A novel hypergraph convolution network for wafer defect patterns identification based on an unbalanced dataset

Abstract

In semiconductor industry, various wafer defect patterns represent different causes of manufacturing failures. Identification of specific defect patterns is important to wafer fabrication process. Recently, many studies concentrate on developing Deep Learning algorithms such as Convolution Neural Network, most of which neglect hyper-relations among dataset. Therefore, in this work, a novel Hypergraph Convolution Network is proposed for the automatic Wafer Defect Identification (HCN-WDI). The main contributions include: (1) The detailed theoretical formulation and NP-Completeness proof of normalized cut for (hyper-)edge segmentation is firstly discussed. (2) The data augmentation techniques are applied to balance the number inequality of different patterns in wafer defect dataset WM-811K. (3) The Hyper Convolution Network is implemented as an end-to-end operator to identify wafer defect patterns and three conventional image classifiers are used as feature extractors and reference baselines for the proposed HCN-WDI model. The experimental results show that the proposed HCN-WDI model outperforms other three fine-tuning conventional image classifiers and obtains the highest $96.44\%$ averaged classification accuracy. Besides, by comparing the results from various combinations of extracted features, it is concluded that the accuracy of the HCN-WDI model is also dependent on the quality rather than quantities of feature extraction.

Appendix A: Proof of NP-completeness

Proof

To prove NP-completeness of optimal normalized cuts in hypergraph, i.e.,

$$\begin{aligned} \mathop {{{\,\mathrm{arg\,min}\,}}}_{\emptyset \ne S \subset V} c\left( S \right) := \textrm{vol}\ \partial S \left( \frac{1}{\textrm{vol}\ S} + \frac{1}{\textrm{vol}\ S^c} \right) \end{aligned}$$

(A1)

Two restrictions are introduced:

• edge weight keeps constant, i.e., $\textrm{w}(e)=1$.

• each hyperedge only connects two nodes.

Based on the two restrictions above, the definitions of volumes of vertex subset S and $S^c$ can be reduced as,

$$\begin{aligned}{} & {} \textrm{Vol}\ S := \sum _{v\in S} \sum _{e \in E} \textrm{w}\left( e \right) h\left( v, e \right) = \sum _{v \in S} \sum _{t \in V} \textrm{w}\left( v,t \right) \\{} & {} \textrm{Vol}\ S^c := \sum _{u \in S^c} \sum _{e \in E} \textrm{w} \left( e \right) h\left( u,e \right) = \sum _{u \in S^c} \sum _{t \in V} \textrm{w} \left( u,t \right) \end{aligned}$$

Then the volume of hyperedge boundary $\partial S$ is defined as

$$\begin{aligned} \textrm{vol}\ \partial S:= & {} \sum _{e \in \partial S} \textrm{w}\left( e \right) \frac{\left| e \cap S \right| \left| e \cap S^c \right| }{\delta \left( e \right) } \\ {}= & {} \sum _{e \in \partial S} \textrm{w}\left( e \right) \frac{\left| e \cap S \right| \left| e \cap S^c\right| }{\sum _{v \in V} h\left( v,e \right) } \\= & {} \frac{1}{2} \sum _{e \in \partial S} \textrm{w}\left( e \right) = \frac{1}{2} \sum _{v \in S, u \in S^c} \textrm{w}\left( u,v \right) \end{aligned}$$

in which, $\left| e \cap S \right| = \left| e \cap S^c \right| =1$ based on the two restrictions above.

Summarily, the Eq. (A1) can be converted to

$$\begin{aligned}{} & {} \mathop {{{\,\mathrm{arg\,min}\,}}}_{\emptyset \ne S \subset V} c \left( S \right) := \textrm{vol} \partial S \left( \frac{1}{\textrm{vol} S } + \frac{1}{\textrm{vol} S^c} \right) \nonumber \\{} & {} \quad = \frac{1}{2} \sum _{v \in S, u \in S^c} \textrm{w}\left( u,v \right) \nonumber \\{} & {} \qquad \left[ \frac{1}{\sum \limits _{v \in S} \sum \limits _{t \in V} \textrm{w}\left( v,t \right) } + \frac{1}{\sum \limits _{u \in S^c} \sum \limits _{t \in V} \textrm{w}\left( u,t \right) } \right] \end{aligned}$$

(A2)

It denotes that minimizing normalized cuts for hypergraph can be reduced to simple graph which is NP-complete.

According to NP-Completeness proof techniques Garey and Johnson (1979), by applying some specifications of additional restrictions on the problem $\Pi$ which is to be proved as NP-Completeness, the modified problem with additional restrictions can be reduced to be identical another problem $\Pi ^{\prime }$ which has been known as NP-Complete.

Therefore, the NP-completeness of the problem that minimizing normalized cuts of hypergraph has been proved.