Quality inspection of aluminum foil products plays an important^{ }role for aluminum foil manufactures. We present a method that^{ }uses input estimate (IE)-based chi-square detectors for defect detection in^{ }aluminum foil. It is assumed that the intensity of the^{ }aluminum foil image is Gaussian distributed, and the distribution of^{ }the defect intensity is different from the normal. Under these^{ }assumptions, Kalman filters with a constant velocity (CV) model are^{ }used to filter the image. We assume there is an^{ }unknown input in the CV model and the unknown input^{ }is estimated in the filtering process. The defects are determined^{ }by the chi-square test of the estimate of the unknown^{ }input. Experiments show that our technique is effective for most^{ }defects in aluminum foil. ©2009 Society of Photo-Optical Instrumentation Engineers^{ }
Optical devices have^{ }been incorporated into mechatronic systems, and many machine-vision-based techniques have^{ }been used for quality control of products equipped with intelligence.^{ }In product quality control of aluminum foil manufacturing, quality inspection^{ }has to resort to machine-vision-based techniques for rigorous manufacture conditions.^{ }There are many machine-vision-based quality control techniques. In Ref. 1,^{ }edges in images are used to detect defects in ceramic^{ }tiles. In the textile industry, the Fourier analysis technique,^{2} morphology-based^{ }technique,^{3} wavelet-based technique,^{4} co-occurrence matrices, and Garbo-filter-based technique^{5} are used^{ }to detect defects in fabric surfaces. In the leather industry,^{ }oriented texture analysis is used to detect defects in leather.^{6}^{ }In Ref. 7, the infrared light reflected by the metal^{ }sheet is used to analyze the defects of metal sheets.^{ }However, each defect detecting technique is designed for a special^{ }target, so those techniques may not be suitable for defect^{ }detection in aluminum foil.^{ }
The techniques suitable for the detection of^{ }defects in aluminum foil should be designed according to the^{ }property of the aluminum foil product line. In our case,^{ }the foil goes through the product line at nearly five^{ }meters per second. The high resolution planar array camera is^{ }not suitable for this case, because high resolution planar cameras^{ }often have low frame rates, and the redundant information in^{ }the image sequence will affect image processing speed. Therefore, a^{ }line scan camera is used, and after every equal time^{ }span an image array can be obtained.^{ }
There are many types^{ }of defects in aluminum foil, some of which are shown^{ }in Fig. 1. Some defects have no apparent edges and^{ }some defects take up no more than 5 pixels [e.g., Figs.^{ }1(a)1(c)1(d)1(g)], so edge-based methods may not perform well. Some defects^{ }have an intensity similar to the normal area [e.g., Fig.^{ }1(a)], so a threshold-based method whose performance is often sensitive^{ }to its parameters may not perform well for this type^{ }of defect. Although some defects have direction properties [e.g., Fig.^{ }1(k)], most defects have no direction property, so Garbo-filter-based techniques^{ }are not suitable. There is less texture and period information^{ }in foil images, so texture- or frequency-analysis-based methods often used^{ }in fabric defect detection are not suitable for our case.^{ }Furthermore, the intensity of the normal area may vary [e.g.,^{ }Fig. 1(e)1(h)1(m)], and the sizes of the defects are different^{ }[e.g., Figs. 1(d)1(g)1(l)1(m)]; these may cause difficulty in defect detection.^{ }
Figure 1.We^{ }collect a large number of foil images from the product^{ }line, and from these images the histogram of intensity is^{ }obtained. The histogram shows that the intensity of the foil^{ }images approximately obeys Gaussian distribution. Then we collect the images^{ }with labeled defects and compute the histogram only using the^{ }pixels labeled as defects. The two histograms show that the^{ }intensity distribution of the defect is different from the normal.^{ }This property can be used to detect defects in aluminum^{ }foil.^{ }
