# A Novel Perceptual Hash Algorithm for Multispectral Image Authentication
## 2 Preliminaries
### Affinity Propagation and Mutual Information-Based Band-Clustering  
Different bands of a multispectral image have different spectral responses and can be distinguished from each other based on their grayscales. Even in the same image, there are big grayscale differences between the bands, especially between the visible band and infrared band. Figure 2 shows several comparison instances of different type regions (mountain and urban) of the Landsat thematic mapper (TM) image data of Band 1 (In\(R_{1}\)) and Band 4 (In\(R_{4}\)). Obviously, there are some differences between mountainous areas and the urban areas. Furthermore, even for the same surface feature, the difference between different bands is obvious, as shown in Figure 2b,d.  
Taking this into account, we adopt band clustering to separate the bands into several groups (band subsets). Band-clustering is used to identify the subset of bands that are as independent as possible, and it is adopted to obtain compactable hash values in this algorithm.  
In this paper, we use AP clustering [17; 18] with MI to divide the bands into several band subsets. The AP algorithm is an exemplar-based clustering algorithm that uses max-sum belief propagation to obtain an optimal exemplar which can determine the number of clusters automatically, with a message exchange approach. The final clustering centers will be generated depending on the given dataset. Compared with traditional clustering algorithms, such as K-means clustering, AP clustering does not need to set the clustering number. Therefore, it is an efficient clustering technique to deal with datasets of many instances because of its better clustering performance over traditional methods.  
The AP clustering algorithm starts with the construction of a similarity matrix \(S\in R^{L\times L}\), in which the element \(s(i,j)(i\
eq k)\) measures the similarity between band \(k\) (In\(R_{k}\)) and band \(i\) (In\(R_{i}\)). In our perceptual hash algorithm, MI is chosen to construct the similarity matrix, i.e., each \(s(i,j)\) denotes the mutual information between the \(i\)-th and the \(j\)-th while \(L\) is determined by the number of bands.  
Figure 2: Ground comparison in thematic mapper (TM) imagery: (**a**) City in band 1; (**b**) Coteau area in band 1; (**c**) City in band 4; (**d**) Coteau area in band 4.  
MI measures the statistical dependence between two random variables and can therefore be used to evaluate the relative utility of each band to classification [19]. Given two random variables \(X\) and \(Y\) with marginal probability distributions \(p(X)\) and \(p(X)\) and joint probability distribution \(p(X,Y)\), their MI is defined as:  
\[I(X,Y)=\sum_{X}\sum_{Y}p(X,Y)\log\frac{p(X,Y)}{p(X)p(Y)} \tag{1}\]  
It follows that MI is related to entropy as:  
\[I(X,Y)=H(X)+H(Y)-H(X,Y) \tag{2}\]  
where \(H(X)\) and \(H(Y)\) are respectively the entropies of \(X\) and \(Y\), and \(H(X,Y)\) is their joint entropy.  
Treating the band's multispectral images as random variables, MI can be used to estimate the dependency between them, and was introduced for band selection in [20; 21]. Using Equation (2), the mutual information between each band of the multispectral image can be calculated, which can be used to evaluate the relative utility of each band to classification.