature signature of the signal. In our experiment, level 3, 4, 5, 6 are chosen. By calculation, around 95% energy is used to represent original signal from the point of energy.
4
If we try to monitor short axis of an ellipse. Assume the process is capable(e.g. Cp=1.3), Upper Specification Limit(USL) and Lower Specification Limit(LSL)of the dimension are (4+0.01, 4-0.01), we can calculate the process standard deviation as follows:
Cp = (USL − LSL) / 6σ σ = 6Cp /(USL − LSL)
So
Suppose some special causes lead to the process shift(in terms of process standard deviation), the shifted sample ellipse data are collected and transformed as the same as the standard one. We search and check coefficients first from coarsest level(A6) in order to detect such shift. It extends to other finer level by adding more detail information (Dj, j=6, 5, 4, 3) if such attempt fails. The procedure stops if the shift has been detected at certain level. The results of all level statistics are shown as Table1: Shift delta 0 1 2 3 4
Tab. 1
A6 mean 45.0527 45.0709 45.0891 45.1069 45.1255
std Dev 0.0147 0.0147 0.0147 0.0149 0.0147
D6 mean 2.00691 2.00811 2.00931 2.01049 2.01172
std Dev 0.00097 0.00097 0.00097 0.00098 0.00097
D5 mean 6.73322 6.73098 6.72873 6.72653 6.72422
std Dev 0.0018 0.00181 0.00181 0.00184 0.00181
D4 mean 8.59569 8.59277 8.58985 8.58699 8.584
std Dev 0.00235 0.00235 0.00236 0.00239 0.00236
D3 mean std Dev 10.1884 0.0015 10.1902 0.0015 10.192 0.0015 10.1938 0.0015 10.1956 0.0014
Mean and Standard Deviation of Each Selected Level with sample size 10
One-Way Analysis Variance is then set up to make pairwise comparisons among A6 levels to check whether there is a corresponding change for each level if process shift occurs. The result(Fig. 4 ) shows that there exists a significant difference in coarse level which shows that only using the coefficients form such level is enough to catch the such process shift. Level a6 0sft a6 1sft
N Mean StDev ---+---------+---------+---------+--10 45.0527 0.0147 (---*---) 10 45.0709 0.0147 (---*---) ---+---------+---------+---------+--Pooled StDev = 0.0147 45.050 45.075 45.100 45.125 Fig. 4 Individual 95% CIs For Mean Based on Pooled StDev of A6 with different shift We also investigate the influence of sample size on the shift detection by reducing the sample size from 10 to 5(see Tab.2). There is a smallest sample size for detecting one standard deviation process shift. After examining tests of different sample size, it is
5
shown that the smallest sample size is 8, which means that we have to use at least sample size 8 to detect the expected process shift if only the coarse level is selected. Sample Size 10
9
8
7
6
5
Tab. 2
Source Factor Error Total Factor Error Total Factor Error Total Factor Error Total Factor Error Total Factor Error Total
DF 1 18 19 1 16 17 1 14 15 1 12 13 1 10 11 1 8 9
SS 0.001656 0.003873 0.005529 0.001491 0.003835 0.005326 0.001325 0.003776 0.005101 0.001159 0.003593 0.004753 0.000994 0.003426 0.004419 0.00083 0.002247 0.003076
MS 0.001656 0.000215
F 7.7
p 0.013
0.001491 0.00024
6.22
0.024
0.001325 0.00027
4.91
0.044
0.001159 0.000299
3.87
0.073
0.000994 0.000343
2.9
0.119
0.00083 0.000281
2.95
0.124
ANOV table of A6 with Different Sample Size
5. CONCLUSION How to set up efficient automatic inspection and process control is a key in modern manufacturing environment. The approach is proposed using wavelet transform coefficients as the quality index for two dimension irregular-shaped digitized parts inspection and analyze process shift instead of using raw data. The smallest of sample size is investigated as to detect the process shift. The experiment results show that such method is promising for two dimension irregular parts inspection and process control. Future work can be done to reduce the sample size for detecting the same process shift with coarse level by adding one or more detail level information and artificial neural network(ANN) also can be used to learn the shifting pattern and do the classification.
6
REFERENCES Burt, P.J (1983),”The Laplacian pyramid transforms for image coding,” IEEE Transactions on Communications, 31:532-540. Chetverikov.D (1992),”A Multiresolution Algorithm for Rotation-Invariant Matching of Planar Shapes”, Pattern Recognition Letters,Vol.13,Sept.,pp669-676 Evans, J. R.(1996), The Management and Control of Quality, West Public Company, St. Paul. Freeman ,H (1974),”Computer Processing of Line-drawing Images,” Computer Surveys, 6, 1, March. Lu, J.S. (1998). “Inspection and Defect Classification Schemes For 2D Irregular Shape Part Using Fuzzy Logic and Neural Network Models.” M.S. Thesis, Kansas State University. Mallat, S.G.(1989),”a Theory for Multiresolution Signal Decomposition: The Wavelet Representation”, IEEE Transaction on PAMI, 11, 7, July. Mokhtarian, F (1986),”Scale-Based Description and Recognition of Planar Curves and two-dimensional Shapes”, IEEE Transaction on PAMI, 8, January, pp34-43 Montgomery, D.C. (1991), Introduction to Statistical Quality Control, 2nd Edition, John Wiley and Sons. Nakamura, Y (1994),”Learning two-dimensional Shapes using Wavelet Local Extrema,” IEEE, pp48-52 Page, E. S. (1964),”Comparison of Process Inspection Schemes, ” Industrial Quality Control,Nov.,pp245-259 Palavajjhala (1994), “Computational Aspect of Wavelets and Wavelets and Wavelet Tranforms,” Wavelet Application in Chemical Engineering. Persoon, E (1976),”Shape Discrimination Using Fourier Descriptor,” IEEE Transaction on SAC, 7, 3, pp170-179 Sekita, I (1992),”Complex Autoregressive Model for Shape Recognition,” IEEE Transaction on PAMI, 14, 4, April. Shewhart, W. A(1925),”The Applications of Statistics as an Aid in Maintaining Quality of a Manufactured Product,” Journal of American Statistical Association, 20, pp546-548. Tieng. Q(1997),”Recognition of 2d Object Contours Using the Wavelet Transform ZeroCrossing Representation”, IEEE Transaction on PAMI, 19, 8, August, pp910-916.
7
