# Spatial Monitoring of Wafer Map Defect Data Based on 2D Wavelet Spectrum Analysis
## 1 Introduction  
Statistical process control (SPC) is one of the monitoring schemes that can be applied to fault detection in manufacturing processes. The basic SPC procedure consists of two phases: the calculation of the statistics to represent the objects' status from multivariate variables and the determination of the control limits that define an abnormal status of the objects. There has been a variety of research with respect to the monitoring of manufacturing processes. [PERSON] [1] introduced multiple indices for process monitoring such as the surface parameters, bulk, and epitaxial layer, which were applicable to the monitoring of the process quality of wafers. Besides, a model for multistage processes from binary data was proposed by [PERSON] et al. [2] by using the hierarchical likelihood approach to illustrate the cumulative effects. [PERSON] et al. [3] suggested a three-stage prognosis for batch manufacturing composed of the extraction of the health index, the variation of representative values using the percentile measure, and the modeling of profiles as gamma processes. As an advanced process monitoring methodology, the machine vision system (MVS) is actively used in the manufacturing area. MVS is an automated two-dimensional inspection system that monitors whether the objects of interest (e.g., products, processes) are in the normal status or not. This system has an advantage over manual investigation in that the inspection can be performed with less time and at a lower cost. Currently, the need for further research on automated two-dimensional (2D) monitoring algorithms is becoming greater to reduce the high inspection costs. MVS is repeatedly operated in the given sequence; obtaining images, analyzing raw data using an appropriate model, and training and updating the constructed model. In the general manufacturing area, MVS is applied to SPC to maintain productivity, diagnose manufacturing processes in real time, and detect faults early [4]. For instance, MVS is utilized in the quality control of wafers, which are the basis of electronic integrated circuits (ICs) as components of semiconductors. As shown in Figure 1, silicon crystal, called ingot, is melted by heat and sliced after hardening. The wafer is then fabricated in the manufacturing process through a series of masking, etching, and polishing to generate the IC. In the process, MVS can be introduced to each processing lot in order to monitor the status of wafers or detect any defective region, which leads to the increase in the yield of the manufacturing process. Though image based multivariate SPC approaches are widely applied to distinguish if the process will have a defective status or not, using a representative value, there are few literature works providing the spatial information of faulty regions; e.g., the defect clustering area in the wafer bin map (WBM) resulting from the semiconductor manufacturing process. [PERSON] et al. [6] applied the traditional [PERSON]'s T\({}^{2}\) chart to WBM data based on the number of defects and defect clustering indices. However, they failed to provide the spatial information of the clustered defects in WBM. [PERSON] [7] used a wavelet transform based multivariate process approach for an automated visual inspection of ripple defects. Therein, he showed that wavelet-based methods had better performance than traditional approaches, which use raw images directly. [PERSON] and [PERSON] [8] proposed the wavelet texture based T\({}^{2}\) control chart, where principal component analysis (PCA) was added in order to extract the feature vector in an efficient manner. To provide spatial information on image data, [PERSON] et al. [9] introduced the spatial \(\overline{X}\) chart to illustrate the density of woven images based on 2D discrete Fourier transform (DFT), which localized wavelet coefficients in the frequency domain. Later, by segmenting a single image matrix into 1D signals, [PERSON] et al.