11.7 Maximum Likelihood Classifier

The maximum likelihood classifier is one of the most popular methods of classification in remote sensing, in which a pixel with the maximum likelihood is classified into the corresponding class. The likelihood Lk is defined as the posterior probability of a pixel belonging to class k.

Lk = P(k/X) = P(k)*P(X/k) / P(i)*P(X/i)
where P(k) : prior probability of class k
P(X/k) : conditional probability to observe X from class k, or probability density function

Usually P(k) are assumed to be equal to each other and P(i)*P(X/i) is also common to all classes. Therefore Lk depends on P(X/k) or the probability density function.

For mathematical reasons, a multivariate normal distribution is applied as the probability density function. In the case of normal distributions, the likelihood can be expressed as follows. where n: number of bands
X: image data of n bands
Lk(X) : likelihood of X belonging to class k k : mean vector of class k k : variance-covariance matrix of class k In the case where the variance-covariance matrix is symmetric, the likelihood is the same as the Euclidian distance, while in case where the determinants are equal each other, the likelihood becomes the same as the Mahalanobis distances. Figure 11.7.1 shows the concept of the maximum likelihood method.

The maximum likelihood method has an advantage from the view point of probability theory, but care must be taken with respect to the following items.
(1) Sufficient ground truth data should be sampled to allow estimation of the mean vector and the variance-covariance matrix of population.
(2) The inverse matrix of the variance-covariance matrix becomes unstable in the case where there exists very high correlation between two bands or the ground truth data are very homogeneous. In such cases, the number of bands should be reduced by a principal component analysis.
(3) When the distribution of the population does not follow the normal distribution, the maximum likelihood method cannot be applied.

Figure 11.7.2 shows an example of classification by the maximum likelihood method.   