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To calculate a bottom-up step traveling through the second quad-tree from the leaves to the root is used. It is based on equation [1.6], in which, besides the priors P(cs), which are known from the first top-down pass, three further probability distributions are necessary: (i) the transition probabilities at the same scale (ii) the parent–child transition probabilities and (iii) the partial posterior marginals

Concerning (i), the algorithm in Bruzzone et al. (1999) is applied to estimate the multitemporal joint probability matrix, i.e. the M × M matrix J, whose (m, n)-th entry is This technique is based on the expectation maximization (EM) algorithm and addresses the problem of learning these joint probabilities as a parametric estimation task. More details can be found in Hedhli et al. (2016). Once J has been estimated, is derived as an obvious byproduct.

With regard to (ii), the parametric model in equation [1.8] is extended as follows:

[1.9]

where θ has the same meaning as in equation [1.8] and is a second hyperparameter.

Concerning (iii), it has been proved that, on all layers except the leaves (Laferté et al. 2000):

[1.10]

for all sites First, is initialized on the leaves of the second quad-tree by setting for Then, is calculated by using equation [1.10] while sweeping the second quad-tree upward until the root is reached. This recursive process makes use of the pixelwise class-conditional PDF whose modeling is discussed in section 1.2.5. After the bottom-up pass, is known on every site s of the second quad-tree.