Statistical Detection of Defect Patterns Using Hough Transform

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Statistical Detection of Defect Patterns Using Hough Transform Qiang Zhou, Li Zeng, and Shiyu Zhou

Abstract—Surface defects on semiconductor wafers often exhibit particular spatial patterns. These patterns contain valuable information of the fabrication processes and can help engineers identify the potential root causes. In this paper, we present a control chart technique to detect spatial patterns of surface defects by using the Hough transform. An approximate distribution model of the monitoring statistic is proposed, and a comprehensive control chart design method is developed. This method is characterized by its intuitive implication and a simple design procedure which relates the statistical performance of the control chart to the design parameters. A case study is presented to validate the effectiveness of this method. Index Terms—Control chart, Hough transform (HT), spatial pattern, surface quality control.

I. Introduction URFACE quality plays a critical role in quality control of manufacturing processes involving products with flat surfaces. Typical examples include hot-rolling processes producing sheet metal, and fabrication processes for semiconductor wafers and liquid crystal display flat panels. Surface defects usually lower product quality and cause higher scrap rate, and therefore should be controlled tightly. In semiconductor industry, particularly, process yield has long been a primary concern, and surface defects on wafers generated during fabrication is generally the main reason for yield loss in integrated circuits (IC) fabrication [1]. Due to the increase of die sizes and decrease of feature sizes, chips become more vulnerable to surface defects. Furthermore, the increasing complexity of manufacturing processes leads to potentially larger process variations. Therefore, reduction of wafer surface defects has become a challenging issue. Inspection systems can often provide not only the number of defects on a surface, but also their spatial locations. Wafer defect maps can often be readily obtained from optical quality inspection systems in semiconductor processes. On the wafer defect map, defects are marked as dots on a white circular background representing the wafer. The spatial distribution of surface defects contains valuable information of the manufacturing processes and can thus be used for root cause

S

Manuscript received February 4, 2009; revised October 4, 2009; accepted February 16, 2010. Date of publication April 22, 2010; date of current version August 4, 2010. This work was supported in part by the National Science Foundation, under Grants 0545600 and 092608. This paper was recommended by Associate Editor R.-S. A. Guo. The authors are with the Department of Industrial and Systems Engineering, University of Wisconsin-Madison, Madison, WI 53706 USA (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TSM.2010.2048959

identification purposes [1]–[6]. Generally, surface defects generated in wafer fabrication processes can be categorized into two types [7], [8]: 1) globally scattered random defects (later referred to as background noise) caused by natural variation of the process; and 2) locally clustered defects due to certain assignable causes. The final surface defects are usually a superposition of these two types. The spatial distribution of the locally clustered defects can often provide hints on root causes. For example, one of the most widely seen patterns on wafer is the line segment, which could be the result of scratches during material handling. Another typical pattern, the edge ring, is normally due to etching problems. Thus, detection of such spatial patterns can help identify potential root causes. The monitoring of wafer surface defects is traditionally conducted using various control charts based on summary measures [9], [10]. For example, a c chart is used to monitor the total number of defects on a wafer and raise alarms when plotted dots fall out of control limit. The underlying assumption for the c chart is that the occurrence of defects follows a Poisson distribution. However, it has been reported that defects on wafer maps tend to cluster [11]–[13]. To solve this problem, some revised distributions, such as Neyman typeA distribution and negative binomial distribution have been proposed [14]–[19]. Generally, control chart techniques are conceptually intuitive and convenient to use for practitioners, but they cannot detect specific spatial patterns of the defects since only the count data are used as monitoring statistics. Because of the importance of detecting spatial patterns, several techniques have been developed. These methods can be roughly put into three categories. 1) Spatial statistics based methods: In these methods, the theory of spatial statistics is utilized to analyze and detect nonrandom patterns [20]. For example, Hansen et al. [7], Friedman et al. [21], and Jeong et al. [22] developed statistics measuring spatial dependency of defects to detect systematic clustering, while Fellows et al. [23] compared two popular spatial randomness models. Although the presence of clusters can be detected, these methods cannot distinguish specific spatial patterns. Moreover, the monitoring statistics often bear relatively complex statistical properties, adding difficulty to implementation of these methods. 2) Data mining methods: These include neural networks, fuzzy rule-based inferences, and various clustering methods, for supervised learning [4], [24], [25], and unsupervised learning [1], [8], [26]–[30]. Many of these methods

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ZHOU et al.: STATISTICAL DETECTION OF DEFECT PATTERNS USING HOUGH TRANSFORM

need a large training dataset, and due to the complexity of the algorithms, most of these methods do not provide a statistically rigorous evaluation of their performances (i.e., Type I error, also called false alarm probability, and Type II error, also called miss detection probability) in pattern detection. 3) Computer vision based methods: Point pattern matching and analysis has been an active research field in computer vision [31]. For spatial pattern detection, one of the most commonly used methods is Hough transform (HT). As a technique for detecting spatial patterns from binary image data [32]–[34], it transforms the binary image into a parameter space and tries to detect the parameterized pattern through a voting process in which each point votes for all the possible patterns passing through it. The patterns with higher votes indicate a higher probability of occurrence on the map. As long as a parameterized model can be established for the spatial pattern of interest, this method can be applied. The HT has long been recognized as a reliable and versatile method with high detection power for spatial pattern detection especially in noisy pictures [35]. However, most of the existing research focuses on performance improvement through deterministic parameterization, quantization of the parameters and image spaces, and computational load reduction. Limited work has been done on statistical evaluation of the performance of the HT under noisy conditions, which is critical in decision making for quality control. In this paper, we propose a statistical pattern detection method based on the HT. This method aims to construct a control chart that is able to monitor the highest number of votes in the Hough space (Hough matrix). This statistic is calculated for each defect map, and an alarm will be raised when it is larger than the control limit, indicating that a pattern has been detected. An approximate distribution of this statistic is developed and the control limit can thus be estimated. Compared to existing techniques, this method has the following characteristics. 1) It focuses on detecting specific spatial patterns of defects instead of only detecting the existence of clustering. 2) The detection is realized through an easy-to-use control chart and the monitoring statistic is intuitive and easy to compute. 3) A quantitative design procedure is also provided, which relates the Type I and Type II errors to the design parameters. In this way, requirements on the statistical performance of the control chart can be achieved through choosing proper values for design parameters. Essentially, this method can be used to detect any parameterized surface patterns. Two types of most commonly observed patterns, linear and circular patterns, will be focused on in this paper as illustrative examples, while the treatment can be extended to other parametric patterns. The rest of this paper is organized as follows. Section II presents the HT-based control chart design procedure for linear

Fig. 1.