We apply an input-estimate-based chi-square detector, which is widely used^{ }in maneuver target tracking, to solve the defect detecting problem^{ }and obtain inspiring results. We assume the intensity of the^{ }aluminum foil image is Gaussian distributed, and use the Kalman^{ }filters with a constant velocity (CV) model to filter the^{ }image. By assuming there is an unknown input in the^{ }CV model, the unknown input is estimated in the filtering^{ }process. The defects are determined by the chi-square significance test^{ }of the estimate of the unknown input. Experiments show that^{ }our technique is effective for most types of defects, and^{ }the processing speed can satisfy real-time applications.^{ }
This work is organized^{ }as follows. Section I is the introduction; some basic^{ }notions are reviewed in Sec. II; Sec. III^{ }is the implementation; and Sec. IV is the experimental^{ }results and the conclusions.^{ }
In this section, we review some basic notions from^{ }the theory of Kalman filters, the constant velocity (CV) model,^{ }as well as the input-estimation-based chi-square detector that we need^{ }in the sequel.^{ }
The Kalman filter^{ }is a linear estimator based on the minimum mean squared^{ }error (MMSE).^{8}^{,}^{9} For the sake of less computation and better^{ }performance, it has been widely used since it was introduced.^{ }Here is a brief review of a Kalman filter.^{ }
Given a^{ }system equation and measurement equation as follows:
where x and z^{ }are state and measurement vectors, respectively; v and w are^{ }zero mean Gaussian noise with covariance Q and R; u^{ }is an input vector that is estimated; F, G, and^{ }H are state, input, and measurement matrices, respectively; and F,^{ }G, H, Q, and R are assumed known and possibly^{ }time varying. The two noise sequences and the initial state^{ }are assumed mutually independent. The previous constitute the linear Gaussian^{ }(LG) assumption.^{ }
The main equations of a Kalman filter are as^{ }follows:
where the expression (k+1|k) denotes the estimate of A^{ }at time k+1 estimated at time k; P and S^{ }are the state covariance and the innovation covariance; and W^{ }and µ are the filter gain and the innovation.^{ }
In the^{ }Kalman filter some kinetic models are often used to describe^{ }the kinetic property of the target, like the constant velocity^{ }(CV) model and the constant acceleration (CA) model.^{9} Given a^{ }target moving with constant velocity, the CV model is suitable^{ }to describe the kinetic property of the target. In the^{ }CV model, the state vector is x=(X,X^{})^{T}, where X is^{ }considered as displacement and X^{} is the differential of X,^{ }i.e., velocity. If the measurement is displacement, the measurement matrix^{ }is H=(1 0). Otherwise, if the measurement is velocity then the^{ }measurement matrix is H=(0 1). In the CV model the state^{ }matrix is
where T is a time interval.^{ }
The input-estimation (IE)-based^{ }chi-square detector is often used to detect target maneuvers in^{ }maneuver target tracking (MTT) applications.^{9}^{,}^{10}^{,}^{11}^{,}^{12}^{,}^{13}^{,}^{14} The detector consists of input^{ }estimation and chi-square significance tests. In the input estimation process^{ }two models are used, one is the nonmaneuvering model [Eq.^{ }(3)] and the other is the maneuvering model [Eq. (4)]^{ }with input u.
Denote the present time by k and assume^{ }the target starts maneuvering at time k-s, that is, the^{ }maneuver onset time is k-s. The unknown inputs during the^{ }interval [k-s,…,k] are u(i), i=k-s,…,k-1. The state from the mismatched^{ }nonmaneuvering filter based on Eq. (3) is denoted by an^{ }asterisk:
with the initial condition ^{*}(k−s|k−s−s)=^{*}(k−s|k−s−1).^{ }
If the inputs are known, the^{ }hypothetical correct filter based on Eq. (4) yields estimates according^{ }to the recursion
The innovations corresponding to the hypothetical correct filter^{ }[Eq. (6)] is
The innovations corresponding to the nonmaneuvering filter [Eq.^{ }(5)] is
Combining Eqs. (5),(6),(7),(8) yields
Assuming the input to be constant^{ }over the interval [k-s,…,k-1], i.e., u(j)=u, j=k-s,…,k-1, yielding
where
Based on Eq.^{ }(10), the input can be estimated via LS from
where
and ,^{ }whose components are innovations [Eq. (7)], is zero mean with^{ }block-diagonal covariance matrix
The input estimate in batch form is
with the^{ }resulting covariance matrix
^{ }
The chi-square significance test is used^{ }to test the significance of the estimation of input. A^{ }maneuver is declared if and only if Eq. (16) is^{ }statistically significant. The significance test for the input is
It is^{ }a chi-square distributed variable, i.e., ~, where n_{u}=dim() is the^{ }dimension of vector . If
then decide if a maneuver occurs,^{ }where 1− is the confidence level.^{ }
