# Multi-Temporal Sentinel-1 Backscatter and Coherence for Rainforest Mapping
## 2 Background
### Texture Features  
In the general interpretation of SAR images, texture has been recognized as an auxiliary feature, able to provide important information about the spatial dependency among neighbouring pixels located within a close vicinity [15; 16; 17]. In our framework, we derive backscatter spatial textures to be used as additional input information features for the Random Forests classification algorithm, with the aim to enhance the final classification accuracy. Among the several methods and techniques based on statistical models, in this paper we derive spatial information by estimating the Sum and Difference Histograms (SADH) textures [18] as follows. A discrete image can be interpreted as the realization of a bidimensional stationary and ergodic process, which means that each pixel of the image, \(u_{x,y}\), can be seen as the observation of a random variable. One of the most used statistical approaches to evaluate this relationship is that of counting the occurrence of the same pixels inside a defined domain \(D\), after a quantization of the original image dynamic using a grey level scale with a fixed number of levels \(N_{g}\). This assumption allows for the generation of a matrix of \(N_{g}\times N_{g}\) co-occurrence elements, known in literature as Gray Level Co-Occurrence Matrix (GLCM) [19; 20]. One of the main properties of the GLCM is its dependency on the relative position of the pixels in the image, \(\delta=(\delta_{x},\delta_{y})\). Indeed, a spatial configuration of the displacement vector defines a precise direction in the co-occurrence counting; in particular, setting the relative position \(\delta=(\delta_{x},\delta_{y})\), the GLCM elements are the results of a comparison between two random variables:  
\[\left\{ \begin{array}{rl}&u_{x,y}\\ &u_{x+\delta_{x},y+\delta_{y}}\end{array} .\right. \tag{2}\]  
Given the GLCM matrix, a set of different textures can be generated. One of the factors that limits the usability of the GLCM is its quadratic computational cost with respect to the amount of gray tones, \(O({N_{g}}^{2})\), that in general introduces constraints in term of memory allocation and computational time. In order to overcome such limitations, an alternative approach for the extraction of the textures is presented in [19] and makes use of the SADH method, which suggests the measurement of directional sum and difference matrices associated with the displacement vector \(\delta=(\delta_{x},\delta_{y})\). Each element of such matrices, \(s_{x,y}\) and \(d_{x,y}\), respectively, is given by:  
\[\left\{ \begin{array}{rl}&s_{x,y}=u_{x,y}+u_{x+\delta_{x},y+\delta_{y}}\\ &d_{x,y}=u_{x,y}-u_{x+\delta_{x},y+\delta_{y}}\end{array} .\right. \tag{3}\]  
This strategy facilitates the computation of spatial textures, since the determination of the co-occurrence matrix is no more needed. Indeed, the second-order joint probability function associated to the \(u_{x,y}\) and \(u_{x+\delta_{x},y+\delta_{y}}\) can be approximated as the product of the first-order probability functions of the sum and difference defined in (3), which, by definition, are uncorrelated random variables [18]. Theoretically, this assessment converges to an identity when the two random variables in (2) are Gaussian. Unfortunately, this does not often happen in real cases, so that the SADH method is chosen as an approximation of the GLCM method, since it reduces the time complexity to a linear computational cost, \(O(2N_{g})\), and limits possible problems of memory allocation required when using the complete bi-dimensional co-occurrence matrix [18]. We can now estimate the probability density functions \(\hat{P}_{s}(i)\) and \(\hat{P}_{d}(j)\), \((i,j=1,...,N_{g})\), of sum and difference, respectively, by normalizing the relative histograms for the total number of counts. Using the SADH approach, we can finally extract nine informative textures as follows [19]:1. Average (AVE), that describes the mean co-occurrence frequencies: \[\text{AVE}=\frac{1}{2}{\sum_{i=1}^{N_{g}}i}\cdot\hat{P}_{s}(i).\] (4)
2.