8.2 Successive N-FINDR (SC N-FINDR)

As noted in the s-SC IN-FINDR in Chapter 7, a major issue in its implementation is computational cost that is expected to be very high. In order to mitigate this problem, a sequential version of s-SC IN-FINDR developed in Section 7.2.3.3 can be derived by replacing simultaneous s endmembers carried out in step 5 of the s-SC IN-FINDR with successive p endmember replacements to achieve more computational efficiency at the expense of optimality. The resulting sequential version of p-SC IN-FINDR is referred to as SuCcessive N-FINDR (SC N-FINDR).

Successive N-FINDR (SC N-FINDR)

1. Preprocessing:

a. Let p be the number of endmembers required to generate.

b. Apply a DR transform such as MNF to reduce the data dimensionality from L to p–1, where L is the total number of spectral bands.

2. Initialization:

Let be a set of initial vectors randomly generated from the data.

3. For , find which yields the maximum volume of defined by (7.3) over all sample vectors r, while fixing other endmembers with and with . That is,

(8.1)

The major difference between s-SC IN-FINDR in Section 7.2.3.3 and SC N-FINDR described above is that SC N-FINDR only executes the inner loop indexed by j in s-SC IN-FINDR without going through s-SC IN-FINDR's outer loop indexed by k. Specifically, the equations used to generate an endmember that yields the maximal simplex volume, that is, (7.8) in s-IN-FINDR and (7.12) in s-SC IN-FINDR, are replaced by (8.1). In addition, even in the best case where the original initial endmembers turn out to be final desired endmembers and no replacements are required, (8.1) still needs to be calculated p times.