Classification of Remotely Sensed Imagery
Using Markov Random Fields
U(w) is defined in (3). Eq. (11) can be regarded as a generalized version of (6).
The idea outlined above is mainly designed to achieve a smooth interpretation of remoteing sensing imagery. In practice, the patterns in an image are only piece-wise continuous. That is, discontinuities (i.e. edges between different patches) are naturally to be found within an image. In such cases, the use of a smooth interpolation operation may smear these discontinuities, causing over-smoothing. We are interested here in the method called line-process(Li, 1995b) because it allows us to mark discontinuities and minimize energy simultaneously.
The basic idea of the line-process is quite simple. Once a discontinuity (edge) between two adjacent pixels has been identified, the interaction (i.e. smooth interpolation operation) between these two neighboring pixels should reduced or set off, and it is reasonable to define some other potential to respond to the presence of the edge. Following this logic, the prior energy in (11) can be modified as follows:
Energy Minimization
Once the posterior energy model and the associated parameters have been determined, the next step is to find out the solution (i.e. start classification). As noted previously, a popular method of pixel labeling is to find the MAP estimate using the Bayesian formulation. The MAP approach is also equivalent to a minimum-energy solution in terms of MRF modeling. If the energy function is convex then the MAP-MRF solution can be obtained by using a search approach, such as the gradient descent technique, because there is only one minimum, which is a global one, in the solution space. However, for a non-convex energy function, the solution space may contain several local minimum. In order to obtain a truly MAP estimate, i.e. to find a global minimum, one has to search the full solution space exhaustively.
Three algorithms, known as Iterated Conditional Modes (ICM), Maximiser of Posterior Marginals (MPM), and Line Process have been proposed in the literature to test classification results. All three algorithms are referred to (Li, 1995b). The genetic algorithm was used for search optimal parameter assignments for the developped model.
Experimental Results and Discussions
The study area is located within the Red Sea Hills of Sudan, The categories are shown in Table 1. Based on (12), only pair-sites cliques parameters (i.e. parameter
b) are designated non-zero. The range of
b was defined as
be [5,-5],
lke [1,-1]and the isotropy assumption (i.e., single value
b,
b1 =
b2 =
b3 =
b4, direction independent) was made. The iterations defined for GA in each of three search experiments was 30,000, and each gene was represented by 7 bits, which results in 2
7.6 (= 2
42, where each candidate solution contains six genes in which five genes for source weighting parameter candidates and one gene for
b; if a non-contextual multisource classification is performed, the search space reduces to 2
7×5 = 2
35) choices in GA search space. The parameters determined by GA for both non-contextual and contextual classification are shown in Table 2(a). The parameters shown on row 2 were further used as inputs for MPM classification algorithms. The accuracy acquired is shown in Table 2(b). With the MPM classification algorithm, a range of values for the parameter k and n were tested. We used different combinations of k and n, with k ranging from 1 to 50 and n from 100 to 500. However, no significant difference was found. The classification accuracy in most cases falls within the range of [80, 80.1].
The source-associated weighting factors and clique potential parameters in Table 2(a) show us some interesting properties. The quantity of clique potential parameters determine how strongly the labeling process for the pixel of interest is affected by its neighbours. In order to achieve a good classification result, it is worthwhile to note again that the values of clique potential cannot be chosen without thought, and GA is a suitable tool for overcoming such parameter-determination difficulty. Under the isotropic assumption, a value of -1.5828 of
b was detected by GA as the best choice in terms of improving classification accuracy. When the assumption of anistropy is concerned, Table 2 row 3 shows a value of -0.8742 for the second
b parameter which indicates a relatively weak contextual effect in terms of vertical orientation. The reason for the lower potential in the vertical direction might be due to the dorminant direction of between-class boundaries as lithological classes have a greater E-W than N-S extent.
The line process mechanism used here is mainly based on (12) using the first order neighbourhood system (4 nearest neighbours) to trigger parameter E
rr'. However, for parameter V
Err', only 4-site cliques are set to be active. Therefore, the value of V
Err' will contain 4 choices (i.e., from the 1-edge case to the 4-edge cases). We further define the process rule as follows.
Within a 3 by 3 window, if a discontinuity between the centre pixel r and its upper nearest neighbour has been detected, this will trigger upper left 4 pixels to carry out line process (i.e., to detect edge patterns). If the discontinuity exists between r and its right nearest neighbour, then the line process will be executed for the upper right 4 pixels. Similarly, a discontinuity between the centre pixel r and its lower nearest neighbour will trigger the lower right 4 pixels, and a discontinuity between pixel r and its left nearest neighbour will trigger the lower left 4 pixels. Compared to the contextual classification patterns without incorporating line process, the line process classification appears to generate more small patches (or holes). The presence of these small patches is mainly due to falsely-marked edges.
