A statistical model of epitome

First we assume the image $X$ consists of a collection of patches $Z=\{Z_k\}_{k=1}^P$, Note here we allow the patches to have overlaps in $X$, but only take on the shape of a square. We let $E$ denote the epitome, and $e=(\mu,\phi)$ the mean and variance of $E$. Finally, we let $T=\{T_k\}_{k=1}^P$ represent the mapping $Z \rightarrow E$ between the patches and their corresponding origins in the the epitome.

We consider the epitome as a generative model of patches by the following dependency graph. Note the patches $Z$ are our observations, the epitome $E$ is the model to be estimated and the mappings $T$ are the set of hidden variables.

Figure 1: Dependency graph of epitome, mappings and patches

We model the conditional probability $p(Z\vert T, e)$ as a Gaussian. For a specific patch, the conditional probability is a product of pixel-wise conditional probabilities in that patch,

$$p(Z_k\vert T_k, e) = \prod_{i\in S_k} N(Z_{i,k}; \mu_{T_k(i)}, \phi_{T_k(i)})$$ (1)

And for a collection of patches,

$$p(Z\vert T, e) = \prod_{k=1}^P p(Z_k\vert T_k, e)$$ (2)

In this way, the joint distribution is formulated as

$$p(Z, T, e)$$ (3)

$$= p(e)p(T\vert e)p(Z\vert T,e)$$ (4)

$$= p(e) \prod_{k=1}^P p(T_k) \prod_{i\in S_k} N(Z_{i,k}; \mu_{T_k(i)}, \phi_{T_k(i)})$$ (5)