In our product line the aluminum foil^{ }goes through the line scan camera at almost five meters^{ }per second. After each equal time span, an intensity image^{ }I with M rows and N columns is obtained. Statistics^{ }from a large number of foil images show that the^{ }intensity approximately obeys Gaussian distribution, and the intensity distribution of^{ }the defects is different from the normal. The intensity of^{ }every column (row) constitutes a time sequence that is Gaussian^{ }distributed, and the time sequence is considered as the measurement^{ }sequence of velocity in the CV model. Under these conditions,^{ }H=(0 1), the state matrix is
where T is a time interval,^{ }and the variance R should be set a bit larger^{ }than the variance of the intensity of the image I.^{ }Since every column (row) is considered as a measurement sequence^{ }of velocity in the CV model, the Kalman filter is^{ }used to filter the column from up to down. There^{ }are N columns in image I, so N Kalman filters^{ }are used correspondingly. Each column relates to one filter. In^{ }the filtering process of each filter, use the input-estimation (IE)-based^{ }chi-square detector to detect maneuvers. The position, where a maneuver^{ }is determined to occur, is considered to be the defect^{ }position. The detecting result is stored in the defect mask^{ }. is an M×N matrix, and if the (i,j)'th^{ }point in image I is detected to be a defect^{ }point, then label the (i,j)'th element of with 1,^{ }otherwise label it with zero. It is found that the^{ }position of the defect areas detected by the IE detector^{ }will be behind the correct defects due to the detection^{ }delay of the IE detector, so it will lead to^{ }imprecise locations of defects. To solve the problem, we detect^{ }image I from the four directions: up to down; down^{ }to up; left to right; and right to left. So^{ }totally 2M+2N filters are used. Denote _{up}, _{down}, _{left}, and^{ }_{right} as the defect mask from the detecting results of^{ }the four directions, respectively, and compute the logical OR of^{ }the four defect mask,
where the operator | is logical OR.^{ }The defect mask is the final defect mask, and^{ }the defect area of image I is labeled in .^{ }
The^{ }Kalman filter needs initialization to determine the first state vector^{ }x(0) and the first state covariance P(0|0). Set x(0)=(0,X)^{T}, where^{ }X is the average intensity of the image I. It^{ }is an alternative choice to set X equal to the^{ }intensity of the pixel where the filter started at if^{ }the defects seldom occur in the marginal area of the^{ }image. Set the first state covariance
It is an alternative choice^{ }to set
which will lead to the convergence of the filter^{ }with a little delay.^{ }
We apply our technique to detect defects^{ }in a defect image database that contains ten types of^{ }defects. The correct detection rate is 95%, and some results^{ }are shown in Fig. 2. Due to the detection delay^{ }of the IE detector, the defect mask will be a^{ }bit larger than the correct defect, but it will not^{ }affect the correct detection. For large defects, there will be^{ }a hole in the defect mask [e.g., Fig. 2(m)]. This^{ }shortcoming can be solved by some morphology operators. In fact,^{ }the sizes of defects in the foil are often small,^{ }so this shortcoming can be ignored for detecting and locating^{ }defects. Moreover, in our technique the four detecting processes of^{ }the four directions are mutually independent, so the four processes^{ }can be carried out in parallel. If carried out in^{ }parallel, the process time can be reduced by three quarters^{ }in theory. We test our technique on a PC with^{ }an Intel E4500 CPU (2.2 GHz) to detect the defect images^{ }of size 1024×768. If in parallel, the average processing time^{ }is less 52 ms, if not, the average processing time is^{ }more than 110 ms.^{ }
Figure 2.There are many maneuver detectors such as measurement-residual-based^{ }chi-square detectors, input-estimate-based Gaussian significance detectors, and generalized likelihood ratio^{ }test detectors. It will be our future work to study^{ }whether these detectors can be used in defect detection for^{ }aluminum foil and compare their performance.^{ }
Fig. 1. Aluminum foil defect images. First citation in article
Fig. 2. Defect^{ }images and defect masks. First citation in article
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