non parametric measures:

Non parametric models for the texture are expressed in terms of histograms. Some of the well known dissimilarity measures for histograms are the following.

• Minkowski-form distance is defined based on the $L_{p}$norm:
  

  $$D(K_{1},K_{2})={\textstyle \left(\sum_{n}\left\vert K_{1}(n;T_{1})-K_{2}(n;T_{2})\right\vert^{p}\right)^{1/p}}$$ (9)

  
  

where $K_{1}(n;T_{1})$ and $K_{2}(n;T_{2})$ are the two distributions and $1\leq p\leq\infty$. $L_{1}$computes the city block or Manhattan distance, $L_{2}$ is the Euclidean distance and finally $L_{\infty}$measures the maximal difference.

• Bhattacharyya distance measures the statistical separability of spectral classes, leading to an estimate of the probability of correct classification and it has a geometric interpretation. For two given distributions $K_{1}(n;T_{1})$ and $K_{2}(n;T_{2})$ this distance is given by:
  

  $$d(K_{1},K_{2})=\left(1-\sum_{n=1}^{N}\sqrt{K_{1}(n;T_{1})*K_{2}(n;T_{2})}\right)^{1/2}$$ (10)

  
  

• The $\chi^{2}$statistic measures the likeliness of one distribution being drawn from another one and is given by:
  

  $$D(K_{1},K_{2})=\sum_{n}\frac{\left(K_{1}(n;T_{1})-K(n)\right)^{2}}{K(n)}$$ (11)

  
  

where $K(n)=[K_{1}(n;T_{1})+K_{2}(n;T_{2})]/2$ is the mean histogram. The main disadvantage of $\chi^{2}$distance is that it yields inaccurate results with low number of observations. Moreover no bin in the histograms can have zero counts.

• The Kullback-Leibler divergence (KL) is regarded as a measure of the extent to which two probability density functions agree and is expressed as:
  

  $$L=-\int p(x)\ln\frac{\tilde{p}(x)}{p(x)}dx$$ (12)

  
  

it can be shown that $L\geq0$. For two discrete distribution the integration becomes a summation over the bins of $K_{1}(n;T_{1})$ and $K_{2}(n;T_{2})$.

• The Jeffrey-divergence (JD) is the symmetric version of KL distance with respect to both $p(x)$ and $\tilde{p}(x)$.
• The Quadric Form (QF). The histogram quadratic distance is designed to measure the weighted similarity between histograms [2]. The quadratic distance provides better results than ``like-bin'' only comparisons at the price of high computational costs. The quadratic distance between histograms and is given by:
  

  $$D(K_{1},K_{2})=\left[\left(K_{1}(n;T_{1})-K_{2}(n;T_{2})\right)^{t}A\left(K_{1}(n;T_{1})-K_{2}(n;T_{2})\right)\right]^{1/2}$$ (13)

  
  

where $A=[a_{ij}]$ is a $N\times N$ matrix and $a_{ij}$is the similarity coefficient between indices (bins) $i$ and $j$. $aij$ is given by $a_{ij}=1-d_{ij}/d_{max}$and $d_{ij}=\left\vert K_{1}(i;T_{1})-K_{2}(j;T_{2})\right\vert$and denotes the similarity between bins $i$ and $j$.

• The Earth Mover's Distance (EMD): This distance considers each value of one distribution as a quantity to be moved (earth) to the other distribution (holes). The EMD is thus defined as the minimum cost of transferring one distribution to another one. The computation of EMD comes from a well-known transportation problem [3,4]. The advantage of EMD is that it works on distributions with a different number of bins.