Texture Estimation

If we already have a prior distribution $T$ for one texture, we can estimate the posterior distribution of the texture descriptor elements $s_{i}$given the observed sequence of the data i.e. $P(s_{i}\mid X,T)$. For example $s_{i}$could be the mean and and covariance of a given region (Gaussian model for texture) or they can be the bins of histogram or the indices of a transition matrix in case of a $1^{st}$order Markovian model of texture. This probability can be expressed using Bayes rule as:

$$P(s_{i}\mid X,T)=K\, P(X\mid s_{i},T)\, P(s_{i}\mid T)$$ (2)

with $K$ being a normalization factor to make sure that the probabilities sum to $1$. The prior distribution determines the term $P(s_{i}\mid T)$. We can eliminate $s_{i}$from the likelihood term $P(X\mid s_{i},T)$ since it doesn't have any effects on the probability of the observed sequence.