Читать книгу Change Detection and Image Time Series Analysis 2 - Группа авторов - Страница 15

Equation [1.5] allows us to calculate the posterior marginal at each site of the second quad-tree (except the root) recursively, as long as the probabilities become known. In particular, the focus on the second quad-tree is consistent with the cascade approach used within the proposed methods: given the input time series, the output classification map is obtained on the leaves of this second quad-tree (or on the leaves of the last quad-tree in the case of a longer time series).

Specifically, under appropriate assumptions, in Hedhli et al. (2016) we proved that, for all

[1.6]

where is the parent–child transition probability across the two quad-trees, P(cs) is the prior probability, P(cs- |cs=) is the transition probability between two sites at the same scale and is the partial posterior marginal probability. Given the obvious constraint equation [1.6] determines uniquely on all sites To determine these probabilities, the approach developed in Hedhli et al. (2016) for the multitemporal single-sensor case is generalized to the multisensor tasks considered here. This process is formalized within three recursive passes across the second quad-tree – one bottom-up and two top-down passes – which are described in the next sections.