texture represented by transition matrix.

In the case of non-parametric representation of the texture and by assuming independent probabilities for the observed pixels ( $1^{st}$ order Markovian model). For a first order Markov process, the 0th order statistics of the samples must be an Eigenvector of $p_{i\vert j}$ with Eigenvalue $1$. Unfortunately, this means that a uniform prior for $T$ over $p_{i\vert j}$ is inconsistent with the uniform prior used in the 0th order case. To re-establish the consistency, it is necessary to choose a $1^{st}$ order prior such that the expected value of a column of the transition matrix $p_{i\vert j}$ is obtained by adding $1/I$ rather than 1 to the number of observations in that column of the transition matrix before normalizing the column to sum to $1$. This means that the transition matrix,

$$E(p_{i\vert j}\vert S_{1}^{c})=\frac{C_{ij}+1/I}{1+\sum _{i}C_{ij}}=\frac{C_{ij}+1/I}{1+o_{j}}$$ (6)

where $C_{i}j$ is the number of times that intensity $i$ follows intensity $j$ in the sequence $X_{1}^{c}$. And hence the expected 0th order distribution (which is the vector $\frac{(o_{j}+1)}{(c+I)}$) has the desired properties since:

$$\sum _{j}E(p_{i\vert j}\vert S_{1}^{c})\frac{(o_{j}+1)}{(c+I)}=\frac{\sum _{j}C_{ij}+1/I}{c+I}=\frac{o_{i}+1}{c+I}$$ (7)

Which is the same as Equation 5. This modification is equivalent to imposing a prior over $p_{i\vert j}$ that favors structure in the Markov process and is proportional to $\prod _{ij}p_{i\vert j}^{(1/I-1)}$). This gives Algorithm 2.