Classification Methods for Remotely Sensed Data, Second Edition

Page 259

For any w_r0 L, the ψ(w_r0, w_Nr) can be obtained by using Equation (6.40), and the joint probability P(w_r0, w_Nr) can be estimated by using a histogram technique. Assume there are a total of y windows of size 3×3 in an image, and a particular configuration of (w_r0, w_Nro) occurs x times. Then:

The relationship between the histogram technique and ML estimation is examined by Gurelli and Onural (1994). Using Equations (6.45) and (6.46), one can obtain numerous equations (parameterised by θ) by counting different configuration for different class labels. For instance, if the number of configurations in Figure 6.15b occurs fifty times, and for Figure 6.15c it occurs 100 times, and if there is a total of 500 windows in an image, one can generate an equation such as:

Once those equations are established, the solution for parameter vector θ can be solved by a least-squares technique (refer to Mather (1976) for a comprehensive descriptions of this approach). In order to reduce estimation bias, Derin and Elliott (1987) suggest that one should discard the case of x=0.

Following from the above definitions, it is shown that the number of equations grows considerably as the number of labels L increases. An alternative approach, which reduces the amount of computation, is the logit model fit method (Dubes and Jain, 1989), which is a simple modification of least-squares estimation. Let H be the collection of all possible configurations of site r₀ and its eight neighbours. Thus, the number of possible configurations in H, denoted by |H|, for a two-label image is equal to 2⁸. Define a relation ‘≈’ on H by h_i ≈ h_i, if h_i, h_j, H and ψ(w_r0, hi)=ψ(w_r0, h_j) (see Equation (6.40)). Relation ‘≈’ partitions H into eighty-one disjoint classes because each component of the four-couple vector ψ can take on three values {−2, 0, 2} (see Equation (6.40)), i.e. 3⁴=81. Parameter estimation can therefore use the same procedure based on the above equations. For instance, if there are other windows containing the pixel values as shown in Figure 6.16 then, based on logit model fit method (Dubes and Jain, 1989), both window configurations will be treated as the same group as that shown in Figure 6.15b because they output the same vector ψ(w_r0, w_Nr)=[−2, 2, 0, 0].