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Line detection by HT. (a) Accumulator array. (b) Defect map.

pattern detection. Section III describes the detection of circular patterns. Section IV provides a case study to validate this method, and Section V concludes the paper and discusses possible future work.

II. HT-Based Control Chart Design for Linear Pattern Detection A. Basics of Hough Transform for Line Detection First, we shall briefly introduce some principles of HT. The notation used here follows [33]. Using normal parameterization, a line in x–y plane can be uniquely defined by its distance ρ from the origin and the angle θ of its norm as x cos θ + y sin θ = ρ

(1)

where θ is restricted within [0, π). This parameterization maps every line in x–y plane to a unique point in θ–ρ plane. A point in x–y plane, e.g., P0 (x0 , y0 ), will be mapped to the θ–ρ plane as a sinusoidal curve defined as x0 cos θ + y0 sin θ = ρ.

(2)

Every point (θ, ρ) on the curve (2) corresponds to a unique line in x–y plane passing through P0 . Therefore, for points lying on the same line in x–y plane, their corresponding sinusoidal curves in θ–ρ plane will pass through a common point. Based on this mapping relationship, a voting process can be designed for detecting collinear points in x–y plane. In this process, θ is equally sampled and ρ is quantized into equal intervals, resulting in an array of accumulators in θ– ρ plane as illustrated in Fig. 1(a). Each accumulator, e.g., the shaded cell (θ = θi , ρj ≤ ρ < ρj+1 ), corresponds to a particular stripe area in the physical defect map shown in Fig. 1(b). Initially, all the accumulators are assigned with value “0.” As previously mentioned, each point in Fig. 1(b) has a corresponding sinusoidal curve in θ–ρ plane in Fig. 1(a). When this curve passes through an accumulator, the accumulator will get a vote and its value increases by “1.” As a result, the final value of each accumulator equals the number of points falling within its corresponding stripe. Therefore, if there is a line on the defect map, the accumulator corresponding to the stripe that overlaps the most with the line will get the highest vote. Assume background noise on the defect map follows a Poisson distribution with density λ. The existence of background noise poses a problem for conventional HT. Expected

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Fig. 3.

Fig. 2.

Maximum entropy quantization for equal-area stripes.

votes of some accumulators are intrinsically higher than others because a stripe near the origin is expected to contain more defects than one near the edge due to its larger area. To ensure a fair comparison among the accumulators, the stripes in Fig. 1(b) must have equal areas. Here, we adopt the maximum entropy quantization suggested in [32]. The angle θ is still sampled at equal distance intervals from 0 to π, i.e., θ = (θ1 , θ2 , . . . , θNθ ), where Nθ , called the sampling parameter of θ, denotes the number of samples. Hence, we have θ1 = 0, θ2 = π/Nθ , . . . , θNθ = (Nθ −1)π/Nθ .ρ will be divided in such a way, as shown in Fig. 2, to produce equal-area stripes. Let R be the radius of the defect map, then ρ will be partitioned into Nρ intervals by ρ0 < ρ1 < · · · < ρNρ , satisfying ρ0 = −R, ρNρ = R

Different degrees of correlation between accumulators.

of Nθ and Nρ combination. In this section, we propose an approximation method which can avoid such simulations while provide a quite accurate control limit. 1) Approximate Distribution of the Monitoring Statistic: According to Section II A, hij follows a Poisson distribution with density λAs when there is only background noise on the wafer, and thus, (hij )max is the maximum value of Nθ × Nρ identically distributed variables. If we assume that these variables are all independent of each other, then the cumulative distribution function of (hij )max can be expressed as Fmax (x) = P[(hij )max ≤ x] = P(all hij ≤ x) Nθ N Nθ N ρ −1 ρ −1   = P(hij ≤ x) = F (x) i=1 j=0 Nθ ×Nρ

= [F (x)] where

(3) F (x) =

and


x  e−λAs (λAs )k k=0

ρi+1

stripe area As =

2



R2 − ρ2 dρ = πR2 /Nρ .

(4)

ρi

Correspondingly, the vote in each accumulator follows the same Poisson distribution with density λAs . Here, Nρ is called the quantization parameter of ρ. Let hij denote the vote received by the accumulator (θ = θi , ρj ≤ ρ < ρj+1 ), i = 1, . . . , Nθ ; j = 0, . . . , Nρ − 1. It is natural to establish a control chart to monitor the largest vote, (hij )max , among all the accumulators. When (hij )max exceeds a predefined threshold, i.e., control limit, an alarm will be raised to warn engineers of the potential presence of linear defect patterns. We call this chart an HT-based control chart. Clearly, for a given defect map with radius R and a background noise level λ, three parameters need to be decided: the upper control limit denoted as UCL, and parameters Nθ and Nρ . This will be discussed in the following sections. B. Control Limit of the HT-Based Control Chart This section deals with the following problem: given parameters (R, λ, Nθ , and Nρ ), how can we specify the control limit for the HT-based chart at a predefined Type I error level (denoted by α)? A straightforward method is through MonteCarlo simulations: simulate the voting process as described in the previous section for a large number of in-control defect maps (i.e., with only background noise), and then obtain the control limit by obtaining the 100(1 − α) percentile of the simulated values of the monitoring statistic. However, this method is computationally intensive, particularly considering that a different set of simulation runs is needed for each choice

(5)

i=1 j=0

k!

