209.

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In such circumstances, if one considers only single-class labelling, one may still choose {P}, because its belief and plausibility are the highest without considering the set {B, F}. However, it may be sensible to collect more evidence rather than to generate a label for the pixel using the single class evidence alone. If other evidence is available, the evidence combination problem will arise. This issue is considered in Section 7.4.3

To investigate further the relationship between belief and plausibility, one can speculate that plausibility is always equal to or greater than belief, that is:

(7.25)

which leads to:

(7.26)

Equation (7.26) is derived on the basis of the observation that both Bel(ψ) and Bel(~ψ) have no subset in common, and Bel(ψ) and Bel(~ψ) are each made by the sum of bpa of its own subsets. If we let ψ=P, it follows that:

(7.27)

The above expression specifies what the interval or uncertainty is, and it is then apparent that 1–Bel(ψ)–Bel(~ψ) can be greater than 0, which confirms Equation (7.26).

7.4.3 Evidence combination

In most cases, more than one set of evidence is available, and the decision to label a pixel is made based on the accumulation of all of the evidence. D-S theory acknowledges such a requirement and provides a formal proposal for multi-evidence management.

The aggregation of multiple belief functions is called Dempster’s orthogonal sum, or Dempster’s rule of combination (Shafer, 1979). Let Bel_a and Bel_b denote two belief functions, and let m_a and m_b be their corresponding bpas. Dempster’s orthogonal sum generates a new bpa, denoted by m_a m_b, which represents the result of combing m_a and m_b. The result of the accumulation of both belief functions, denoted by Bel_a Bel_b, is derived from m_a m_b. If m(ψ) denotes the new aggregated bpa, the combination rule can be specified by:

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