6.8 Constrained Band Selection

By re-inventing a wheel Chang and Wang (2006) recently developed a new approach to BS, called constrained band selection (CBS). Its idea was derived from the Linearly Constrained Minimum Variance (LCMV) adaptive beamforming (Frost, 1972) and can be described as follows.

Let denote the selected band subset where for each the kth spectral band in Ω_BS is represented by a column vector denoted by b_BS,k. Furthermore, let b_l represent the lth spectral band as a column vector. Then the entire image correlation matrix is defined by . Following the LCMV approach outlined in Chang (2002b; 2003a) we can consider a CBS problem by finding a Finite Impulse Response (FIR) filter specified by a weighting matrix W_BS that linearly constrains the selected bands in Ω_BS with a constraint matrix, C, while minimizing the band correlation matrix Q through the following optimization equation:

(6.78)

where , , and C is a constraint matrix. In light of the J(Ω_BS) in (6.72) the constrained objective function is defined by

(6.79)

The solution to (6.79), is given by

(6.80)

Alternatively, we can exclude the selected band image correlation matrix from the entire image correlation matrix Q and further define as the selected band image dependence matrix. Replacing Q in (6.78) or (6.79) with results in a similar CBS problem given by

(6.81)

where J(Ω_BS) in (6.72) is now defined by

(6.82)

The solution to (6.82), is

(6.83)

that is exactly the same as the one in (6.80) with Q replaced by .

Obviously, the major obstacle in using BS is to solve (6.72) for any given number of bands needed to be selected, n_BS. When n_BS is increased, the bands selected by (6.72) for a smaller number n_BS cannot be either included or used as prediction. The whole process of solving (6.72) must be re-carried out again. This is a significant disadvantage of using BS. In Chapter 21, a new concept called band prioritization is developed to resolve this issue where bands are prioritized and selected in accordance with their priorities calculated by a certain criterion, in which case previously selected bands in a smaller band subset are always included in a larger band subset without repeatedly solving (6.72) for different n_BS.