alt="images"/> is a hyperparameter of the method. Experiments conducted in Hedhli et al. (2016) indicated limited sensitivity of the result of MPM on multiple cascaded quad-trees to the value of this hyperparameter. We also note that equation [1.8] implicitly yields a stationary model for the considered transitions, i.e. the probability
depends on the pair
of classes, but not on the specific site location
After the first top-down pass, the prior P(cs) is known on every site s of the second quad-tree.
1.2.4.3. Bottom-up pass
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:
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.
1.2.4.4. Second top-down pass
Finally, based on equation [1.5], the posterior marginal is initialized at the root of the second quad-tree as
Then, given the probabilities that have been determined or modeled within the previous stages,
is obtained on all sites
of all other layers through equation [1.5], by sweeping the second quad-tree downward in a second top-down pass.
1.2.4.5. Generation of the output map
The aforementioned stages lead to the computation of the posterior marginal
on every site s of the second quad-tree. In principle, site s could directly be given the label
i.e. the label that maximizes
over the set Ω of classes. However, this strategy is often avoided in the literature of hierarchical MRFs because of its computational burden and of the risk of blocky artifacts (Laferté et al. 2000; Voisin et al. 2014). As an alternative, the case-specific formulation of the modified Metropolis dynamics (MMD) algorithm (Berthod et al. 1996), which was combined with MPM in Hedhli et al. (2016) for the case of multitemporal single-sensor classification, is generalized here to the multisensor case. We refer the reader to Hedhli et al. (2016) for more detail. In the case of both proposed methods, after this integrated MPM–MMD labeling, the classification result on the leaves of the second quad-tree provides the output classification map at the finest of the observed resolutions.
1.2.5. Probability density estimation through finite mixtures
For each class, layer and quad-tree, a finite mixture model (FMM) is used for the corresponding pixelwise class-conditional pdf. This means that the function
for
is supposed to belong to the following class of pdfs:
[1.11]
where
is a pdf model depending on a vector
of parameters that takes values in a parameter set
alt="images"/> is a hyperparameter of the method. Experiments conducted in Hedhli et al. (2016) indicated limited sensitivity of the result of MPM on multiple cascaded quad-trees to the value of this hyperparameter. We also note that equation [1.8] implicitly yields a stationary model for the considered transitions, i.e. the probability 
