183.

[Cover] [Contents] [Index]

Page 262

Table 6.1 Specified and estimated parameters

Figure Number

Description

Number of Equations

β1

β2

β3

β4

6.l7(a)

Specified

2803

0.40

0.40

0.40

0.40

6.l7(b)

Estimated

 

0.25

0.4

0.32

0.37

6.l7(c)

Specified

5954

−1.00

1.00

−1.16

1.00

6.l7(d)

Estimated

 

−0.99

0.46

−1.13

0.48

6.l7(e)

Specified

7355

0.50

0.50

0.50

−0.50

6.l7(f)

Estimated

 

0.35

0.28

0.34

−0.36

MAP estimate. This is equivalent to a minimum-risk solution with a 0− 1 loss function. The MAP approach is also equivalent to a minimum-energy solution in terms of MRF modelling (see Equation (6.18)). A more detailed description of Bayes theory is given by Walpole (1982).

If the energy function is strictly convex (i.e. bowl-shaped, with one minimum point), a MAP-MRF solution can be obtained using a basic search approach such as a gradient decent technique, because there is only one minimum in the solution space. However, for a non-convex energy function, there may be many local minima. Therefore, in order to obtain a truly MAP estimate, i.e. to find a global minimum of the function shown in Equation (6.18), one has to search all local minima over the entire solution space, and also show that there are no more local minima. It is evident that such a search process can be very lengthy.

Three algorithms, known as simulated annealing (Metropolis et al., 1953; Geman and Geman, 1984); iterated conditional modes (ICM) (Besag, 1986); and maximiser of posterior marginals (MPM) (Marroquin et al., 1987) have been proposed in the literature. These methods aim to approximate MRF-MAP estimates. All three algorithms are iterative in nature. There is also an alternative way to apply the MRF contextual concept based on only one trial. Readers may refer to Haslett (1985) and Pickard (1980) for details. More detailed descriptions of the MRF-MAP framework are contained in Geman and Geman (1984), Dubes and Jain (1989), Szeliski (1989) and Geman and Gidas (1991).

6.5.1 Iterated conditional modes

Iterated conditional modes (ICM) is a local optimisation method proposed by Besag (1986). This algorithm will converge to a local minimum of the energy function. The idea of ICM is based on two assumptions, the first of which is that the observation components d1, d2,…, dm (m is the number of pixels) are class-conditional independent, and each dr has the same known conditional density function P(dr|wr) dependent only on wr, i.e. the label on the pixel r. Thus, the following equation holds:

[Cover] [Contents] [Index]


Classification Methods for Remotely Sensed Data
Classification Methods for Remotely Sensed Data, Second Edition
ISBN: 1420090720
EAN: 2147483647
Year: 2001
Pages: 354

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