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we have Q images, each with a continuous class label. An MRF model can then be formed follows. We define
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and then (7.20) Assuming that f, is a softmax function, ignoring the time step, and setting the
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/3s to 1, it can be seen that models (7.18) and (7.20) are identical, both having the
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Thus model 3 (equation 7.18) offers a natural way of implementing a sequential updating scheme.
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7.5.2 At the hidden layer
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In a similar fashion, neighbor model 2 can be used to implement an Ising spin glass model a t each hidden layer unit. Consider
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While it is possible to train any of the models (including the neighbor--subnetworks) by any of the usual methods, it is not advisable to do so. All models involve combining information from two different sources to make a decision. Unfortunately, training the MLP on homogeneous regions means that both sources of information the spectral values and the class labels of - will be giving essentially the same information. In this case, an unconstrained MLP may totally ignore one source of information. Because of this, it is necessary to impose some constraints on the MLP during the training process.
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The way we have overcome these problems is to: train the R weights for the spectral data alone; set the weights for the neighbor-subnetwork to the same values as the R weights for the spectral connections;
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Kiiveri and Campbell (1992) suggest that there is no advantage to using models more complex than (7.4) and in their examples take 40 = 0 and 8, = 1. We set 01 = 42 = p3 = 0 = - o 4 : P so that the network is summing the probabilities of the neighbors being in class S. If the sum is positive, the unit returns a value 0.5), otherwise it returns a - value. Ripley (1986) suggests some simple geometric arguments for choosing a reasonable range of values for Even with this approach, model 1 requires split training for the matrix as both A and A are estimating and so the MLP may ignore one of them in training. In this case, the procedure we have used is to train one set of weights at a time (either the spectral or the neighbor weights) and then combine the weights to get the final MLP model. Because of this, and because model 1 does not lend itself in a natural way to the Hopfield extension discussed in Section 7.5, it is not attractive a model either 2 or 3.
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We reconsider the Martin s farm example from Section 6.4 (page 76). The example here differs in that the selected training sites are different, there are now 8 ground cover classes, and we use the pc penalty term. The training classes and the number of pixels in each class are given in Table 7.1. The ground cover class and the number of pixels in the training data for each ground cover class for the Martin s farm example.
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ground cover class water primary salt secondary salt remnant vegetation wheat lupins medic pasture dark meen
number of pixels 9 96 67 54 78 66 72 168
The initial task-based MLP was of size 6.28.8 and was selected according to the algorithms described in 6 (p. 69). This was pruned down to a network of size 6.6.8, and as the pruning algorithm stopped here and an inspection of the eigenvalues of the Hessian matrix of the weights indicated that a minimum had
been found in weight space, this was accepted the final MLP. The neighbor modifications were then made to the network and a series of iterations performed on the image. There is a marked improvement in the smoothness of the resulting image, but without any apparent loss of class delineation. Figure 7.2 gives the hidden layer outputs as semi-classified images. The weight simplification procedure discussed in Section 6.4.5 (p. 85) was applied giving the simplified matrix 15
0 0 0 -10
0 -20 25 -15 0
-10 0 , 0
0 -10 0 0 -20 15 0 0 -15 0 20 - 10 0 5 0 0 15 0 0 0 0 0 0 0
-10 0 0 -10 0 -10 10 0
0 0 0 0
It can be seen that the ground cover classification medic pasture is formed by semi-classified image 5 minus semi-classified image 1. Figure 7.5 shows the classification achieved with the standard MLP model. Figure 7.6 shows the classification achieved with neighbor model 3. Figure 7.7 gives the hidden layer outputs semi-classified images updated by neighbor model 2. In this example it was necessary to retrain the matrix after applying neighbor model 2 to the hidden layer images the proportion of the images classified wheat shrunk markedly. Figure 7.8 shows the action of a neighbor MLP on a small section of the image.