(6)

is the cumulative distribution function of a Poisson distribution with density λAs . Plugging (6) into (5) yields  x N ×N  e−λAs (λAs )k θ ρ Fmax (x) = . (7) k! k=0 However, the hij s in the accumulator array are in fact correlated. This can be illustrated using the example in Fig. 3, where the four accumulators, A1 , A2 , A3 , and A4 , correspond to stripes S1 , S2 , S3 , and S4 , respectively. For example, h1 and h2 (the votes received by A1 and A2 ) are not independent of each other because S1 and S2 share a common area. From Fig. 3, we can summarize the following characteristics of the correlations among accumulators: 1) accumulators in the same column, i.e., with the same θ value, are independent, e.g., A1 and A4 (or S1 and S4 ); and 2) in general, closer accumulators have higher correlations, e.g., A1 and A3 (or S1 and S3 ), and the degree of correlation is determined by the ratio of the overlapped area over the stripe area. Clearly, it is very difficult, if not impossible, to obtain the exact distribution of (hij )max by accounting for the complicated correlation structure of all the accumulators. Instead, we propose an approximate model based on the independent model given in (7)  x N ×N ×K  e−λAs (λAs )k θ ρ Fmax (x) = (8) k! k=0 where K ∈ (0, 1] is an adjusting factor depending on the overall (average) degree of correlation among all the accumulators. A lower value of K represents a higher correlation among

ZHOU et al.: STATISTICAL DETECTION OF DEFECT PATTERNS USING HOUGH TRANSFORM

Fig. 4.

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Correlation due to stripe overlapping.

accumulators and K = 1 corresponds to the independent case. The idea is that the Nθ × Nρ correlated variables can be “equivalently” viewed as a smaller number, i.e., Nθ × Nρ × K, of i.i.d. variables when we want to find the maximum value of these variables. To get some insight about the overall degree of correlation, the general case of two stripes’ geometric relation is considered, as shown in Fig. 4. To be representative, assume that the two stripes are of the same length, L, the average length of all stripes, and the same width, ρ = 2R/Nρ . The angle θ = π/Nθ is the smallest angle difference between two stripes. Consequently, the correlation can be described as overlapped area ρ2 / sin θ ρ/ sin θ = = . stripe area Lρ L (9) Recall that θ is the sampling resolution of θ, which should not be coarse in practice. Therefore, we limit it to θ ≤ π/36 = 5°, which results in the approximation sinθ ≈ θ. Moreover, since the relationship between L and R is almost linear, i.e., L ≈ tR, where t is a constant. Hence, (9) can be further simplified as

correlation =

ρ/ sin θ ρ/θ ≈ L tR (2R/Nρ )/(π/Nθ ) 2 Nθ Nθ = = =C tR πt Nρ Nρ

correlation =

(10)

where C is a constant. Equation (10) suggests that K is related with the ratio Nθ /Nρ . Hence, we can estimate K as a function of Nθ /Nρ K = f (Nθ /Nρ ).

(11)

Fig. 5.

Simulation results: Ln(K) against Nθ /Nρ at different α levels.

Consequently, K can be estimated by minimizing | UCLα − UCLα | .

(15)

Through extensive simulations, we can obtain the relationship between the estimated K and Nθ /Nρ at different α levels, as shown in Fig. 5. In the simulations, R ∈ [50, 200], λ ∈ [0.0002, 0.01], Nθ ∈ [36, 180], Nρ ∈ [20, 200], and α = 0.01, 0.05, and 0.1. Clearly, Ln(K) has an approximately linear relationship with Nθ /Nρ . R and λ are not much related with K because they are irrelevant to the correlation structure. Therefore, we can obtain the fitted results of K as α = 0.01 : Ln(K) = −0.091(Nθ /Nρ ) − 0.002 α = 0.05 : Ln(K) = −0.110(Nθ /Nρ ) − 0.015 α = 0.10 : Ln(K) = −0.123(Nθ /Nρ ) − 0.038.

(16)

At the 0.05 level, we further simplify it as

Thus, we have



Fmax (x) =

x  e−λAs (λAs )k k=0

Nθ ×Nρ ×f (Nθ /Nρ )

k!

K = exp(−0.11Nθ /Nρ ). .

(12)

2) Empirical Formula for Estimation of Control Limits: The function in (11) will be decided in this section. Let UCLα be the “true” control limit obtained through MonteCarlo simulations with Type I error α, which can be expressed as function UCLα = g1 (R, λ, Nθ , Nρ ).

(13)

Similarly, the estimated control limit, denoted as UCLα , can be obtained based on (8) and expressed as UCLα = g2 (R, λ, Nθ , Nρ , K).

(14)

(17)

Substituting (17) into (8) yields the approximate model  Fmax (x) =

x  e−λAs (λAs )k k=0

k!

Nθ ×Nρ ×exp(−0.11Nθ /Nρ ) .

(18)

To show how good this approximation is, Table I compares UCLα and UCLα at α = 0.05 for ten new sets of parameters. It can be clearly seen that the estimated control limits from (18) predict the simulated results quite well. Moreover, compared with the i.i.d. model given in (7), the approximate model provides much better estimations, especially when Nθ /Nρ is large, i.e., the correlation is high. It should be noted that (hij )max follows a discrete distribution, and the values of the control

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TABLE I Comparision of Simulated and Estimated Control Limits Parameter Set Nθ /Nρ Simulated UCLα UCLα from (18) UCLα from (7) | UCLα − UCLα | UCLα − UCLα Parameter Set Nθ /Nρ Simulated UCLα UCLα from (18) UCLα from (7) | UCLα − UCLα | UCLα − UCLα

| |

| |

1 1.8 25.96 25.92 26.19 0.04 0.23 6 4 36.64 36.62 37.31 0.02 0.67

2 2 23.31 23.40 23.67 0.09 0.36 7 4.5 30.65 30.49 31.13 0.16 0.48

3 2.4 26.14 26.16 26.55 0.02 0.41 8 5.1 21.92 21.89 22.57 0.03 0.65

4 3 30.31 30.20 30.70 0.11 0.39 9 6 24.04 23.92 24.74 0.12 0.70

5 3.6 26.53 26.48 26.92 0.05 0.39 10 6 36.98 36.80 37.84 0.18 0.86

Fig. 7. Simulated distribution of (hij )max when R = 100, λ = 0.01, Nθ = 180, and Nρ = 200.

