310.

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Page 78

One can thus allocate pixel i to the class k which has the largest value of the term P(w_k|x_i) in Equation (2.20). The classification criterion can be expressed as:

(2.21)

where arg denotes ‘argument’. The criterion shown in Equation (2.21) is called the maximum a posteriori (MAP) solution, which maximises the product of conditional probability and prior probability. However, on some occasions, the prior probability P(w) is also set to be uniformly distributed (because of lack of prior knowledge, or because one does not know the true distribution). In this case, Equation (2.20) reduces to:

(2.22)

If we allocate pixel i to that class k which maximises expression (2.22), the result is called the maximum likelihood solution.

Normally, the conditional probability P(x_i|w_j) is assumed to follow a Gaussian (or normal) distribution assumption. P(x_i|w_j) can then be expressed as:

(2.23)

where C_j is the covariance matrix of class w_j with dimension ρ, μ_j is the mean vector of class w_j, and |·| denotes the determinant.

In practical applications, Equation (2.23) can be further reduced to following expression by taking the natural logarithm:

(2.24)

Equation (2.24) avoids the computation of the exponential term. Since the term ρln(2π) is the same for all classes, it can be regarded as a constant and dropped from the equation without affecting the final ranking of the values of ln[P(x_i|w_j)]. Finally, the expression in Equation (2.24) is multiplied by the constant ‘–2’ to give:

(2.25)

It is clear that maximising expression (2.23) is equivalent to minimising Equation (2.25). Note that the second term of Equation (2.25) is the Mah

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