Читать книгу Change Detection and Image Time Series Analysis 2 - Группа авторов - Страница 13

Let us first define the multiple quad-tree structure associated with the first proposed method. The K images in the series are included in the finest-scale layers (i.e. the leaves) of K distinct quad-trees. The coarser-scale layers of each quad-tree are filled in by applying wavelet transforms to the image on the finest-scale layer (Mallat 2008). The roots of the K quad-trees are assumed to correspond to the same spatial resolution. The rationale of this hierarchical structure is that each image in the input series originates from a separate multiscale quad-tree, generally with a different number of layers and the input image on the leaves, and that the roots of these quad-trees share a common spatial resolution (see Figure 1.4). This graph topology implicitly means that the spatial resolutions of the input images in the series are in a power-of-2 mutual relation. In general terms, this is a restriction but when concerning current high-resolution satellite missions, this condition is easily met up to possible minor resampling.

Let be the image associated with the -th layer of the k-th quad-tree in the series. We will index the common root with = 0 and the leaves of the k-th quad-tree with coincides with the original input image The images in the other layers have been obtained through wavelets from The whole time series of multiscale images, either acquired by the considered sensors or obtained through wavelets, will be denoted as .

We will also indicate as the pixel lattice of the -th layer of the k-th quadtree . We will denote as s = (p, q) the coordinate pair of a generic pixel in one of these layers . Following the literature of hierarchical MRFs, the site will be named s in the following. Sites in the described quad-tree structure are linked by parent–child relations – within each quad-tree and across consecutive quad-trees – as a function of their spatial scale. Specifically, if is a site in the -th layer of the k-th quadtree and is not on the root layer, then indicates its parent node in the same quad-tree (k = 1, 2,..., K). Similarly, if with i.e. s is not on the leaves layer, then denotes the set of its four children nodes in the same quad-tree. Finally, if with and i.e. if s is neither in the first quad-tree of the series nor in the root of the other quad-trees, then indicates its parent node in the (k – 1)-th 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 define the corresponding edges.

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 among the sites. In particular, the quad-trees are meant to be in cascade, consistently with the input time series. Let be the class label of site and let be the corresponding time series of random fields associated with all multiscale layers. Each realization of corresponds to a set of classification maps for all images in the series and all scales in the corresponding quad-trees.

The key assumption in the hierarchical MRF model is that the random fields are Markovian, both across scales and time (Kato and Zerubia 2012):

[1.1]

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 (Li 2009; Kato and Zerubia 2012):

[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 are also assumed to be conditionally independent on the labels in :

[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 ) and the class label of site and cs, respectively, dropping the explicit dependence on k and for the sake of clarity. For this reason, we will explain the formulation of the first proposed method in the case of a series composed of K = 2 images acquired by two different sensors and at two different resolutions on the considered area. In this case, two quad-trees in cascade are used. The extension to the case K > 2 is straightforward.

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 the class label that maximizes the posterior marginal probability i.e. the distribution of its own individual label, given all feature vectors in the image series (Li 2009; Kato and Zerubia 2012). This decision rule is especially advantageous in the case of hierarchical graphs because it penalizes classification errors as a function of the scale at which they occur. Intuitively, an error on a site in the leaves layer only directly affects the corresponding pixel, whereas an error in a single pixel in the root layer may propagate into many erroneously labeled pixels on the leaves layer. MPM correctly penalizes the latter scenario more strongly than the former (Laferté et al. 2000).

As proven in Laferté et al. (2000) and Hedhli et al. (2016), under suitable conditional independence assumptions, the posterior marginal can be recursively expressed as a function of the posterior marginal of the parent node s– in the same quad-tree and, in the case k = 2, also of the posterior marginal of the parent node s⁼ in the previous quad-tree:

[1.5]

where collects all sites except the root in the k-th quad-tree (k = 1, 2) and X_s 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