Fig. 8.

Fig. 6. Comparisons of estimated distributions. Histogram: simulated results. (a) * is calculated based on (7); (b) * is calculated based on (18), K = 0.47.

limits in Table I are obtained through linear interpolation. For example, if P(x ≤ 6) = 0.91 and P (x ≤ 7) = 0.97 by (18), then UCLα is approximately 6.67. The strength of the approximate model can be better seen from Fig. 6, which shows a typical distribution of (hij )max . The parameters in this case are R = 100, λ = 0.01, Nθ = 180, and Nρ = 25. Clearly, the approximate model with an appropriate K value fits the simulated results better than the i.i.d. model, especially at the upper tail which is crucial for determining the control limit. 3) Modified Control Limit Due to Discretization: Although Type I error of a control chart is given as a fixed value, e.g., α = 0.05, this value is hard to be reached precisely since the underlying distribution of (hij )max is discrete. Therefore, directly using the estimated value UCLα as the control limit may cause unexpected problems. For example, Fig. 7 shows the simulated distribution when R = 100, λ = 0.01, Nθ = 180, and Nρ = 200. In this distribution, P(x ≤ 9) = 82.46%, and P(x ≤ 10) = 97.26%. The estimated control limit from (18) is UCLα = 9.85 (at α = 0.05). Using 9.85 as the control limit, the actual Type I error will be as high as αactual = 1 − P(x ≤ 9.85) = 1 − P(x ≤ 9) ≈ 0.18.

(19)

To avoid such problems, we suggest using the following revised control limit: UCLcc = [UCLα ] + 0.5

(20)

where [x] denotes the integer that is closest to x, and if x is exactly in the middle of two integers, then either integer can

Linear defect pattern on the wafer defect map.

be selected. By using (20), the actual Type I error αactual will be the closest possible Type I error to the given α. The benefit of selecting this control limit can be seen from the following lemma. Lemma 1: Given a discrete random variable X and a value α ∈ (0, 1). Assume m is a positive integer such that P1 = P(X ≤ m) ≤ 1 − α and P2 = P(X ≤ m + 1) > 1 − α. Define C = m + (1 − α − P1 )/P2 − P1 ) and CL = [C] + 0.5. Then, | 1 − α − P(X ≤ CL) |= min(1 − α − P1 , P2 − (1 − α)) and | 1 − P(X ≤ CL) |≤ 2α. The proof of this lemma is given in Appendix I. In our case, we let (hij )max be the discrete random variable X and then from this lemma, we can see that CL provides a control limit whose actual Type I error is the closest to the specified value α. The value of CL is determined by the value of C in the lemma. In practice, the true value of C for (hij )max is unknown because the true distribution of (hij )max is unknown. However, as we discussed in previous session, UCLα is a very good approximation to C (we call it UCLα in previous sections) and thus, we can simply substitute C with UCLα . Hence, we propose to use (20) as the final control limit. Using this control limit, αactual in the example in Fig. 7 becomes αactual = 1 − P(x ≤ [9.85] + 0.5) = 1 − P(x ≤ 10.5) ≈ 0.0274 (21) which is much closer to 0.05 than 0.18. C. Selection of Nθ and Nρ for the HT-Based Control Chart To decide the parameters Nθ and Nρ , we need to consider the Type II error of the control chart defined as β = P (no line detected | there is at least one line). For simplicity, we assume that there is at most one line on the wafer, and the profile of collinear points is rectangular, as shown in Fig. 8. The following parameters are needed to uniquely define such a line as in Fig. 8: 1) point density of the line λl , λl > λ; 2) length of the line L;

ZHOU et al.: STATISTICAL DETECTION OF DEFECT PATTERNS USING HOUGH TRANSFORM

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3) width of the line d; 4) line direction γ with respect to x-axis; 5) location of the line center C(x, y). Consequently β = f (R, λ, Nθ , Nρ , λl , L, d, γ, C).

(22)

Influences of R, λ, λl , L, d, γ and C on β are as follows. 1) The radius of a defect map is not truly a parameter because we can always normalize it as “1” in our analysis, e.g., a map with R = 100 and λ = 0.01 is essentially the same as a map with R = 1 and λ = 100. 2) Larger ratio λl /λ or larger values of L and d will lead to more easily detectable patterns and thus, result in smaller Type II errors. 3) Because of isotropy, γ and C will not have significant influence on Type II error, but may cause minor fluctuations due to discretization. The influences of Nθ and Nρ , however, are more complicated. As parameters of the HT, they can be arbitrarily chosen to achieve different performances of detection, leading to different Type II errors even for the same defect pattern. Essentially, the other seven parameters in (22) are determined once the wafer is given, and can thus be categorized as uncontrollable parameters, while Nθ and Nρ can be categorized as controllable parameters. How these parameters influence the Type II error will be discussed in the follows. 1) Impact of Nθ and Nρ on Type II Errors: Extensive simulation results with various sets of parameters suggest that Nθ and Nρ have very large influence on the Type II error. As a typical case, Fig. 9(a) shows their impact when R = 100, λ = 0.001, λl = 0.03, L = 100, d = 5, γ = π/6, C(130, 130) and Type I error α = 0.05. It is obvious that the Type II error varies largely (from 2% to 20% in this case) with different values of Nθ and Nρ . In the figure, each curve rises up and drops down along the horizontal axis, and small Type II errors can always be found right after curves drop down. Taking advantage of this property, proper combinations of Nθ and Nρ can be identified to obtain small Type II errors. Fig. 9(b) shows the corresponding control limits defined in (20). For example, when Nθ = 90 and Nρ = 110, from Fig. 9(b) we know that the control chart uses a control limit UCLcc = 5.5, and from Fig. 9(a) we know that its corresponding Type II error is approximately 0.16. A careful comparison of these two figures reveals that a Type II error curve in Fig. 9(a) always drops down when its corresponding control limit curve in Fig. 9(b) drops down. For example, when Nθ = 90 (line with triangles), both curves drop down near Nρ = 60 and Nρ = 120. Extensive simulations show that this phenomenon is not merely coincidence in this particular example, but common among all parameter sets. To understand this phenomenon, we need to take a close look at what happens when UCLcc drops. Fig. 10 shows the distributions of (hij )max of a wafer having a line pattern in two cases with Nρ = n and Nρ = n + 1 (all other parameters being the same). Assume that UCLcc drops from m + 0.5 to m − 0.5 when Nρ increases from n to n + 1, where both m and n are integers, and n should not be small, e.g., n > 20.

