Classification Methods for Remotely Sensed Data, Second Edition

Page 287

where K is a normalising constant, which is defined by:

The accumulation of bpa m_a: [m_a(ξ₁), m_a(ξ₂),…, m_a(θ)] and bpa m_b: [m_b(ζ₁), m_a(ζ₂),…, m_a(θ)] is calculated by considering all products in the form of m_a(ξ)×m_b(ζ) where ξ and ζ are individually varied over all subsets of Bel_a and Bel_b. Note that the resulting sum of m_a(ξ)×m_b(ζ) is equal to one, as is required by the definition of a bpa.

Equations (7.28) and (7.29) illustrate that m(ψ) is formed by the orthogonal summation of all products m_a(ξ)×m_b(ζ) which have an intersection ψ. The normalising constant K is formed by the reciprocal of the sum of all products m_a(ξ)×m_b (ζ) for which there is no intersection. This normalising constant ensures that no contribution is committed to the null set (i.e. ξ ζ=Ø). The evidence combination process is illustrated by two examples, described below.

In the first example, suppose that one observation supports {B, F} to the degree of 0.7 (i.e. m_a), whereas another observation disconfirms {F} to the degree of 0.6 (i.e. m_b) (note that the situation here is the same to confirm {B, P} to the degree of 0.6). For illustrative purposes, it is convenient to use an intersection table with the values assigned by m_a and m_b along the rows and columns, respectively, and only non-zero values are taken into consideration. We define the entry (r, c) in the table as the intersection of the subsets in row r and column c. The result of the rule combination is shown in Table 7.1.

In the example, each subset appears only once in the intersection table and no null intersection occurs. Hence m_a m_b for each intersection is easily calculated as shown in each entry. Once the calculation of m_a m_b is completed, the belief and plausibility of the combination Bel_a Bel_b can be derived as:

Table 7.1 Example of Dempster’s rule combination