# Spatial Monitoring of Wafer Map Defect Data Based on 2D Wavelet Spectrum Analysis
## 4 Application and Discussion
The distribution was split into normal and suspicious parts by the valley, that is the defective region belonged to the distribution that had a greater distance from the local minimal value, because the suspicious area had a further distance from the background than the other. The probability of faults was calculated by the cumulative distribution function (CDF), which was more than the valley. If the probability was more than the pre-specified standard for fault detection, the wafer was determined to have failed. The visualization of the distance matrix and density estimation from the given images having circle and vertical defects and half defects is shown in Figure 5. If the degree of defects in a specific window was serious, the window had a greater distance from the background. The kernel density function was separated by each valley, which lied at 0.5564 and 0.6219 in the WBM with circle and vertical defects and half defects, respectively. The probabilities of failure from two images were estimated as 0.2928 and 0.3394. Since we set the threshold for failure to 0.3 in this application, the half defective WBM would be classified as the faulty state, and the UCL for the spectral control chart was defined as the maximizer of a suspicious distribution. In this example, UCL was set to 1.7461 to detect the failure parts exceeding the control limits. Figure 4: Energy profiles of windows from wafer bin maps: (**a**) normal; (**b**) defective. Figure 5: (**a**) Circle and vertical defective WBM; (**b**) spectral reconstruction of the distance matrix of (**a**); (**c**) density estimation of the distance matrix of (**a**); (**d**) half defective WBM; (**e**) spectral reconstruction of the distance matrix of (**d**); (**f**) density estimation of the distance matrix of (**d**). Based on the matrix composed of the statistics and upper control limit, a spectrum-based 3D spatial control chart was constructed. The spatial SPC implemented from the WBM image given in Figure 5 is provided in Figure 6, where three axes, X, Y, and Z, stand for the column, the row index of windows, and the control statistics. Because the defective region had a greater distance from the background, it would have higher values of the control statistics, and the statistics exceeding UCL were detected as faulty regions. The plane for UCL could be generated to detect the faulty location easily with higher values of the control statistics, and the plane was parallel to the XY-plane. The result showed that WS-2 DSPC conducted the detection of the faulty region and calculated the degree and strength of the defects with an efficient computation speed (it took approximately 2.17 s for a single image with a size of \(2^{10}\times 2^{10}\) on an Intel(r) Core(tm) i-5 6400 CPU @ 2.70 GHz and 8.00 GB of RAM, including the feature exportation and visualization). If the application of WS-2 DSPC can be extended to multiple online images in a timed sequence, a system for stereotaxic and spatial monitoring, as well as the diagnosis of defects can be implemented without a bottleneck in the manufacturing process.