Fig. 9. Illustration of Type II error depending on Nθ and Nρ (α = 0.05). (a) Type II error with respect to Nθ and Nρ . (b) Corresponding UCLcc with respect to Nθ and Nρ .

Fig. 10. Illustration of control limit change due to increase of Nρ . (a) Case 1: Nρ = n. (b) Case 2: Nρ = n + 1.

The solid lines in Fig. 10 indicate the control limits used. Because the only difference between the two cases is that Nρ has a slight increase of 1, the two distributions of (hij )max will be almost identical, as can be seen clearly from Fig. 10. In case (a), Type II error β1 = P(X ≤ m + 0.5), and in case (b), β2 = P(X ≤ m−0.5). Apparently, β2 is smaller than β1 by the amount of P(X = m). Therefore, we can conclude that those sudden drops of Type II error curves are caused by the loss of the portion P(X = m) in Type II error when UCLcc drops from m + 0.5 to m − 0.5. Based on this intuitive understanding, we can obtain the following insights. 1) UCLcc will keep dropping down as Nρ increases. As Nρ increases, the area of each stripe decreases and so does the expected number of points falling within each stripe, i.e., E(hij ). Therefore, the null (without lines) distribution of (hij )max will keep moving/leaning toward left, and thus, UCLcc will become smaller. 2) The Type II error will keep rising with Nρ between two dropping points of UCLcc . UCLcc remains as a constant between two dropping points, while the alternative (with a line) distribution of (hij )max keeps moving toward left as Nρ increases, causing larger Type II errors.

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3) The Type I error can also change when UCLcc drops because the distribution of the monitoring statistic is discrete. In practice, we should keep the Type I error as a constant when we try to reduce the Type II error by adjusting the system parameters. But clearly in this case, the selection of Nθ and Nρ will influence both the Type I and Type II errors. Fortunately, a close examination of the change pattern in Type I error reveals the following fact: When UCLcc drops, the actual Type I errors before and after the dropping point deviate approximately the same amount from the specified α level and their difference is not larger than 2α. A brief description of the proof procedure of this result is given in Appendix II. This result provides a bound of the change in Type I error and enables us with the following guidelines in selecting Nθ and Nρ . 2) Guidelines on the Selection of Nθ and Nρ : According to the above analysis, given the radius of defect maps, R, and background noise level, λ, the key to implement the HT-based control chart is choosing proper values of the two parameters, Nθ and Nρ . This can be done following the guidelines below. 1) Based on (18) and (20), calculate UCLcc at the specified α level for different combinations of Nθ and Nρ , and plot them on a figure like Fig. 9(b). 2) Choose a proper combination of Nθ and Nρ . This can be done by identifying those dropping points of UCLcc curves. They can be chosen at the point after the curve drops while far from its next drop point to avoid a high Type II error. Other useful information, such as the observed width of lines we intend to detect, can also be taken into consideration when selecting Nθ and Nρ . After Nθ and Nρ are chosen, their corresponding UCLcc can be used as the control limit, and the actual Type I error can be estimated based on (18).

III. HT-Based Control Chart Design for Circular Pattern Detection In this section, we will briefly present the extension of the proposed methodology for detecting circular patterns including circles (rings) and arcs. A. Basics of the Circular Hough Transform To detect circular patterns, the circular Hough transform (CHT) will be used [33], [37]. A circle can be parameterized by its center O(a, b) and radius r as (x − a)2 + (y − b)2 = r2 .

(23)

The voting process for circular pattern detection on the defect map is illustrated in Fig. 11. The accumulator (Oi , rj < r ≤ rj+1 ) in Fig. 11(a) uniquely corresponds to an annulus in Fig. 11(b) with Oi being the center, and rj and rj+1 being the inner and outer radius, respectively. In other words, the number of votes, hij , for this accumulator equals the number of defects falling within such an annulus. With the same assumption that background noise is Poisson distributed,

Fig. 11. wafer.

Circle detection by HT: (a) accumulator array; (b) annulus on the

Fig. 12.

Quantization of circle centers.

Fig. 13.

Imaginary extension for the wafer.

hij is Poisson distributed with density Ac λ, where Ac is the 2 area of the annulus, i.e., Ac = π(rj+1 − rj2 ). Similar to linear pattern detection, sampling of the center O and quantization of the radius r are needed. Without loss of generality, here we limit our interest to circular patterns with: 1) RL < r ≤ RH , 0 < RL < RH ; and 2) O is within the wafer, i.e., a2 + b2 ≤ R2 . The range of r to consider, i.e., (RL , RH ], will be divided into Nr intervals by r 0 < r 1 < . . . < rNr , where r0 = RL , rNr = RH . The area of any annulus is the same as Ac = π(R2H − R2L )/Nr . Na and Nb are the quantization parameters along a and b axis for center O, and we let Na = Nb for simplicity. This gives a sampling resolution of  = 2R/Na , as shown in Fig. 12. The resulting centers are O1 , O2 , . . . , ONo , where NO is the total number of sampled points. When R is large compared to , NO ≈ πR2 /2 = πNa2 /4. One problem will arise when implementing the above scheme, that is, the annulus might be partially outside the wafer, as shown in Fig. 13, where the black solid circle repre-

ZHOU et al.: STATISTICAL DETECTION OF DEFECT PATTERNS USING HOUGH TRANSFORM

sents the wafer. A simple solution is to add an imaginary extension to the wafer so that all the annuluses of interest will be encompassed. Under the two assumptions, i.e., RL < r ≤ RH and O is within the defect map, the radius of the extended edge (the gray circle in Fig. 13) is R + RH . Background noise with density λ will be added into the extended region, so that pattern detection will be conducted within the gray circle as if we have a “wafer” with a radius of R + RH . Besides the above problem, another problem will arise in the implementation is that if a small value of r is allowed, then the width of the corresponding annulus will be large in order to keep the area the same as those of larger r values. Under this situation, a small cluster of points can be mistakenly included in this annulus and identified as a circle. To avoid this problem, we will make the lower bound of the radius sufficiently large. Particularly, in this paper, we set RL = R/2. In semiconductor manufacturing, most circular patterns have large radii (e.g., the radius of an edge ring is close to R). Therefore, this assumption will not limit the applicability of the proposed method.

