Stationarity and Ergodicity

The adaptive modeling strategy in this dissertation relies on certain assumptions on the MRF. These are, namely, the stationarity and ergodicity properties.

A strictly stationary [161] random field on an index set $\mathcal{T}$, defined on a Cartesian grid, is a random field where all the joint CDFs are shift-invariant, i.e.,

$$F ( X_{t_1}, \ldots, X_{t_n} ) = F ( X_{t_1 + S}, \ldots, X_{t_n + S} ); \forall n, \forall t_1, \ldots, t_n, \forall S.$$ (93)

If the CDFs are differentiable, then it implies that all the joint PDFs are also shift invariant, i.e.,

$$P ( X_{t_1}, \ldots, X_{t_n} ) = P ( X_{t_1 + S}, \ldots, X_{t_n + S} ); \forall S, \forall n, \forall t_1, \ldots, t_n.$$ (94)

A strictly-stationary MRF implies that the Markov statistics are shift invariant, i.e.,

$$\forall t \in \mathcal{T}, P ({\bf Z}_t) = P ({\bf Z}).$$ (95)

Such a MRF is also referred to as a homogenous MRF. In this dissertation, all references to stationarity imply strict stationarity.

In this dissertation, we also refer to a piecewise-stationary random fields, similar to the references in [175]. Through this terminology, we actually mean that the image comprises a mutually-exclusive and collectively-exhaustive decomposition into $K$ regions $\{ \mathcal{T}_k \}_{k=1}^{K}$, where the data in each $\mathcal{T}_k$ are cut out from a different stationary random field.

Ergodicity allows us to learn ensemble properties of a stationary random field solely based on one instance of the random field. We use this property to be able to estimate the stationary Markov PDF $P(Z)$ from an observed image. A strictly-stationary random field ${\bf X}$, defined on an $m$D Cartesian grid, is mean ergodic [161] if the time average of $X_t$, over $t$, converges to the ensemble average $E [ X_t ] = \mu_X$ asymptotically, i.e.,

$$\lim_{S \rightarrow \infty} \frac {1} {(2S)^m} \int_{-S}^{S} \ldots \int_{-S}^{S} X_t dt = \mu_X.$$ (96)

A strictly-stationary random field ${\bf X}$ is distribution ergodic [161] if the indicator process ${\bf Y}$ defined by

$$Y_{x,t} = H (x - X_t)$$ (97)

is mean ergodic for every value of $x$. This implies that RVs in the random field are asymptotically independent as the distance between them approaches infinity [161]. This behavior is also captured in the notion of a mixing random field. A random field ${\bf X}$ on an index set $\mathcal{T}$ is strongly mixing if two RVs become independent with as the distance between them tends to infinity, i.e.,

$$\lim_{\parallel u - v \parallel \rightarrow \infty} \vert P (X_u, X_v) - P (X_u) P(X_v) \vert = 0; \forall X_u,X_v \in {\bf X}.$$ (98)

In this dissertation, all references to ergodicity imply distribution ergodicity.