# Research on Multi-Scale Fusion Method for Ancient Bronze Ware X-ray Images in NSST Domain
## 2 Materials and Methods
### MSMG-Based High-Pass Subband Fusion Rule  
High-pass subband fusion is an image processing technique that extracts precise details such as edges and textures from the source image through spectral analysis and filtering. The primary objective of high-pass subband fusion is to preserve the intricate information of the source image and produce a fused image that is superior in clarity, richness, and depiction of the detailed attributes within the image. In the process of high-pass subband fusion, the essential factor is to optimize all the detailed characteristics of the source image. The multi-scale morphological gradient (MSMG) [16] shares similar functionality with common edge detection operators (e.g., Prewitt, Sobel, Laplacian) in detecting image edges. However, common edge detection operators enhance the noise in the image and degrade the image quality when extracting the image edges [17]. For the detection of boundaries, it is very important in the fusion of bronze mirror X-ray images [18]. The MSMG is an operator that can extract gradient information from an image. It represents the contrast difference between a pixel and its neighboring pixels. Consequently, the MSMG is frequently utilized in scenarios such as edge detection and image segmentation [17]. This method of transformation can maintain a low enhancement rate in noise while extracting image edges, effectively decreasing the noise's impairment of the results of edge detection. To enhance the informational value of the high-pass subband image and create a fused image with detailed edges and textures, this study utilizes the multi-scale morphological gradient operator (MSMG) in combination with the eight-neighborhood-based weighted and modified Laplace operator (WSEML) [9]. This approach significantly enhances the clarity of contour edges in the fused image, preserves detailed information, and leads to improved fusion of the high frequency subbands. The single-scale morphological gradient is defined in Equation (1):  
\[G_{l}[(x,y)]=\left\lfloor f(x,y)\oplus g_{l}(x,y)-(f(x,y)\odot g_{l}(x,y))\right\rfloor \tag{1}\]  
Figure 2: The schematic diagram of the three-level NSST decomposition. where \((x,y)\) denotes the pixel position, \(l\) stands for the number of multi-scale levels, \(f(x,y)\) represents the source image, \(g_{l}(x,y)\) refers to the structural elements at the \(l\) level scale, \(G_{l}[(x,y)]\) represents the structural information at the \(l\) level scale, and \(\oplus\) and \(\odot\) denote the dilation and erosion operations, respectively. In summary, the \(MSMG\) is calculated as in Equation (2):  
\[MSMG(x,y)=\sum_{n=1}^{n}w_{q}\times G_{q}(x,y) \tag{2}\]  
where \(w_{q}\) denotes the gradient weight of the \(G_{q}\) level, which can be expressed as in Equation (3):  
\[w_{q}=\frac{1}{2q+1} \tag{3}\]  
The multi-scale morphological gradient of the high frequency subbands \(I_{l,k}^{A}\) and \(I_{l,k}^{B}\) in each layer are as follows:  
\[\left\{ \begin{aligned}  MG_{l,k}^{A}&=MSMG\left(abs\left(I_{l,k}^{A} \right),n\right)\\ MG_{l,k}^{B}&=MSMG\left(abs\left(I_{l,k}^{B} \right),n\right)\end{aligned} \right. \tag{4}\]  
In Equation (4), \(MG_{l,k}^{A}\) and \(MG_{l,k}^{B}\) are the multi-scale morphological gradient of the \(l\) high frequency subbands of the \(k\) layer of the bronze mirror X-ray images A and B, respectively. \(WSEML\) is the metric used for extracting details and is defined by the following equation:  
\[WSEML_{lr}(i,j)=\sum_{m=-r}^{r}\sum_{m=-r}^{r}w(m+r+1,n+r+1)\times EML(i+m,j+n) \tag{5}\]  
\(w\) is the weighting matrix of \((2r+1)\times(2r+1)\), where \(EML\) is defined as follows:  
\[ \begin{aligned} EML_{lr}(i,j)&=|2C_{lr}(i,j)-C_{lr }(i-1,j)-C_{lr}(i+1,j)|+\ |\ 2C_{lr}(i,j)-\\ C_{lr}(i,j-1)-C_{lr}(i,j+1)\Big{|}+\frac{1}{\sqrt{2}}\Big{|}2C_{lr} (i,j)-C_{lr}(i-1,j-1)-C_{lr}(i+1,j+\\ 1)\Big{|}+\frac{1}{\sqrt{2}}\Big{|}2C_{lr}(i,j)-C_{lr}(i-1,j+1)-C _{lr}(i+1,j-1)\ |\end{aligned}