# A Full-Spectrum Registration Method for Zhuhai-1 Satellite Hyperspectral Imagery
## 3 Methods
### Evaluation Criteria
#### 3.3.2 Similarity Measures  
A block strategy was used to generate the similarity measures for two bands. First, the two bands were divided into blocks, whose sizes were determined based on the size of the image. Generally, the block size can be set as \(50\times 50\), \(100\times 100\), or \(200\times 200\) pixels, and it was set as \(100\times 100\) pixels in this study. The similarity measures for corresponding blocks were then calculated to evaluate the distribution of accuracy in the entire image. In this study, two commonly used similarity measures, structural similarity index (SSIM) and cosine similarity (CS), were selected for the accuracy assessment of image registration.  
The structural similarity measure is an objective image quality evaluation standard that is in line with the characteristics of the human visual system [34]. This indicator measures image similarity based on three aspects: brightness (mean), contrast (variance), and structure. Equation (17) is given as:  
\[SSIM(P,Q)=l(P,Q)^{\alpha}\cdot c(P,Q)^{\beta}\cdot s(P,Q)^{\gamma}, \tag{17}\]  
where  
\[l(P,Q)=\frac{2 upu_{Q}+c_{1}}{u_{P}^{2}+u_{Q}^{2}+c_{1}}, \tag{18}\]  
\[c(P,Q)=\frac{2\sigma_{P}\sigma_{Q}+c_{2}}{\sigma_{P}^{2}+\sigma_{Q}^{2}+c_{2}}, \tag{19}\]  
\[s(P,Q)=\frac{2\sigma_{PQ}+c_{3}}{\sigma_{P}\sigma_{Q}+c_{3}}, \tag{20}\]  
\[\sigma_{PQ}=\frac{1}{N-1}\sum\
olimits_{i=1}^{N}(P_{i}-\mu_{P})\big{(}Q_{i}- \mu_{Q}\big{)}, \tag{21}\]  
where \(P\) and \(Q\) are the reference image and the image to be registered, respectively; \(\mu_{P}\), \(\mu_{Q}\), \(\sigma_{P}^{2}\), \(\sigma_{Q}^{2}\), and \(\sigma_{PQ}\) represent the mean, variance, and covariance of images \(P\) and \(Q\), respectively; \(c_{1}\), \(c_{2}\), and, \(c_{3}\) are small normal numbers that are used to avoid instability due to the zero denominator in Equations (18)-(20); the parameters \(\alpha\), \(\beta\), and \(\gamma\) are all greater than zero and are used to adjust the proportions of the three components in the model; and the value range of SSIM is [0,1], with larger values indicating higher image similarities. The following equation is given when \(\alpha=\beta=\gamma=1\) and \(c_{3}=c_{2}/2\):  
\[SSIM(x,y)=\frac{\Big{(}2u_{x}u_{y}+c_{1}\Big{)}\Big{(}2\sigma_{xy}+c_{2}\Big{)} }{\Big{(}u_{x}^{2}+u_{y}^{2}+c_{1}\Big{)}\Big{(}\sigma_{x}^{2}+\sigma_{y}^{2} +c_{2}\Big{)}}, \tag{22}\]  
Cosine similarity measures the similarity between two vectors by measuring the cosine value of their inner product space, which is suitable for the vector comparison of any dimensions [35]. Digital images contain more feature codes that belong to high-dimensional space, and thus represent the application domain of the cosine similarity algorithm. The satellite image is regarded as a vector,and the cosine distance between the vectors is calculated to characterize the similarity between two images (Equation (23)) as:  
\[\cos(\theta)=\frac{\sum_{i=1}^{n}(X_{i}\times Y_{i})}{\sqrt{\sum_{i=1}^{n}(X_{i}) ^{2}}\times\sqrt{\sum_{i=1}^{n}(Y_{i})^{2}}}, \tag{23}\]  
where \(X_{i}\) and \(Y_{i}\) represent the image vectors of the reference and sensed image, respectively. The normalized cosine similarity values range from 0 to 1, with larger values representing higher image similarity.