quad-tree, i.e. in the quad-tree associated with the previous image of the series (see Figure 1.4). From a graph-theoretic perspective, if the sites in the quad-trees are meant as nodes in a graph, then the pairs (s, s–), (s, s=) and (s, r) with 
Figure 1.4. Quad-trees associated with the input multisensor and multiresolution time series and related notations. For a color version of this figure, see www.iste.co.uk/atto/change2.zip
Given this multiple quad-tree topology, a probabilistic graphical model based on a hierarchical MRF is defined. It is made of a series of random fields associated with the various scales and connected by transition relations associated with the links
The key assumption in the hierarchical MRF model is that the random fields
where P(·) indicates the probability mass function (pmf) of discrete random variables and fields. Equation [1.1] implies that the distribution of the labels in each layer of each quad-tree, conditioned on the labels in all above layers of the same quad-tree and of the previous quad-trees in the series, can only be restricted to the distribution conditioned on the labels of the upper layers in the same and previous quad-trees. Furthermore, this distribution factorizes in a conditionally independent fashion – a common assumption in the area of latent Markov models
[1.2]
In the case of the first quad-tree in the series, these Markovianity and conditional independence assumptions are naturally adapted as follows
[1.3]
Finally, the feature vectors in the image time series
[1.4]
where p(·) denotes the PDF of continuous random variables and fields. Again, this assumption is widely accepted in the literature of latent MRF models (Li 2009; Kato and Zerubia 2012).
To ease the notations, in the following, we will simply write the feature vector
The formulation of MPM defined in Hedhli et al. (2016) with regard to the case of multitemporal classification of single-sensor multiresolution imagery is generalized here to the case of multisensor data. The MPM decision rule assigns site
As proven in Laferté et al. (2000) and Hedhli et al. (2016), under suitable conditional independence assumptions, the posterior marginal
where
1.2.3. Hierarchical