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Fig. 14.

Control limit UCLcc versus Nθ and Nρ .

Fig. 15.

Estimated Type I error versus Nθ and Nρ .

B. Distribution of (hij )max in Circular Pattern Detection Under the CHT defined in the previous section, the control limit in circular pattern detection is studied. Similar to the line detection, an approximate distribution of the form as (8) is assumed for (hij )max among the accumulators in Fig. 11(a), and the adjusting factor, K, is determined through simulations. Not surprisingly, the relationship of K and the two design parameters, Nr and Na , depends on the interested range of circular patterns. When RL = R/2 and RH = R, K can be approximated by Ln(K) = −0.000047Na2 /Nr − 0.2794

(24)

at α = 0.05. Consequently, the approximate distribution model of (hij )max is Fmax (x)  x π/4×Na2 ×Nr ×exp(−0.000047Na2 /Nr −0.2794)  e−λAc (λAc )k = . k! k=0 (25) This result shows that, Na2 and Nr play similar roles in circular pattern detection as Nθ and Nρ in linear pattern detection. We can also see that when Na2 /Nr is not very large, K can be approximated by a constant exp(−0.2794). The intuition is that in this case the overlapping among annuluses is relatively small, and thus, the correlation among accumulators is not sensitive to the design parameters. We also studied the Type II error when using the control limits determined by (25) to detect circular patterns through extensive simulations. Note that in implementing the control chart, an extended region as shown in Fig. 13 needs to be artificially added to each defect map. The simulation results exhibit similar characteristics as those in linear pattern detection, rising up and dropping down along Nr . Small Type II errors can be achieved by carefully choosing the combination of Na and Nr .

IV. Case Study In this section, we will use a case study to validate the effectiveness of the proposed methodology. The defect maps are generated through simulations with the basic parameters extracted from real surface defect maps described in [24]. This is a common way for validation of such methods, e.g., [37]. A. Control Charts for Detecting Linear and Circular Patterns 1) Detection of Linear Patterns: Let’s first consider the use of the HT-based control chart for detecting linear patterns. In this example, R = 100 and λ = 0.0114. The linear pattern has parameters λl = 0.2444, L = 25, and d = 4. Two sets of defect maps are generated: one with only background noise, and the other with both background noise and a linear defect pattern. Each group has 5000 defect maps. A proper range is assigned to Nρ , i.e., 50 ≤ Nρ ≤ 100, and the angle resolution is θ = 5° , i.e., Nθ = 180° / 5° = 36. Together with parameters R, λ, and the specified α = 0.05, UCLcc can be calculated and the corresponding Type I errors can be estimated based on (18). The results are shown in Fig. 14 and Fig. 15. The choice of Nθ is based on the desired accuracy of how well a line can be matched in terms of its angle. Here, we choose Nθ = 180, i.e., θ = 1° . Therefore, based on Fig. 14 and the method described in Section II, proper Nρ values can be identified. In this example, potentially good choices are Nρ = 52, 57, 63, 69, 77, 87, and 100. When a combination of Nρ and Nθ has been selected, their corresponding Type I error can be identified from Fig. 15. To choose from these multiple combinations, other information can be taken into consideration. For example, Nρ should not be too small (quantization is too coarse) or too large (unnecessary

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TABLE II Results Based on Selected Parameters (Nθ = 180) Nρ Type Type Nρ Type Type

I error II error I error II error

52 0.0502 0.0466 51 0.0272 0.08

57 0.0576 0.045 56 0.0216 0.0836

63 0.0594 0.0528 62 0.021 0.1046

Fig. 16. HT-based control chart for linear pattern detection (Nθ = 180, Nρ = 53).

69 0.0684 0.0568 68 0.0244 0.1022

78 0.0604 0.0734 76 0.024 0.1306

87 0.073 0.076 86 0.0208 0.1644

100 0.0658 0.10 99 0.0146 0.2132

Fig. 18. Circular pattern detection (Na = 100, Nr = 50). (a) HT-based control chart. (b) Corresponding defect map of the out-of-control point. TABLE III Alarming Rates of the Two Detectors

Line detector Circle detector

Fig. 17.

Corresponding defect maps of points I and II in Fig. 16.

computational load). Our experience suggests that a rule of thumb in the selection of Nρ is to make the width of the quantized strip on the map comparable with the width of the line we intend to detect. To show the advantages of these chosen parameters, simulations have been conducted based on the generated defect maps, and UCLcc is used as control limit. For comparison, bad choices of Nρ , e.g., 51, 56, 62, 68, 76, 86, and 99, are also tested during the simulation. The results are given in Table II. As can be seen clearly from the table, the error rates vary considerably under different choices of Nρ , and the selected values provide significantly smaller Type II errors. To construct the control chart, we should first select one of the suggested parameter combinations, e.g., Nθ = 180 and Nρ = 52. According to Fig. 14, the upper control limit is 20.5 in this case. The HT is then applied to each defect map. From the resulting accumulator array, we obtain the highest number of votes and mark it on the control chart. Fig. 16 shows the chart with 12 monitored statistics (showing two out-of-control points) and Fig. 17 shows their corresponding defect maps. 2) Detection of Circular Patterns: To define a circular pattern on the wafer, four parameters are needed: 1) point density of the circle λc (λc > λ); 2) center of the circle C(x, y); 3) inner radius rin ; 4) outer radius rout ;

