Detection and classification of defect patterns on semiconductor wafers

IIE Transactions, Dec, 2006 by Chih-Hsuan Wang, Way Kuo, Halima Bensmail

Fixed the number of clusters as G

Use K-means to initialize the Gaussian EM algorithm Repeat

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M step: compute maximum-likelihood parameter estimates given initialization:

[^.n.sub.k] [left arrow] [[summation].sub.i=1.sup.n] [^.t.sub.ik] (updating the members [n.sub.k] in every cluster)

[^.p.sub.k] [left arrow] [^.n.sub.k]/n (updating the mixture proportion [p.sub.k] of each cluster)

[^.[mu].sub.k] [left arrow] [[[summation].sub.j=1.sup.n] [^.t.sub.ik] [x.sub.i]]/[^.n.sub.k] (updating the mean of [[mu].sub.k] each cluster)

[^.[SIGMA].sub.k] = [[[summation].sub.i=1.sup.n] [^.t.sub.ik] ([x.sub.i] - [^.[mu].sub.k])([x.sub.i] - [^.[mu].sub.k])[.sub.t]]/[^.n.sub.k] (updating the covariance [[SIGMA].sub.k] of each cluster)

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E step: compute [^.t.sub.ik] given the parameter estimates from the M step:

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[^.t.sub.ik] [left arrow] [[^.p.sub.k][f.sub.k]([x.sub.i]|[^.[mu].sub.k], [^.[SIGMA].sub.k])]/[[[summation].sub.j=1.sup.G][^.p.sub.j][f.sub.j]([x.sub.i]|[^.[mu].sub.j],[^.[SIGMA].sub.j])](MAP: choosing the k value to maximize [t.sub.ik])

Repeat until a convergence criterion is satisfied.

3.4. Principle of the spherical-shell algorithm

The Gaussian EM algorithm is unable to classify nonconvex rings, and thus we adopt the spherical-shell algorithm of Krishnapuram et al. (1992) to classify rings and estimate their prototype parameters. If the cluster resembles a hyper-spherical shell, then its prototype [[theta].sub.k] consists of two parameters ([[mu].sub.k], [r.sub.k]), with [[mu].sub.k] being the center of the hyper-sphere and [r.sub.k] its radius. In order to avoid dealing with nonlinear coupled problems (Dave, 1992), we define the distance from defect [x.sub.j] to a prototype [[theta].sub.k] = ([[mu].sub.k], [r.sub.k]) as (see Fig. 4):

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[d.sub.ik.sup.2] = [d.sup.2]([x.sub.i], [[theta].sub.k]) = ||[x.sub.i] - [[mu].sub.k]||[.sup.2] - [r.sub.k.sup.2]. (6)

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Equivalently, the objective function to be minimized can be defined as [[summation].sub.k=1.sup.G] [[summation].sub.i[member of][G.sub.k]][d.sub.ik.sup.4]. For the purpose of simplification, we rewrite the distance as:

[d.sub.ik.sup.4] = [P.sub.k.sup.T][M.sub.i][P.sub.k] [V.sub.i.sup.T][P.sub.k] [B.sub.i], (7)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Obviously, the ring center [[mu].sub.k] and ring radius [r.sub.k] can be obtained through [P.sub.k]. And the optimal [P.sub.k] that minimizes the objective function can be derived as follows:

[P.sub.k] = -0.5([H.sub.k])[.sup.-1][W.sub.k], (8)

where [H.sub.k] = [[summation].sub.[x.sub.i][member of][G.sub.k]] [M.sub.i] and [W.sub.k] = [[summation].sub.[x.sub.i][member of][G.sub.k]]. [V.sub.i]. The resulting spherical-shell algorithm can be summarized as:

Fix the number of clusters as G

Use K-means to initialize the spherical shell algorithm Repeat

Calculate [H.sub.k] and [W.sub.k] using the above formula

Update [P.sub.k] for each cluster