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Neighbor information could be included in a standard MLP model by simply augmenting the spectral values for pixel with the values for the pixels in a fully--connected MLP. However, this would lead to an MLP with a large number of weights and would, moreover, violate assumption (7.2) that the label of a pixel depends on the spectral values of neighboring pixels only via their class labels. Such a model would in fact be using texture information to perform the classification and, such, may cause confusion on class boundaries3. The architecture of the MLP can be modified to model the contextual information in the labels of neighboring pixels by augmenting a pixel with the spectral values of its neighbors. However, the appropriate model should have certain weights set to 0 and certain sets of weights held a t the same value to avoid violating assumption (7.2). To incorporate neighbor information into the MLP, we return to the theme of task-based MLPs considered in 6 , p. 69. Each output unit of an MLP estimates for some class gq versus the other classes. Similarly, by construction, each hidden-layer unit in a task-based MLP can be considered to be discriminating between two classes or super-classes formed by the union of classes. If we designate a super class by SJ, then the j t h hidden-layer unit estimates I z , w h ) versus the alternative 1 I, w h ) . These estimates, I may be poor; for example, they may be small in magnitude across the entire range of the training data. However, the deficiencies in scale can be rectified by the Y matrix of weights, which can be viewed combining and rescaling the probabilities of superclass membership to find I for the classes of interest. If, without loss of generality, we drop the j superscript and consider a particular can hidden-layer unit, and take S, to be the label of the i t h pixel, then each be considered a binary random variable. An estimate of = 1 1 is produced by the hidden-layer unit. If we consider the grid of these estimates S = = l ) , i E I}, then we can see that each hidden-layer unit produces a [0, 11 grey-scale image. We refer to this as a semi-classified image. These images were briefly discussed in 6, p. 69. See Figure 7.2 for a full set of semi-classified images from an MLP of size 6.6.8 used to classify Martin s farm. We ignore the spectral information for the moment, and just consider the information contained in the labels of the neighboring pixels For pixel we take
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3See Bischof et al. (1992) for an example of such a classification scheme applied to remotely sensed images. While using texture information may cause confusion on the class boundaries, it may in some circumstances have compensating benefits.
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Figure The hidden layer images from the standard M L P of size 6.6.8. Compare with Figure 7.7, p. 112.
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yn(t) =
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= 1 ) . Let
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yn(l)is then the output of a singlelayer MLP of size P.1, referred to
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a neighbor sub-network . These neighbor sub-networks will be used in the construction of the neighbor MLP model. Assumption (7.2) resolves into
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(7.12)
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(7.13) (7.14)
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Using the arguments of Section 7.1.2.2 (p. we model equation (7.12) as (7.13) where is a normalizing constant, and is the number of neighbors of i with label S. We then specialize this to (7.14). We can estimate by a sum of the For class label S and, say, four neighbors, we have four units indicating, by an output in the range of [0,1], the probability that neighbor has class label S. Then
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