None 1.46% 2.72%

1 Line 99.52% 12.08%

1 Circle 5.92% 95.02%

1 Line + 1 Circle 99.58% 95.58%

where rin < rout . In the simulation, we consider a case with λc = 0.2444, C(100, 0), rin = 51, and rout = 51.8. Following similar steps, we have found an appropriate choice of design parameters to be Na = 100 and Nr = 50. With RL = R/2 and RH = R, the approximated control limit is 24.5. Fig. 18 shows the control chart with 12 samples and the corresponding defect map of the identified out-of-control point. B. Simultaneous Detection of Linear and Circular Patterns In practice, linear patterns and circular patterns could coexist on a wafer. Thus, the HT-based control chart for linear patterns and that for circular patterns should be applied simultaneously. It needs to be pointed out that if multiple patterns of the same type, e.g., three lines, two circles, etc., are present, the most “obvious” pattern will be identified because the monitoring focuses on the accumulator with highest votes in the parametric space. To show the performance of the two detectors together, four sets of defect maps are generated: the first set with only background noise λ = 0.0114, the second set with one line, the third set with one circle, and the last set with both a line and a circle. The parameters of the line are: λl = 0.1629, L = 150, and d = 1; the parameters of the circle are: λc = 0.07, C(0, 0), rin = 99.5, and rout = 100. Each set contains 5000 defect maps. The linear detector and circular detector are applied to each set with design parameters Nθ = 180, Nρ = 200, and Na = 100, Nr = 75, respectively. The alarming rates (percentage of cases where the detector alarms) are listed in Table III. The results suggest that the two detectors are very powerful in detecting corresponding patterns, and quite robust to other

ZHOU et al.: STATISTICAL DETECTION OF DEFECT PATTERNS USING HOUGH TRANSFORM

types of patterns. Nonetheless, in general, the performance of the two detectors is a complex function of the characteristics of the patterns on the wafer.

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Furthermore, from 1 ≥ P 2 > Pα we have |Pα − P2 | ≤ α.

(A5)

Based on (A4) and (A5), we obtain V. Conclusion and Discussion In this paper, we proposed an HT-based control chart to statistically detect the linear and circular patterns on a defect map. Using this method, the largest number in the accumulator array of the HT was monitored. An approximate distribution model for the monitoring statistic was developed and used to identify the control limits. The impact of two important parameters of HT, Nθ , and Nρ (Na and Nr for CHT), on the Type I and Type II errors were also studied, and guidelines on how to choose their values were provided based on our findings. The effectiveness of the proposed technique was validated using a case study. In Section III, we extended the proposed method to detect circular patterns based on CHT. The proposed methodology can also be extended to other spatial patterns in a similar way. However, as the pattern gets more complex, it becomes increasingly hard to use the conventional HT method because of the high dimensionality of parameters. A more promising way to extend this method for detecting various commonly observed patterns is to analyze “Hough signatures” of different patterns in the Hough matrix of the linear HT. The idea is that a particular type of defect pattern, after applying the HT, corresponds to a unique “signature” in the Hough matrix, and thus, the identification of such patterns can be realized through analysis of its signature pattern [38]. However, more statistical analysis on properties of these signatures is needed to provide rigorous results of pattern detection, which will be studied in the future. Acknowledgment The authors gratefully thank the editors and referees for their valuable comments and suggestions.

|1 − P(X ≤ CL) − α| ≤ α

(A6)

which means |1 − P(X ≤ CL)| is not more than 2α.

Appendix II Proof of the Result Without loss of generality, we can take the two cases in Fig. 10 as example. As mentioned previously, the null distributions of them are almost the same. In case (a), the actual Type I error is αactual1 = 1 − P(X ≤ m + 0.5) = 1 − P(X ≤ m)

(A7)

while in case (b) αactual2 = 1 − P(X ≤ m − 0.5) = 1 − P(X ≤ m − 1). (A8) Assume αactual1 = α − δ, where 0 ≤ δ ≤ α. Since UCLcc drops from m + 0.5 to m − 0.5 when Nρ has a slight increase of 1, the interpolated UCLα must be slightly larger than m − 0.5 when Nρ = n, and slightly smaller than m − 0.5 when Nρ = n + 1, meaning UCLα ≈ m − 0.5 in both cases. Since UCLα is obtained by linear interpolation, this indicates P(X ≤ m) ≈ (1 − α) + δ

(A9)

P(X ≤ m − 1) ≈ (1 − α) − δ.

(A10)

and

Therefore, αactual2 ≈ α + δ and αactual2 − αactual1 ≈ P(X = m) = 2δ ≤ 2α.

(A11)

Appendix I

References

Proof of Lemma

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Let Pα = 1 − α and m as (Pα − P1 )/(P2 − P1 ), we have Pα − P1 Pα − P1 m = = P2 − P1 (P2 − Pα ) + (Pα − P1 ) 1 = . (A1) (P2 − Pα )/(Pα − P1 ) + 1 If Pα − P1 ≤ P2 − Pα , then m ≤ 0.5. Consequently |Pα − P(X ≤ CL)| = Pα − P(X ≤ m) = Pα − P1 .

(A2)

If Pα − P1 > P2 − Pα , then m > 0.5, CL = [m + m] + 0.5 = m + 1.5 and |Pα − P(X ≤ CL)| = P(X ≤ m + 1) − Pα = P2 − Pα . (A3) Therefore |(1 − α) − P(X ≤ CL)|= min(1 − α − P1 , P2 − (1 − α)). (A4)

