# Research on Wavelet Transform Modulus Maxima and OTSU in Edge Detection
## 3 Materials and Methods
### Theory
#### 3.1.3 Information Entropy and SSIM  
Information entropy is a concept introduced by [PERSON], a famous American mathematician and founder of information theory. Information entropy is used as a measure of uncertainty--that is, the probability of discrete random events occurring. Briefly, the more chaotic things are, the higher the information entropy, and conversely, the lower the information entropy [35; 36; 37; 38; 39][PERSON]  
For an image, random variable described by the information entropy is the pixel value of the image. The information entropy of an image is a statistical form of a feature and reflects the size of the average amount of information in the image. Typically, the greater the information entropy of an image, the more informative and detailed it is, and the better the image quality. The one-dimensional information entropy of an image can show the concentration characteristics of the grey-scale distribution in the image, but it cannot reflect the spatial characteristics of the grey-scale distribution. On the basis of the one-dimensional entropy, characteristic quantities which can reflect the spatial characteristics of the grey-scale distribution can be introduced to compose the two-dimensional entropy of an image [40]. The neighborhood grey mean of the image is chosen as the spatial characteristic quantity of the grey scale distribution, and combated with the pixel grey scale of the image to form into the feature binary (\(i\), \(j\)). Equations (7) and (8) are the calculation methods of image information entropy [41]. \(f(i,j)\) is the frequency of occurrence of (\(i\), \(j\)), N is the scale of the image.  
\[H=-\sum_{i}\sum_{j}P_{i,j}\log P_{i,j} \tag{7}\]  
\[P_{i,j}=f(i,j)/N^{2} \tag{8}\]  
SSIM (Structural Similarity Index Measure) is a measure of the degree of similarity between two images, and the structural similarity between the two can be seen as a quality measure of distorted images. Compared to traditional image quality measures, SSIM ismore in line with the human eye's judgment of images. Given two signals \(x\) and \(y\), the structural similarity between the two is defined [42; 43; 44] as shown in Equations (9) and (10).  
\[SSIM=[l(x,y)]^{\beta}[c(x,y)]^{\beta}[s(x,y)]^{\gamma} \tag{9}\]  
\[l(x,y)=\frac{2\mu_{x}\mu_{y}+C_{1}}{\mu_{X}^{2}+\mu_{Y}^{2}+C_{1}},c(x,y)= \frac{2\sigma_{x}\sigma_{y}}{\sigma_{x}^{2}+\sigma_{y}^{2}+C_{2}},s(x,y)=\frac {\sigma_{xy}+C_{3}}{\sigma_{x}\sigma_{y}+C_{3}} \tag{10}\]  
where \(l(x,y)\) is used to compare the brightness of \(x\) and \(y\), \(c(x,y)\) to compare the contrast of \(x\) and \(y\), and \(s(x,y)\) to compare the structure of \(x\) and \(y\). \(\alpha\), \(\beta\), \(\gamma\) are parameters to adjust the relative stability of \(l(x,y)\), \(c(x,y)\), and \(s(x,y)\). \(\mu_{X}\) and \(\mu_{Y}\) are the mean values of \(x\) and \(y\), \(\sigma_{x}\) and \(\sigma_{y}\) are the standard deviations of \(x\) and \(y\), respectively, and \(\sigma_{xy}\) is the covariance of \(xy\). \(C_{1}\), \(C_{2}\) and \(C_{3}\) are all constants used to maintain the stability of \(l(x,y)\), \(c(x,y)\) and \(s(x,y)\). In practice, it will generally default to, \(\alpha=\beta=\gamma=1\) and \(C_{3}=0.5\times C_{2}\), so Equation (9) can be simplified to Equation (11).  
\[SSIM=\frac{(2\mu_{x}\mu_{y}+C_{1})(2\sigma_{xy}+C_{2})}{(\mu_{x}^{2}+\mu_{y}^{ 2}+C_{1})(\sigma_{x}^{2}+\sigma_{y}^{2}+C_{2})} \tag{11}\]