Change Detection and Image Time Series Analysis 2 - Группа авторов

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rel="nofollow" href="#ulink_a6f8a152-c8ef-5dfc-be03-cbc3dfdc2911">[1.5]

      where

collects all sites except the root in the k-th quad-tree (k = 1, 2) and Xs is a vector collecting the features of all of the descendants of site s. Through this formulation, MPM takes into consideration the information conveyed by the input multisensor data within the labeling of each site

      1.2.3. Hierarchical model associated with the second proposed method

      In the second proposed method, the case-specific problem of jointly classifying a multimission, multifrequency (radar X band, radar C band and optical VNIR) and multiresolution series of images acquired by COSMO-SkyMed, RADARSAT-2 and Pléiades is addressed. All of these sensors support multiple spatial resolutions, up to 0.5 m for Pléiades, approximately 1 m for COSMO-SkyMed and 1 m × 2 m for RADARSAT-2.

      Figure 1.5. Cascaded quad-trees associated with the second proposed method. In this example, we assume that the input time series includes Pléiades data at a 0.5 m resolution, COSMO-SkyMed spotlight data at a 1 m resolution and RADARSAT-2 data sampled on a 2 m pixel lattice. Accordingly, the Pléiades image is included in the leaves layer of both quad-trees. The COSMO-SkyMed and RADARSAT-2 images are inserted in the intermediate layers of separate quad-trees, according to their resolutions. The empty layers of both quad-trees are filled in with wavelet transforms of the Pléiades imagery

      1.2.4. Multisensor hierarchical MPM inference

      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).

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.

      1.2.4.1. Initializing on the first quad-tree

      First, in order to initialize the process, classification is performed using only the data included in the first quad-tree through a classical MPM on a single quad-tree. We refer the reader to Laferté et al. (2000) for more detail. Here, we only briefly recall that the algorithm is initialized by choosing the prior distribution on the root of this first quad-tree. In order to favor spatial regularity, this prior is selected here according to a Potts MRF model (Li 2009; Kato and Zerubia 2012). Details can be found in Hedhli et al. (2016). From the perspective of the classification of the input series of multisensor images, the key point is that, after this initialization stage, the posterior marginal

is known for each site
of the first quad-tree. Furthermore,
is also obtained as an intermediate byproduct on the same sites.

      1.2.4.2. First top-down pass

      In the first top-down pass, the second quad-tree is swept downward from the root to the leaves to calculate the prior P(cs) recursively. This prior is initialized in each site

of the root of the second quad-tree as
is the site of the root of the first quad-tree with the same spatial location as s. The partial posterior marginal
has been derived in the aforementioned initialization. Then, the top-down pass travels along the other layers until it reaches the leaves

      [1.7]

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