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[9] D. C. Montgomery, Introduction to Statistical Quality Control, 5th ed. Hoboken, NJ: Wiley, 2005. [10] C. J. Spanos, “Statistical process control in semiconductor manufacturing,” Proc. IEEE, vol. 80, no. 6, pp. 819–830, Jun. 1992. [11] C. H. Stapper, “The effects of wafer to wafer defect density variations on integrated-circuit defect and fault distributions,” Int. Bus. Machines J. Res. Develop., vol. 29, pp. 87–97, Jan. 1985. [12] C. H. Stapper, “The defect-sensitivity effect of memory chips,” IEEE J. Solid-State Circuits, vol. 21, no. 1, pp. 193–198, Feb. 1986. [13] C. H. Stapper, “Simulation of spatial fault distributions for integrated circuit yield estimations,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol. 8, no. 12, pp. 1314–1318, Dec. 1989. [14] S. L. Albin and D. J. Friedman, “The impact of clustered defect distributions in IC fabrication,” Manage. Sci., vol. 35, pp. 1066–1078, Sep. 1989. [15] A. V. Ferrisprabhu, “A cluster-modified Poisson model for estimating defect density and yield,” IEEE Trans. Semicond. Manuf., vol. 3, no. 2, pp. 54–59, May 1990. [16] I. Koren, Z. Koren, and H. Stapper, “A unified negative-binomial distribution for yield analysis of defect-tolerant circuits,” IEEE Trans. Comput., vol. 42, no. 6, pp. 724–734, Jun. 1993. [17] C. H. Stapper, “On yield, fault distributions, and clustering of particles,” Int. Bus. Mach. J. Res. Develop., vol. 30, pp. 326–338, May 1986. [18] C. H. Stapper, F. M. Armstrong, and K. Saji, “Integrated-circuit yield statistics,” Proc. IEEE, vol. 71, no. 4, pp. 453–470, Apr. 1983. [19] L. I. Tong, “Modified process control chart in IC fabrication using clustering analysis,” Int. J. Quality Reliab. Manage., vol. 15, no. 6, pp. 582–598, 1998. [20] P. Diggle, Statistical Analysis of Spatial Point Patterns, 2nd ed. London, U.K.: Arnold, 2003. [21] D. J. Friedman, M. H. Hansen, V. N. Nair, and D. A. James, “ Model-free estimation of defect clustering in integrated circuit fabrication,” IEEE Trans. Semicond. Manuf., vol. 10, no. 3, pp. 344–359, Aug. 1997. [22] Y. S. Jeong, S. J. Kim, and M. K. Jeong, “Automatic identification of defect patterns in semiconductor wafer maps using spatial correlogram and dynamic time warping,” IEEE Trans. Semicond. Manuf., vol. 21, no. 4, pp. 625–637, Nov. 2008. [23] H. H. Fellows, C. M. Mastrangelo, and K. P. White, “An empirical comparison of spatial randomness models for yield analysis,” IEEE Trans. Electron. Packag. Manuf., vol. 32, no. 2, pp. 115–119, Apr. 2009. [24] T. P. Karnowski, K. W. Tobin, S. S. Gleason, and F. Lakhani, “The application of spatial signature analysis to electrical test data: Validation study,” in Proc. Int. Soc. Opt. Eng., 1999, pp. 530–541. [25] K. W. Tobin, S. S. Gleason, F. Lakhani, and M. H. Bennett, “Automated analysis for rapid defect sourcing and yield learning,” in Future Fab International, vol. 1. London, U.K.: Technology Publishing Ltd., 1997, p. 313. [26] C. H. Wang, S. J. Wang, and W. D. Lee, “Automatic identification of spatial defect patterns for semiconductor manufacturing,” Int. J. Prod. Res., vol. 44, no. 23, pp. 5169–5185, Dec. 2006. [27] C. H. Wang, “Recognition of semiconductor defect patterns using spatial filtering and spectral clustering,” Expert Syst. Appl., vol. 34, no. 3, pp. 1914–1923, Apr. 2008. [28] C. H. Wang, “Separation of composite defect patterns on wafer bin map using support vector clustering,” Expert Syst. Appl., vol. 36, pp. 2554– 2561, Mar. 2009. [29] T. Yuan and W. Kuo, “A model-based clustering approach to the recognition of the spatial defect patterns produced during semiconductor fabrication,” IIE Trans., vol. 40, no. 2, pp. 93–101, Feb. 2008. [30] T. Yuan and W. Kuo, “Spatial defect pattern recognition on semiconductor wafers using model-based clustering and Bayesian inference,” Eur. J. Oper. Res., vol. 190, pp. 228–240, Oct. 2008. [31] B. Li, Q. Meng, and H. Holstein, “Point pattern matching and applications: A review,” in Proc. IEEE Int. Conf. Syst. Man Cybern., 2003, pp. 729–736. [32] M. Cohen and G. T. Toussaint, “Detection of structures in noisy pictures,” Pattern Recognit., vol. 9, pp. 95–98, Jul. 1977.

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Qiang Zhou received the B.Eng. degree in vehicle engineering and the M.Eng. degree in mechanical engineering, both from Tsinghua University, Beijing, China, in 2005 and 2007, respectively. He is currently working toward the Ph.D. degree in industrial engineering and the M.S. degree in statistics, both from the University of Wisconsin-Madison, Madison. He is a member of INFORMS.

Li Zeng received the B.S. and M.S. degrees in optical engineering from Tsinghua University, Beijing, China, in 2002 and 2004, respectively, the M.S. degree in statistics, and the Ph.D. degree in industrial engineering, both from the University of Wisconsin-Madison, Madison, in 2007 and 2010, respectively. She is currently a Research Associate with the Department of Industrial and Systems Engineering, University of Wisconsin-Madison. Her current research interests include statistical modeling and analysis in complex systems, and quality measure and control in healthcare.

Shiyu Zhou received the B.S. and M.S. degrees in mechanical engineering from the University of Science and Technology of China, Hefei, China, in 1993 and 1996, respectively, the M.S. degree in industrial engineering in 2000, and the Ph.D. degree in mechanical engineering, both from the University of Michigan, Ann Arbor. He is currently an Associate Professor with the Department of Industrial and Systems Engineering, University of Wisconsin-Madison, Madison. His current research interests are the in-process quality and productivity improvement methodologies by integrating statistics, system and control theory, and engineering knowledge. Dr. Zhou’s research is sponsored by the National Science Foundation, Arlington, VA, the Department of Energy, Washington D.C., the National Institute of Standards and the Technology-Advanced Technology Program, Gaithersburg, MD, and industries. He was a recipient of the CAREER Award from the National Science Foundation in 2006. He is a member of the Institute of International Education, INFORMS, the American Society of Mechanical Engineers, and Society of Manufacturing Engineers.