Classification Methods for Remotely Sensed Data, Second Edition

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7.3.2 Bayesian multisource classification mechanism

The model described in this section is based on Lee et al. (1987). In the general multisource case, a set of observations from n (n>1) different sources is used. Let x_i, i [1, n], denote the measure of a specific pixel from measurement source i. The aim is to derive the probability of the pixel belonging to each of c information classes ω_j, j [1, c]. It is possible to get some information about the prior probability of class ω_j, denoted by P(ω_j), i.e. the probability that an observation will be a member of class ω_j. A useful model for prior probability is context, as described in Chapter 6. The approach using contextual assumption to perform multisource classification is considered later.

According to Bayesian probability theory, the law of conditional probabilities states that:

where P(ω_j| x₁, x₂…, x_n) is known as the conditional or posterior probability that ω_j is the correct class, given the observed data vector (x₁, x₂, …, x_n). P(x₁, x₂,…, x_n|ω_j) is the probability density function (p.d.f.) associated with the measured data (x₁, x₂,…, x_n) given that (x₁, x₂,…, x_n) are members of class ω_j. P(x₁, x₂,…, x_n) is the p.d.f. of data (x₁, x₂,…, x_n). Assuming class conditional independence among the sources, one obtains P(x₁, x₂,…, x_n|ω_j)=P(x₁|ω_j)·P(x₂|ω_j)·…·P(x_n|ω_j), and Equation (7.1) becomes:

Again, following the law of conditional probabilities,

If Equation (7.3) is substituted into Equation (7.2), one obtains:

If the intersource independence assumption has been made such that P(x₁)·P(x₂)·…·P(x₁)=P(x₁, x₂,…, x_n), then Equation (7.4) results in: