texture represented by histograms.

In the case of non-parametric representation of the texture and by assuming independent probabilities for the observed pixels ($0^{th}$order Markovian model) we obtain:

$$P(X\mid T)=\prod_{k=1}^{c}P(x_{k}\mid T)=\prod_{j=1}^{I}P(s_{j}\mid T)^{O_{j}}$$ (3)

where $O_{j}$ is the number of occurrence of $s_{j}$ in the observation vector $X$, so that we have $\sum O_{j}=c$. This would lead to the following equation for the posterior probability of the texture parameters, $S$:

$$P(s_{i}\mid X,T)=K\prod_{j=1}^{I}P(s_{j}\mid T)^{O_{j}}\, P(... ...,j\neq i}^{I}P(s_{j}\mid T)^{O_{j}}\, P(s_{i}\mid T)^{O_{i}+1}$$ (4)

If on the other hand there is no a priori knowledge about $T_{1}$or equivalently $S$, the above formula should be integrated over the set of all possible textures, $T$:

$$\begin{array}{c} P(s_{i}\mid X)=K\int\prod_{j=1,j\neq i}^{I}... ...ots dp_{s_{1}}\vspace{0.6cm}\\ =\frac{O_{i}+1}{c+I}\end{array}$$ (5)

where the last equality come from the law of succession.This result can be applied recursively to the whole sequence to calculate the conditional probability of each new observation given the preceding observations Algorithm 1.