Change Detection and Image Time Series Analysis 2

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]

This formulation encourages parent and children sites to share the same class label, although it does not deterministically enforce this condition. It also implies a model for the statistical relations between labels in consecutive layers. Here, the parent–child transition probability is expressed using the parametric model in Bouman (1991):

[1.8]

where 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.

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.