22.3 Signature Discriminatory Probabilties

As noted above, the key to materialize the concept of DDA is to find a means of interpreting source alphabet probabilities used in source coding in terms of hyperspectral signatures. Let the entire hyperspectral data be considered as an information source with a set of hyperspectral signatures that correspond to source alphabets . We now interpret relative occurrence frequencies among J source alphabets, as relative spectral discriminatory powers among the n_S signatures, , then the source alphabet probabilities can be interpreted as signature discriminatory probabilities among , denoted by which can be obtained as follows.

To begin with, we select a spectral similarity measure, denoted by such as spectral angle mapper (SAM), and spectral information divergence (SID) (Chang, 2003a). Technically, SID may be a better candidate than SAM since it is a criterion designed to measure discrepancy between two probability distributions. However, if SAM is used, the cosine value, cos θ, will be used instead of the values of angle, θ. Next, we choose a reference signature s as a benchmark against which each signature of will be compared and computed for finding their relative spectral discriminatory probabilities, for all . Normalizing by the constant of a probability vector can be obtained by

(22.1)

that is a probability of difficulty level of discriminating the jth signature s_j with respect to the reference signature s.

There are three candidates can be used for selection of the reference signature s, data sample mean μ, signature mean and any signature from . Which one is a better choice depends upon applications (Chang et al., 2010; Wang et al., 2010; Wang and Chang, 2007).