## 5 Methodology
### Texture Feature Extraction
#### 5.1.3 Calculation of Co-Occurrence Features  
The GLCMs obtained from the detailed images of HL2 and LH2 further calculate the co-occurrence features in 0\({}^{\circ}\) and 90\({}^{\circ}\), respectively. Both of the GLCMs in 45\({}^{\circ}\) and 135\({}^{\circ}\) obtained from the detailed image of HH2 further calculate the co-occurrence features in the double-diagonal orientations. The 7 aforementioned co-occurrence features can be obtained using the following texture descriptors:  
Angular Second Moment (ASM): The uniformity of distribution of the gray level in an auspicious pattern image can be represented thus [35]. The texture descriptor of ASM is expressed as follows:  
\[\sum_{i=1}^{n}\sum_{j=1}^{n}P_{ij}^{2} \tag{1}\]  
The value of ASM ranges from \(1/n^{2}\) to 1. When the value is 1, it means a constant image. Here, \(P_{ij}\) is calculated as \(\frac{M_{ij}}{\sum_{i=1}^{n}\sum_{j=1}^{n}M_{ij}}\).  
Entropy (ENT): A statistical measure of randomness determines the textural interference in an auspicious pattern image [24; 35]. The texture descriptor of ENT is obtained as follows:  
\[-\sum_{i=1}^{n}\sum_{j=1}^{n}P_{ij}\cdot\log_{10}^{P_{ij}} \tag{2}\]  
The larger the ENT value, the higher the textural complexity of the auspicious pattern. Thus, there should be a highly negative correlation between ASM and ENT.  
Contrast (CON): The local variations in an auspicious pattern image can be measured by this descriptor, which is calculated as follows:  
\[\sum_{i=1}^{n}\sum_{j=1}^{n}(i-j)^{2}\cdot P_{ij} \tag{3}\]  
The higher the CON, the higher the image contrast will be. Hence, a CON of 0 means a constant image.  
Homogeneity (HOM): HOM can also be called an inverse differential moment, which measures the similarity between the distributions of elements in a GLCM and those in the diagonal GLCM [24]. The textural descriptor is defined as follows:  
\[\sum_{i=1}^{n}\sum_{j=1}^{n}\frac{P_{ij}}{1+(i-j)^{2}} \tag{4}\]  
The representation of HOM is contrary to that of CON--the higher the HOM, the lower the image contrast (or the more the image homogeneity) will be.  
Dissimilarity (DIS): The degree of dissimilarity of the gray levels in an auspicious pattern image is now measured. DIS is very sensitive to the arrangement of gray-level values or tones in an imagery space, and is expressed as follows:  
\[\sum_{i=1}^{n}\sum_{j=1}^{n}|i-j|P_{ij} \tag{5}\]  
A higher DIS value means a greater dissimilarity of gray levels in the image.  
Correlation (COR): The spatial dependencies between the image pixels can be defined [24] to indicate the textural directionalities of the auspicious patterns [36] by this textural descriptor, which is calculated as follows:  
\[\frac{\sum_{i=1}^{n}\sum_{j=1}^{n}(i\cdot j)P_{ij}-\mu_{x}\mu_{y}}{\sigma_{x} \sigma_{y}} \tag{6}\]  
In other words, COR is a correlation coefficient of GLCM. The higher the COR is, the greater is the spatial dependency of gray levels in a certain direction.  
Cluster tendency (CLU): The degree of the textural clustering in an auspicious pattern image is measured and can be obtained as:  
\[\sum_{i=1}^{n}\sum_{j=1}^{n}\bigl{(}i-\mu_{x}+j-\mu_{y}\bigr{)}^{2}\cdot P_{ij} \tag{7}\]  
It should be noted that a lower CLU value means higher textural clustering. Moreover, there is an interrelationship between CLU and COR. When the values of CLU and COR are large, there will be an indefinite textural directionality or a wide distribution of gray levels. Here, \(\mu_{x}\), \(\mu_{y}\), \(\sigma_{x}\), and \(\sigma_{y}\) are calculated as follows:  
\[\left\{\begin{array}{c}\mu_{x}=\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}i \cdot P_{ij}\\ \mu_{y}=\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}j\cdot P_{ij}\\ \sigma_{x}=\sqrt{\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}(i-\mu_{x})^{2} \cdot P_{ij}}\\ \sigma_{y}=\sqrt{\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}(j-\mu_{y})^{2} \cdot P_{ij}}\end{array}\right. \tag{8}\]